Matthieu Robert De Saint Vincent

Hamza Chraïbi

Jean-Pierre Delville
email: jean-pierre.delville@u-bordeaux.fr

Optical flow focusing: Light-induced destabilization of stable liquid threads

Keywords: numbers: 47.55.dm, 47.20.Dr, 47.55.db

Confinement of flowing liquid threads by solid walls makes them stable with respect to the Rayleigh-Plateau instability. We demonstrate here that light can break this stability, by forcing locally the deformation of the liquid interface through thermally-induced Marangoni stresses. Depending upon the confining conditions and fluid properties, this optocapillary deformation either pinches or inflates the thread, which may in both cases lead to its localized fragmentation into droplets. In the pinching regime, the laser beam behaves as a wall-free constriction that flow focuses the thread, leading to successive regimes of single and multiple periodicity. Light-driven local Marangoni stresses may prove an elegant contactless alternative to control reversibly the thread-todroplet transition for digital microfluidics.

I. INTRODUCTION

Liquid threads or jets [START_REF] Eggers | Physics of liquid jets[END_REF] have been shown naturally unstable since the seminal work of Savart [START_REF] Savart | Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi[END_REF], later clarified by Plateau [START_REF] Plateau | Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires[END_REF] and Lord Rayleigh [START_REF] Strutt | On the instability of jets[END_REF][START_REF] Strutt | On the capillary phenomena of jets[END_REF][START_REF] Strutt | On the instability of a cylinder of viscous liquid under capillary force[END_REF], who identified the minimization of surface energy as the driving mechanism. By geometrically constraining the volume available for surface deformation, rigid boundaries slow down their development, improving stability [START_REF] Frischknecht | Stability of cylindrical domains in phase-separating binary fluids in shear flow[END_REF][START_REF] Son | Suppression of capillary instability of a polymeric thread via parallel plate confinement[END_REF][START_REF] Humphry | Suppression of instabilities in multiphase flow by geometric confinement[END_REF][START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF] up to full stabilization when the thread is squeezed [START_REF] Miguel | Phase separation in two-dimensional binary fluids[END_REF]. Thus, liquid threads are either unstable (convectively or absolutely [START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF]) when they are three dimensional, or absolutely stable when behaving as ribbons, depending on the confinement. This confinement-induced stability is a key issue in two-phase microfluidics [START_REF] Seemann | Droplet based microfluidics[END_REF]. On the one hand, droplet production from thread instability, also called digital microfluidics, is essential for a considerable number of processes involving droplets from micro-chemistry and physical chemistry to biotechnologies [START_REF]Encyclopedia of Microfluidics and Nanofluidics[END_REF], or ink-jet printing [START_REF] Brown | Impulsively actuated jets from thin liquid films for high-resolution printing applications[END_REF]. On the other hand, stable co-flows are more efficient to transport fluids over long distances due to the reduction of the friction compared to a confined droplet flow. Consequently, it became crucial for lab-on-a-chip applications to trigger the thread-to-droplet instability at well-defined locations and produce droplets of size appropriate to the application. The first approach consisted in designing the geometry of the contact between the immiscible fluid streams coming from different channels [START_REF] Christopher | Microfluidic methods for generating continuous droplet streams[END_REF]. Three main designs were investigated: T-junctions [START_REF] De Menech | Transition from squeezing to dripping in a microfluidic t-shaped junction[END_REF], axial co-flows [START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF] and flow-focusing through a constriction [START_REF] Lee | Role of geometry and fluid properties in droplet and thread formation processes in planar flow focusing[END_REF]. The drop size can be actuated using the size and the shape of the channels, the inlet flow rates and the interfacial properties between the immiscible liquids. However, while extremely reliable, some applications require versatility in flow and droplet management which cannot be fulfilled with a rigid design of the chip and the unavoidable transient regimes when changing flow characteristics [START_REF] Robert De Saint | Real-time droplet caliper for digital microfluidics[END_REF]. Active control of the thread instability was thus proposed to tune the drop size in flow focusing and T-junctions with mechanical [START_REF] Willaime | Arnold tongues in a microfluidic drop emitter[END_REF][START_REF] Choi | Designed pneumatic valve actuators for controlled droplet breakup and generation[END_REF], electrical [START_REF] Link | Electric control of droplets in microfluidic devices[END_REF][START_REF] Belloul | High-throughput formation and control of monodisperse liquid crystals droplets driven by an alternating current electric field in a microfluidic device[END_REF], acoustical [START_REF] Yin | Characterization of acoustic droplet formation in a microfluidic flow-focusing device[END_REF][START_REF] Schmid | Saw-controlled drop size for flow focusing[END_REF] or optical breakdown [START_REF] Park | High-speed droplet generation on demand driven by pulse laser-induced cavitation[END_REF] actuation, to cite a few.

The step forward consists in controlling reversibly the thread-to-droplet transition in confined geometries to either convey stable viscous co-flows or dispense droplets on demand and at any location on a chip. This issue has already been explored using UV exposure of specific photosensitive cis-trans surfactants [START_REF] Diguet | Photoreversible fragmentation of a liquid interface for micro-droplet generation by light actuation[END_REF], or DC electric fields to form Taylor cones on droplets and trigger the droplet-to-thread transition at high voltage [START_REF] He | Low-frequency ac electro-flow-focusing microfluidic emulsification[END_REF], both approaches being dedicated to lab-on-chip applications and requiring specific fabrications or responsive chemicals. In the present investigation we address this question considering a totally different situation in the sense that our approach does not require dedicated microfabrication or specific surfactant other that those classically used in microfluidics. As surface tension depends on temperature, a thermal gradient sets a surface tension gradient and therefore produces tangential stresses along the interface separating two immiscible fluids [START_REF] Levich | Surface-tension-driven phenomena[END_REF]. These Marangoni stresses in turn induce interfacial flows and may cause interface deformations [START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF][START_REF] Loulergue | Interface deflections induced by the Marangoni effect: an application to infrared-visible image conversion[END_REF]. Thermal modulations of surface stresses were actually employed on free-surface threads to force a specific mode of the Rayleigh-Plateau instability or conversely to oppose to the most unstable mode and delay the instability onset [START_REF] Furlani | Temporal instability of viscous liquid microjets with spatially varying surface tension[END_REF][START_REF] Nahas | Control of surface tension flows: instability of a liquid jet[END_REF][START_REF] Cordero | Quantitative analysis of the dripping and jetting regimes in coflowing capillary jets[END_REF].

We demonstrate experimentally and numerically that optocapillarity, i.e. light-driven Marangoni stresses, can deform and eventually destabilize stable liquid threads, as conceptually illustrated on Fig. 1. Lasers are indeed very appealing to trigger thermocapillary flows because optical focusing in absorbing liquids yields very large thermal gradients for a given overheating [START_REF] Cordero | Time-resolved temperature rise in a thin liquid film due to laser absorption[END_REF][START_REF] De | Thermocapillary migration in small-scale temperature gradients: Application to optofluidic drop dispensing[END_REF]. Using a 1D model, we first illustrate that the interfacial deformation induced by a thermal gradient nonlinearly depends on both the viscosity and the confinement of the liquid layers involved. This is then used to demonstrate how optocapillarity can locally pinch, and more surprisingly inflate, a confined liquid thread, and trigger instability despite the confinement. We finally show that optocapillary pinching behaves as a wall-free tunable flow focusing which controls the thread fragmentation into droplets regimes of different periodicity reminiscent of various nonlinear systems [START_REF] Clanet | Transition from dripping to jetting[END_REF][START_REF] Ambravaneswaran | Theoretical analysis of a dripping faucet[END_REF][START_REF] Marginean | Order-chaosorder transitions in electrosprays: The electrified dripping faucet[END_REF][START_REF] Garstecki | Nonlinear dynamics of a flow-focusing bubble generator: An inverted dripping faucet[END_REF][START_REF] Hashimoto | Formation of bubbles in a multisection flow-focusing junction[END_REF][START_REF] Nguyen | Spatiotemporal dynamics in a train of rising bubbles[END_REF][START_REF] Schmit | Cooperative breakups induced by drop-to-drop interactions in one-dimensional flows of drops against micro-obstacles[END_REF].

II. OPTOCAPILLARY DEFORMATION OF LIQUID SHEETS AND THREADS

The interface deformation of confined threads driven by Marangoni stresses can be unveiled with a simple three-layer 1D model. A layer of liquid 1, of height 2H 1 , is sandwiched between two immiscible layers of liquid 2, of height H 2 each. We solve the Stokes equation, -∇p i + η i ∇ 2 u i + ρ i g = 0, in each phase (i = 1, 2), in the upper half-system as sketched on Fig. 2 (a). Here p and u denote the pressure and velocity fields, g is the gravitational acceleration and ρ and η are the liquid density and viscosity, respectively. We assume incompressible liquids with a lubrication approximation, and set the following boundary conditions: (i) u x,2 (z = H 2 ) = 0 (no slip at the wall), (ii) (du x,1 /dz) z=-H1 = 0 (by symmetry), and (iii) u x,1 (z = 0) = u x,2 (z = 0) (continuity). In addition, the incompressibility imposes, in a finite system, ∫ 0 -H1 u x,1 (z)dz = ∫ H2 0 u x,2 (z)dz = 0. This condition is imposed either by the remote vertical boundary of the system, or by mirror symmetry of the interfacial tension gradient along the x axis, as in our experiment. This bounding of the system creates a deep layer counterflow as sketched on Fig. 2 (a) that prevents any net flow through either phase.

An interfacial tension gradient, such as dσ/dx = η 1 (du x,1 /dz) z=0 -η 2 (du x,2 /dz) z=0 , drives interfacial flows in both phases, that deform the interface of profile h(x). Solving the Stokes equation for dh/dx ≪ 1 with the above assumptions yields

(b) (c) Water Oil Oil (a) h(x) z x ρ 1 > ρ 2 , η 1 ρ 2 , η 2 dσ/dx H 2 -H 1 0 g A A A A = 0.4 = 0.4 = 0.4 = 0.4 A A A A = 2 = 2 = 2 = 2 A A A A = 1.2 = 1.2 = 1.2 = 1.2 A A A A = 6.
dh dx = ℓ 2 c H 1 H 2 σ 0 η 1 H 2 2 -2η 2 H 2 1 η 1 H 2 + 4/3η 2 H 1 dσ dx (1)
with ℓ c = √ σ 0 /(ρ 1 -ρ 2 )g the capillary length. The deformation at the point of largest interfacial tension should thus be oriented from liquid 1 to liquid 2 if (ρ 1 > ρ 2 )

A = H 2 H 1 √ η 1 2η 2 > 1. ( 2
)
For a given interfacial stress, the direction of the deformation is thus determined by the relative thicknesses and viscosities of the fluid layers. Note finally that the amplitude and orientation of the interfacial tension gradient do not influence the basic mechanism of interfacial deformation; reversing the sign of the interfacial tension gradient should thus only reverse the orientation of the deformation.

We force optocapillary deformation of confined waterin-oil threads into PDMS microfluidic channels [START_REF] Duffy | Rapid prototyping of microfluidic systems in poly(dimethylsiloxane)[END_REF]. Water (η 1 = 1 mPa s) and hexadecane (η 2 = 3.3 mPa s) are injected at constant flow rates (Q 1 and Q 2 , respectively), meet at a cross-shaped junction, and co-flow alongside further downstream. The rectangular cross-section of the channel (typically, 100-µm wide, 50-µm high) ensures a vertical confinement that increases thread stability [START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF]. To favor thread formation at low flow rates, the interfacial tension is reduced by adding two surfactants, sorbitan monooleate (Span 80, 0.3% w/w) in oil and sodium dodecyl sulfate (SDS, 2.5% w/w) in water, both above the critical micelle concentration (CMC). The equilibrium interfacial tension is σ 0 ∼ 10 -4 N m -1 [START_REF] Robert De Saint | Dynamic interfacial tension effects in the rupture of liquid necks[END_REF]. Appropriate choices of flow rates thus produce stationary water-in-oil threads, stable during minutes to hours over the whole length of the channel (1 cm, see Fig. 1); the weak vertical squeezing allows to consider a circular cross-section of diameter 2R 0 similar to the channel height.

Optocapillary deformations were achieved by focusing a continuous TEM 00 Ar + laser (wavelength in vacuum λ 0 = 514.5 nm) on the centerline of the flow as illustrated on Fig. 2 (b,c). The laser position was sufficiently far from the junction (0.8 mm) to assume parallel flow. A dye, uranine (disodium fluorescein), is added in water (0.1% w/w) to ensure heating occurs by light absorption [START_REF] De | Thermocapillary migration in small-scale temperature gradients: Application to optofluidic drop dispensing[END_REF]. A ×20 microscope objective both focuses the beam onto the liquid interface (beam waist ω 0 = 1.3 µm), and images the flow in the channel. Finally, the deformation dynamics was recorded by a CMOS fast camera (Lightning RTD 16000).

Using surfactants above the CMC may reverse the direction of thermocapillary flows compared to classical cases, most likely due to subtle couplings between thermal and molecular transport of surfactants along the interface [START_REF] Khattari | Migration of a droplet in a liquid: effect of insoluble surfactants and thermal gradient[END_REF]. This is accounted for by introducing an 'effective' coefficient of variation of interfacial tension with (∂σ/∂T ) eff , that turns out to be positive in this case. The point of highest interfacial tension therefore coincides with the hot spot.

Continuously shining the thread at moderate laser power deflects the interface as represented on Fig. 2 (b). From Eq. 2, one expects thick (resp. thin) threads to pinch (resp. bulge), the transition being governed by the ratio of viscosities. We first varied the cross section of a water-in-hexadecane thread via the flow rates, as shown on the left and right images of Fig. 2 (b), top row. Pinching occurs when H 1 /H 2 ≃ 1.1 (left, A = 0.4), while a thinner thread (H 1 /H 2 ≃ 0.2) slightly humps (right, A = 2). We then changed the viscosity ratio by adding sucrose in water (60% w/w, η 1 ≃ 60 mPa s). Similar behavior is seen when changing the liquid thickness ratio, as illustrated on the bottom row of Fig. 2 tions that decay within 3-4 wavelengths (see Video 1 and Fig. 1, middle image). This observation illustrates the absolute stability of the confined thread. Increasing the laser power may actually break the thread periodically, producing consecutive droplets. Surprisingly, this process occurs for either pinched or bulged threads [Fig. 2 (c)]. In the former case, the fragmentation is directly initiated near the laser spot when the pinched neck becomes unconfined [START_REF] Dollet | Role of the channel geometry on the bubble pinch-off in flow-focusing devices[END_REF]. The mechanism in the latter case is more subtle. Starting from the continuous thread, some liquid accumulates into the growing bulge, that partly obstructs the channel. The resulting overpressure forces breakup upstream of the hot spot [START_REF] Guillot | Stability of parallel flows in a microchannel after a T junction[END_REF]. Once broken, the upstream advancing front of the thread is blocked by the laser [START_REF] De | Thermocapillary migration in small-scale temperature gradients: Application to optofluidic drop dispensing[END_REF], allowing another droplet to grow and eventually break upstream through the same process.

H H H H 1 1 1 1 / / / /H H H H 2 2 2 2 = 2 = 2 = 2 = 2 A A A A = 1.41 = 1.41 = 1.41 = 1.41 H H H H 1 1 1 1 / / / /H H H H

III. PINCHING DYNAMICS

The reversal of the optocapillary deformations can be well understood by the effect of confinement on viscous liquid layers. We first use the Boundary Element Method [START_REF] Chraïbi | Optohydrodynamics of soft fluid interfaces: Optical and viscous nonlinear effects[END_REF] to perform numerical simulations on a stationary thread (initially cylindrical), surrounded by an axisymmetric liquid shell as represented on Fig. 3 (a). A Gaussian heating sheet, of transverse profile ∆T (r, x) ∼ e -2x 2 /ω 2 th and width ω th = 2H 1 , imposes a transient interfacial stress ∂σ/∂x ∼ ∆σte -t/τscan in a cross-sectional plane [∆σ = (∂σ/∂T ) eff ∆T ]. Defining τ scan = 2ω th /v 0 , this form reproduces the temperature field produced by a laser beam scanned at the velocity v 0 [START_REF] Sanders | Temperature distributions produced by scanning Gaussian laser beams[END_REF]. Considering the tangential stress condition at the interface between liquids 1 and 2, we also define a characteristic deformation time, τ def , as

τ def = ηω th H 1 H 2 ∆σ . ( 3
)
Figure 3 (b) shows for α = ∆σ/σ 0 = 0.25 that the reversal behavior of deformations versus thicknesses and viscosities, obtained both experimentally (Fig. 2) and theoretically in 1D (Eq. 2), is retrieved numerically. Moreover, Fig. 3 ). Not surprisingly, the stronger the stress, the faster the deformation; note that the final acceleration is due to the Rayleigh-Plateau instability because the simulated threads are not absolutely stabilized by the confining walls. The right inset of Fig. 3 (c) finally shows the pertinence of τ def to capture the initial dynamics of interface deformation as the pinching dynamics obtained at different α collapse onto a single behavior in reduced time at first stage. The deformation then slows down as the heating decays, and the Rayleigh-Plateau instability finally develops. However, as experiments are performed on flowing threads, we compare τ def with the other time scales involved: heating time τ heat = ω 2 th /D, viscous time τ visc = ρR 2 0 /η, capillary time τ cap = ρv 0 R 2 0 /σ 0 , and advective time τ adv = R 0 /v 0 . Assuming a temperature distribution as calculated in Ref. [START_REF] De | Thermocapillary migration in small-scale temperature gradients: Application to optofluidic drop dispensing[END_REF], we find ω th ≃ 5 µm for ω 0 ≃ 1.3 µm. We choose η = (η 1 + η 2 )/2 the average viscosity; D ≃ 10 -7 m 2 s -1 is the thermal diffusivity, and v 0 is the initial velocity of the thread. For waterhexadecane pinching threads, we estimate τ heat ≃ 0.25 ms, τ visc ≃ 0.3 ms, τ cap ∼ 0.05 ms, and τ adv ≃ 2-5 ms.

Furthermore, due to the vertical confinement, the deformation rate of the flowing thread is limited by the draining of the outer fluid through lubrication films of initial thickness ε imposed by the disjoining pressure. The deformation time therefore becomes τ def = η 2 ω th R 0 /(ε∆σ). Assuming (∂σ/∂T ) eff in the order of 10 -5 N m -1 K -1 , a temperature increase of 1 K per mW [START_REF] De | Thermocapillary migration in small-scale temperature gradients: Application to optofluidic drop dispensing[END_REF], and setting ε ∼ 300 nm [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF], we estimate τ def ∼ 2-40 ms in the power range investigated. From these time scales (τ heat , τ cap , τ visc ≪ τ def ), we conclude that the thermocapillary deformation of the confined thread mainly competes with its advection by the flow.

We experimentally investigate this interplay by mea- suring the pinching dynamics in response to a stepshaped, 16.7-ms wide laser pulse. The temporal evolution of the minimal thread radius is represented in dimensionless terms on Fig. 4 (a) for laser powers ranging from 3 to 58 mW and a fixed thread velocity (v 0 ≃ 1 cm s -1 ). Up to 55 mW, the deformation initially increases, following a common behavior, then saturates before the thread has become unstable and finally attenuates. The larger the beam power, the later the deviation from the common trend. The collapse of all initial dynamics in reduced time, consistent with simulations, emphasizes that the tangential stresses drive the first stages of the deformation in presence of confinement. As suggested by the estimated time scales, the later saturation should result from the advection of the deformation at longer time as the thread flows. We therefore varied the flow rates Q 1 and Q 2 , keeping their ratio constant so the initial confinement remains unchanged. Figure 4 (b) depicts nine dynamics, gathered into three groups of constant τ adv /τ def ∼ P/v 0 ratio -higher ratio yielding to more pronounced deformations and a delayed deviation from the common pinching behavior. In each group, the three dynamics overlap quite well, supporting that the pinching dynamics of a stable flowing thread is controlled by the interplay between optocapillarity and the fluid advection.

IV. THREAD FRAGMENTATION BY OPTICAL FLOW FOCUSING

Above a threshold power [P 58 mW in Fig. 4 (a)], the deformation amplifies, and the pinching follows a specific dynamics leading to the thread breakup [START_REF] Robert De Saint | Dynamic interfacial tension effects in the rupture of liquid necks[END_REF]. The thread indeed becomes locally unstable when the neck narrows down to a diameter significantly smaller than the smallest length of the channel, here its height [START_REF] Dollet | Role of the channel geometry on the bubble pinch-off in flow-focusing devices[END_REF]. This rupture occurs slightly downstream of the laser spot, and releases droplets as represented on the left image of Fig. 2 (c). Note however that the thread remains continuous upstream of the laser spot, the optocapillary pinching behaving as a wall-free tunable flow focusing which locally forces the thread fragmentation.

To investigate this optical flow focusing effect, we characterized under continuous forcing exposure and over long times (up to several minutes [START_REF] Robert De Saint | Real-time droplet caliper for digital microfluidics[END_REF]) the sequences of droplets emitted on a wide power range. Distinct emission patterns emerge depending on the laser power (Fig. 5). At low power (below 40 mW), the droplet production is simply periodic, period-1, resulting in streams of highly monodisperse droplets (left inset). In an intermediate power range (here, between 40 and 60 mW, second and third insets), two distinct droplets, one long and one short, are successively produced. This period-2 regime may coexist close to the bifurcation (especially, at P = 41 mW) with more complex transient regimes, involving three (one short, one intermediate, one long) or four droplets (two distinct period-2 sequences entangled) per sequence. Finally, further increasing the laser power (above 60 mW) brings the system back to a period-1 regime (right inset).

The alternation between sequences of single and mul- tiple periodicities results from the geometric confinement of the thread near the breakup point. As shown on the left panel of Fig. 2 (b,c), the liquid expelled from the pinched area accumulates both upstream and downstream. The water thread may therefore touch the lateral walls and obstruct the channel upstream, altering the pinching dynamics due to the slow draining through the outer oil films. More specifically, in the low-and highpower period-1 regimes, the widest part of the thread after breakup respectively never and permanently touches the wall (Videos 2 and 3). Thus, in both cases, boundary conditions do not vary during the pinching, resulting in periodic droplet emission. In contrast, in the regime of higher periodicity the bulged part of the thread strongly evolves during a full sequence, alternatively touching (when a long droplet forms) and not touching (hence releasing a short droplet) the wall (Video 4). This nonlinear behavior is reminiscent of feedback mechanisms occurring either at the leading edge of a thread, as in the widely-studied dripping faucet [START_REF] Clanet | Transition from dripping to jetting[END_REF][START_REF] Ambravaneswaran | Theoretical analysis of a dripping faucet[END_REF] or into microfluidic constrictions [START_REF] Garstecki | Nonlinear dynamics of a flow-focusing bubble generator: An inverted dripping faucet[END_REF][START_REF] Hashimoto | Formation of bubbles in a multisection flow-focusing junction[END_REF]. The simplified period-1-period-2-period-1 bifurcation diagram observed here results from the interplay between the pinching and the obstruction due to the channel confinement. Moreover, contrary to classical flow focusing cases when inertia may be crucial, optical flow focusing allows to control the drop formation at low Reynolds (Re = τ visc /τ adv ∼ 10 -1 ) and Weber (We = τ cap /τ adv ∼ 10 -2 ) numbers.

V. CONCLUDING REMARKS

In this investigation, we have demonstrated that optocapillarity, i.e. light-driven local Marangoni stresses, can trigger the capillary instability of confinement-stabilized liquid threads. We demonstrated that depending on the interplay between the fluid confinement and viscosities the optocapillary instability forces either bulging or pinching of the confined thread and local breakup into droplets above a critical Marangoni stress. In the bulging case, the breakup of the thread occurs upstream of the laser hot spot due to Marangoni flow-induced accumulation leading to partial obstruction of the channel; the resulting overpressure forces breakup. In optocapillary pinching, the fragmentation is directly initiated near the laser spot when the pinched neck becomes locally unconfined. Optocapillary pinching thus behaves as a wall-free flow focusing device that can be used to control and finetune the thread-to-droplet transition while leaving infusing flow rates constant. We also showed numerically and experimentally that the early stage pinching dynamics obtained for different flow and heating conditions collapse onto a single behavior when time is reduced with a characteristic deformation time. Finally, we investigated the efficiency of this new optical flow focusing device by characterizing the sequences of droplets produced under continuous optical forcing and over long periods of time. At low Marangoni stress the droplet production is period-1 with a high monodispersity. At intermediate Marangoni stresses, a period-2 bifurcation emerges with two distinct droplets. Finally, further increase of the stress brings the droplet emission back to a period-1 regime. This bifurcation by period doubling, which is quite common in dripping, results here from the interplay between the optical pinching and the channel obstruction due to Marangoni flows. Two important clues may be raised for applications. Contrary to classical flow focusing situations, even in microfluidics, no inertia is required to produce drops optically; optical flow focusing thus offers an alternative at low Reynolds and Weber numbers. Then, the period-2 regime, and strictly speaking any nperiod regime with n 2, are generally not desirable for applications as the drop size usually represents the ele-mental digit for quantitative experiments (micro-reactor in chemistry, final dot in ink-jet printing, etc.). Nonetheless, fast droplet production requires large flow rates and at the same time the resulting inertia may alter the final drop distribution. Our experiment shows that although optical flow focusing does not prevent droplet production from bifurcation, finite-size effect due the confinement reduces the parameter extension of the bifurcation region and then preserves monodispersity over very large variations of control parameters. As a final remark, the efficiency of optical flow focusing requires interface deformation and then co-flows at low interfacial tensions. We overcome this difficulty by adding two surfactants, a non-ionic (resp. ionic) one in oil (resp. water). This specificity may appear a drawback at first sight, but surfactants are in fact already used in the large majority of digital microfluidic applications, and the enormous panel of surfactants available offers easy and affordable combinations.

In conclusion, from a practical standpoint, our new optical approach may prove an elegant contactless alternative to microfabrication-intensive approaches such as resistive [START_REF] Chwalek | A new method for deflecting liquid microjets[END_REF] or electrocapillary [START_REF] He | Low-frequency ac electro-flow-focusing microfluidic emulsification[END_REF] methods.

Figure 1 .

 1 Figure 1. (Color online) Concept of optical flow focusing: A laser beam focused onto a stable liquid thread flowing within a microfluidic channel (top image) can destabilize it (middle image) and reversibly force its fragmentation (bottom image). The white dot corresponds to the location of the laser beam.

Figure 2 .

 2 Figure 2. (Color online) (a) Sketch of the three-layer 1D model system; only the upper half is represented. (b) Reversal of the optocapillary deformation when varying the viscosity ratio or the confinement. Thick threads (A < 1) pinch while thinner ones (A > 1) rather bulge as indicated by the red arrows. (c) For large enough power, the thread eventually breaks in either case. The black arrows indicate the breakup location. Flow rates and laser power (Q1, Q2; P ) in µL min -1and mW are respectively (2, 1; 21), (0.2, 0.2; 13), (1.5, 2; 36), (0.03, 0.2; 12), (2, 1; 37) and (0.03, 0.2; 18) from top to bottom; the white dot is the laser spot location observed by the fluorescence of the water-dye solution. Scale bar, 100 µm.

  (b) [H 1 /H 2 ≃ 2.5 (left, A = 1.2) and 0.4 (right, A = 6.3)].

Figure 2 (

 2 b) also shows the limitations of the 1D model close to the transition value A = 1: as expected, large powers are necessary for interface deformation but a bulge was anticipated at A = 1.2 instead of a weak pinching.Moreover, for pinched water threads, the deformation propagates downstream, leading to subsequent modula-

Figure 3 .

 3 Figure 3. (Color online) (a) Sketch of the simulated geometry. (b) Longitudinal profile of the thread interface when varying the viscosity ratio and the degree of confinement; α = 0.25, t/τ def ≃ 3 except for A = 0.35 (t/τ def ≃ 6). (c) Pinching dynamics for four values of α, with η1/η2 = 0.1 and H1/H2 = 3.33, in response to an interfacial stress ∂σ/∂x evolving as represented in Left Inset. Right Inset: Same dynamics, normalized by τ def .

Figure 4 .

 4 Figure 4. (Color online) Pinching dynamics of a stable thread: normalized neck radius R/R0 versus reduced time t/τ def . (a) Fixed thread velocity (v0 ≃ 1 cm s -1 ), increasing laser powers. Dashed line: thread pinching up to rupture. (b) Comparison of the pinching dynamics for the three flow rate couples investigated, for three laser powers selected accordingly.

Figure 5 .

 5 Figure 5. (Color online) Bifurcation diagram: period-one (red circles), period-two (dark blue squares), period-three (green triangles), period-four (light blue diamonds) droplet production patterns. Error bars are ±1 standard deviation. Insets: examples of sequences measured over one minute.

Video 3 .

 3 Movie illustrating the period-1 fragmentation of a confined liquid thread under strong continuous laser forcing. The thread interface permanently touches the vertical walls. The real-time dynamics is ten times faster; P = 62 mW, (Q1, Q2) = (3, 1.5) µL min -1 . Video 4. Movie illustrating the period-2 fragmentation of a confined liquid thread under continuous laser forcing. The thread interface alternatively does and does not touch the lateral walls. The real-time dynamics is ten times faster; P = 54 mW, (Q1, Q2) = (3, 1.5) µL min -1 .

ACKNOWLEDGMENTS

The authors wish to thank Nicolas Bertin and Alexandre Vilquin for early-stage experiments, and Isabelle Cantat and Benjamin Dollet for discussions. This work was partly supported by the CNRS and the Conseil régional d'Aquitaine through Grant No. 20061102030. The authors would also like to acknowledge networking support by the COST Action MP1205 (http://costmp1205.eu).


Vincent Vercamer
email: vincent.vercamer@upmc.fr

Gérald Lelong
email: gerald.lelong@impmc.upmc.fr

Hiroyuki Hijiya

Yuki Kondo

Laurence Galoisy

Georges Calas

An optical absorption and X-band/Q-band EPR study

Keywords: iron redox, clustering, optical absorption spectroscopy, X-band/Q-band EPR/ESR, site distortion

Introduction

Iron, an impurity encountered in most industrial glasses, may be used to control glass coloration and to give rise to functional glasses, by controlling energy transmission and thermal insulation.

Understanding the influence of Fe on glass properties may then help develop innovative functional glasses. Iron usually occurs under two oxidation states Fe 2+ (ferrous) and Fe 3+ (ferric). The relative proportion of these two oxidation states defines the redox state of iron (R = Fe 2+ /Fe tot ). R is a function of glass composition, elaboration temperature and oxygen fugacity. [1][START_REF] Mysen | Silicate Glasses and Melts: Properties and Structure[END_REF][START_REF] Kress | [END_REF] Ferrous and ferric ions are localized in specific environments within glass structure, giving diluted glasses a blue (Fe 2+ ), yellow (Fe 3+ ) or green (mixed oxidation states) color. Ultraviolet-Visible-Near-Infrared (UV-Visible-NIR) spectroscopy provides a direct correlation between the colorimetric properties of transition metalcontaining glasses through their transmittance functions, and the structural surrounding of these transition elements, through the sensitivity of optical absorbance functions to crystal field effects, charge-transfer transitions or magnetic coupling. [START_REF] Burns | Mineralogical applications of crystal field theory[END_REF] Optical absorption spectroscopy has been used to study Fe-speciation and redox properties in glasses. [START_REF] Sakaguchi | [END_REF][6][7] Among transition elements, optical properties of Fe 3+ in glasses are still a matter of discussion, due to the low intensity of their spin-forbidden optical transitions and the field-independent character of the most intense transitions. In addition, these transitions are superimposed either to the tail of Fe 2+ bands in the visible range (at least 10 times more intense) or to the tail of the oxygen to metal charge transfer (OMCT) transitions (100 to 1000 more intense) in the near UV. In a sodium-silicate glass with higher than 3 wt% Fe 2 O 3 , Fe 3+ is not only in tetrahedral sites ( [4] Fe 3+ ), ~30% of ferric ions are in a higher coordination. [8] The presence of higher coordinated Fe 3+ in glasses has been deduced from Xray absorption spectroscopy, neutron diffraction and numerical simulation studies [9][10][11][12] although, their influence on optical absorption spectra in iron-doped glasses is still speculative.

Since the pioneering work of Castner et al. [13], electron paramagnetic resonance (EPR) has proven to be a sensitive technique to probe the environment of Fe 3+ in glasses (the short spin-lattice relaxation time of Fe 2+ precludes direct observation of Fe 2+ in glasses with conventional EPR). The Xband EPR spectrum of Fe 3+ in silicate glasses is characterized by an asymmetric signal at g = 4.3 * , accompanied by two weaker features near g = 8 and g = 2. The resonances at g = 4.3 and g = 8 originate from paramagnetic transitions of isolated Fe 3+ ions in rhombic (C 2v with E/D ~ 1/3) and axially (E/D ~ 0) distorted sites, respectively. [14][15][16] The attribution of the feature at g = 2 is less clear. [17] Different scenarios have been proposed: (i) a signal due to exchange-coupled pairs or clusters of more than two atoms [18] involving edge-or face-sharing Fe 3+ sites [16,19,20]. These clusters could be trimers [21] up to nano-clusters as in ferritine [22], nanoparticles [23] or obsidian wt.% Wt.% glasses [7,24]; (ii) a paramagnetic signal due to Fe 3+ in axially distorted sites (E/D ~ 0) [19], this signal being enhanced by dipole-dipole interactions. [START_REF] Abragam | Electron Paramagnetic Resonance of Transition Ions[END_REF] The proportion of these two overlapping contributions depends on glass composition and Fe-redox state. [16,[START_REF] Elvers | [END_REF] The g = 2 signal can also be assigned to Fe 3+ in regular site with high symmetry, e.g. tetrahedral or octahedral [15,27]. However, this latter interpretation is not coherent with the long tail above 350 mT that goes back to 0 around 600 mT. Low-temperature EPR measurements have shown the presence of paramagnetic clusters in Fe-doped borate glasses, because small clusters exhibit an antiferromagnetic behavior, whereas the EPR signal of diluted Fe 3+ exhibits a paramagnetic behavior. [28] Thus, the disappearance of the g = 2 signal when temperature decreases is used as evidence of the presence of Fe 3+ containing clusters, as observed in ferritin [22] or nanoparticles [23].

The aim of this study is to improve our understanding of the local environment of Fe 3+ in silicate glasses at low Fe 3+ concentration and interpret the origin of the most important Fe 3+ optical transitions by comparing variable temperature EPR spectroscopy with optical absorption spectroscopy of glasses and crystalline references. We report the first Q-band EPR data on Fe 3+ in silicate glasses, showing that Fe 3+ site distortion is limited. Regarding the results, the presence of Fe 3+ -containing clusters varies with redox and composition and seems to do not affect the optical absorption. Moreover, Fe 3+ shows a structural and chemical preference for larger alkaline earth environment.

Experimental details

Sample preparation

Three series of silicate glasses containing 0.5 wt% of Fe 2 O 3 were prepared using reagent-grade materials (Fe 2 O 3 , SiO 2 , Na 2 CO 3 , MgO and CaCO 3 ) with the nominal molar compositions 16Na 2 O-xCaO-(10-x)MgO-74SiO 2 (x = 0, 5, 10). They will be denoted as NMS05-X, NCMS05-X and NCS05-X, respectively. X describes the redox state R = Fe 2+ /Fe tot : Med for medium (R ≈ 25%), Red for reduced (R ≈ 99%), and Ox for oxidized (R ≈ 5%). The redox was adjusted using three experimental protocols:

 Normal conditions (Med) -R ≈ 25%: 400 g of batch was melted and stirred at 1500°C for 1 h in a Pt crucible under air atmosphere. Samples were poured and quenched onto preheated plates, annealed at 580°C for 1 h to release internal stresses, and finally cooled down to room temperature at 1°C/min.  Reducing conditions (Red) -R ≈ 99%: 20 g of batch was melted at 1500°C for 1 h in a carbon crucible under N 2 atmosphere. Glasses were not annealed to avoid oxidation.

 Oxidizing conditions (Ox) -R ≈ 5%: 5 g of batch was melted at 1200°C at 200°C/h in a Pt crucible, under O 2 atmosphere, then kept at 1200°C for 2 h and cooled down at 150°C/h. Glasses were not annealed to avoid reduction.

Wt.%

The amount of Fe 2+ and total iron was determined by wet chemical analysis based on standard test method [START_REF]Test Methods for Chemical Analysis of Soda-Lime and Borosilicate Glass[END_REF] using the absorption band of the complex formed between o-phenanthroline and Fe 2+ at 510 nm. The measured redox states are presented in Table 1. The vitreous nature of the glasses was checked using X-ray diffraction. Sample homogeneity was checked using an electron probe microanalyzer (CAMECA SXFive) at the Camparis Facility (Université Pierre et Marie Curie, Paris, France). Analyses were made with a 15 kV accelerating voltage and a 4 nA sample current. Actual and nominal compositions (in wt%) are shown in Table 2.

Optical absorption spectroscopy

Optical absorption spectroscopy was carried out using a Perkin-Elmer® Lambda 1050 UV-Visible-NIR spectrophotometer in transmission mode in the range 196 -3333 nm (51000 -3000 cm -1 ). Double-sided polished samples (2.5 mm thick) were used. In order to estimate the absorption background related to reflection phenomena on surfaces, 2.5 and 5.5 mm thick samples were measured for the three compositions NCS05-Med, NCMS05-Med and NMS05-Med. The absorption difference gives the equivalent spectrum of a 3 mm thick sample corrected from reflection. The intensity of Fe 3+ bands was measured after the removal of the UV edge using two Gaussian curves. The procedure explained in [START_REF] Volotinen | [END_REF] was used from 27500 to 31000 cm -1 in order to fit the tail of the OMCT bands before absorbance saturation. The fitted Fe 3+ bands (above 20000 cm -1 ) are not influenced by the Fe 2+ tail (below 20000 cm -1 ).

Electron paramagnetic resonance (EPR) spectroscopy

EPR spectra were measured using a Bruker ESP300 spectrometer at X-band (9.5 GHz) and Q-band (34 GHz), with a field modulation of 100 kHz and an amplitude modulation of 1 mT. Lowtemperature measurements were carried out using helium Oxford cryostats allowing measurements down to 4K. Samples were machined to have a cylindrical form: 15 mm long by about 2 mm diameter for X-band and 2 mm long by 1.5 mm diameter for Q-band. Samples were placed in pure silica tubes (Suprasil) for X-band or stuck on a pure silica rod for Q-band. The measurements were performed at different microwave power to adjust a convenient signal to noise ratio without saturation phenomena. EPR X-band spectra were normalized to sample mass, gain, amplitude modulation and square root of power (in mW).

Results and discussion

Optical absorption spectroscopy

The shape and position of the main Fe 3+ bands (near 22800, 24000 and 26300 cm -1 ) are similar in NCS05-Med and NCS05-Ox together with a weak and broad band at around 20500-21200 cm -1 (Figure 1). After removal of the OMCT edge, a Gaussian fit of the NCS05-Ox spectrum (Figure 2 and Table 3) was performed by comparison with three Fe 3+ -bearing crystalline references: (1) Feorthoclase [START_REF] Burns | Mineralogical applications of crystal field theory[END_REF]31], with substituted 4-fold Fe 3+ in a tetrahedral site, (2) yoderite [32,33], a 5-fold Fe 3+ Wt.% bearing mineral and (3) andradite [START_REF] Burns | Mineralogical applications of crystal field theory[END_REF]34], a 6-fold Fe 3+ bearing garnet.

According to the transitions observed in crystalline references (Table 4), the three resolved bands at around 22800, 24000 and 26300 cm -1 are commonly assigned to tetrahedral [4] Fe 3+ . [START_REF] Volotinen | [END_REF] However, a good fit may only be obtained with at least three additional contributions at around 21200, 25500 and 27200 cm -1 , otherwise Gaussian shape residuals are present at these positions. A weak contribution at around 27200 cm -1 can be attributed to uncertainties related to edge subtraction. The band at around 25500 cm -1 , which has a similar absorbance as the bands at 22800 and 24000 cm -1 , is absent from optical spectra of Fe-orthoclase ( [4] Fe 3+ ) or andradite ( [6] Fe 3+ ). Moreover, the important integrated area is indicative of a non-centrosymmetric site, such as in [5] Fe 3+ , which is consistent with the position and intensity of the 6 A 1 (S) → 4 T 2 (D) transition of [5] Fe 3+ in yoderite (Table 4). [32] The correspondence of this band between glass and mineral supports the existence of 5-fold Fe 3+ in glasses.

The contribution around 20500-21200 cm -1 has been assigned to the 6 A 1 (S) → 4 T 2 (G) transition of tetrahedral Fe 3+ . [35] For the glasses under investigation, this band is broader than in Fe-orthoclase and no decrease of the absorbance is observed between the signal at 20500 cm -1 and the band at 22800 cm - 1 . This can be explained by the overlapping of absorption bands of higher coordinated Fe 3+ , such as [START_REF] Sakaguchi | [END_REF] Fe 3+ , whose bands at 20700 and 21500 cm -1 in yoderite are assigned to the transition 6 [32] The optical evidence of [5] Fe 3+ in glasses may indicate a Fe 3+ -site distribution from 4-to 5coordinated sites, as shown for Fe 2+ . [36,37] However, there is no evidence of 6-fold coordinated Fe 3+ in the investigated glasses in the 12000-17000 cm -1 range expected for the crystal field dependent transitions of [6] Fe 3+ ( 6 A 1g (S) → 4 T 1g (G) and 6 A 1g (S) → 4 T 2g (G) as in andradite: Table 4). In this range, there is no significant contribution superimposed to the tail of Fe 2+ absorption band. The presence of [4] Fe 3+ and [5] Fe 3+ is sufficient to account for the visible-UV transitions between 20000 and 26000 cm -1 . In this range, optical absorption bands of [6] Fe 3+ occur almost at the same positions as 4-and 5-fold coordinated Fe 3+ with similar intensities. By contrast, band intensity is expected to be weak in the absence of a 3d-4p hybridization for centrosymmetric geometries such as O h, D 4h or axially strained site geometries as in andradite (C 3i ) (see [38] and references therein).

A 1 → [ 4 A 1 , 4 E(G)].
The spectrum of NCS05-Red (Figure 1) does not show evidence of Fe 3+ d-d transitions: the band at 26300 cm -1 is not discernible, even after the UV edge removal. The absence of Fe 3+ bands confirms the high Fe 2+ /Fe tot ratio (R = 99%). In the spectral region of interest for Fe 3+ , i.e. between 18000 and 28000 cm -1 , the background is slightly higher than in the NCS05-Ox and two weak bands are visible around 21500 and 23400 cm -1 . These two weak bands, which can not be ascribable to residual Fe 3+ (the Fe 3+ concentration is estimated here at ~50 ppm), are assigned to the Fe 2+ spin-forbidden transitions 5 T 2 (D) → 3 T 1 (H) and 5 T 2 (D) → 3 T 2 (H) as suggested in Fe 2+ (H 2 O) 6 [START_REF] Burns | Mineralogical applications of crystal field theory[END_REF] and in a sodiumalkaline earth silicate glass. [START_REF] Glebov | Proc. 18th Int. Congr. Glass[END_REF] Grandidierite ( [5] Fe 2+ ) shows similar weak features between 20000 and Wt.% 25000 cm -1 . [START_REF] Rossman | [END_REF]41] Our measurements suggest that the background of the optical spectra of reduced silicate glasses containing iron is constituted by a succession of poorly resolved bands with low intensities related to Fe 2+ spin-forbidden transitions that could lead to overestimate the background if the minimum of the spectrum around 18000 cm -1 is considered as the background value.

The Ca:Mg ratio influences the optical absorption spectra as shown in Figure 3. When replacing Ca by Mg, the background-subtracted Fe 3+ spectra keep a similar shape, the NCMS05-Med spectrum being closer to NCS05-Med rather than to NMS05-Med. At the same time, the 26300 cm -1 and 25500 cm -1 bands become more intense and weaker, respectively. This could arise from an increasing proportion of [4] Fe 3+ in the oxidized and medium Mg-rich samples (NMS) compared to Ca-rich samples (NCS).

X-band electron paramagnetic resonance (EPR) spectroscopy

NCS05-Med and -Ox glasses give typical EPR signals for a Fe 3+ -bearing glass (Figure 4) with two main lines at 160 mT (g = 4.3) and at 340 mT (g = 2) characteristic of the site repartition presented in the introduction. However, NCS05-Red (50 ppm Fe 3+ ) gives rise to a peculiar EPR signal (Figure 4) with (i) a zero field signal, (ii) a sloping background below 100 mT, (iii) a weak signal at g = 8 (around 80 mT), (iv) appearance of shoulders just before and after the g = 4.3 line (at 130 mT and 170 mT), and (v) a split signal at g = 4.3 (See the inset of Figure 4). Moreover, two signals are superimposed at g = 2: a wide signal (marked A in Figure 4) related to clustering of Fe 3+ ions and a sharp one at 350 mT (marked B in Figure 4) related to defects trapped on silicate groups that will not be studied here. The split signal at g = 4.3 observed in the reduced samples is due to the separation of the three EPR Kramer's doublet transitions d xx , d yy and d zz , which are not overlapping each other. This unusual splitting indicates that the distribution of the isolated Fe 3+ sites is relatively narrow and does not include sites with a pure rhombic distortion. By contrast, in oxidized samples, Fe 3+ (>1000 ppm) the broad site distribution among different site geometries includes sites with rhombic distortion, which is consistent with optical spectroscopy. Site geometry does depend on Fe 3+ concentration: in that sense, residual Fe 3+ appears to probe the specific ("defective") sites of the structure of reduced glasses, as it probes an average local structure broader site distribution at higher concentration.

Low-temperature X-band EPR

The EPR spectrum of NCS05-Red at 4K (Figure 5) shows an intense and broad line centered on 60 mT (g~13), which follows a Curie's law (I pp is proportional to 1/T). This signal may be caused by a ferromagnetic resonance due to traces of metallic Fe [16], and is at the origin of the zero field signal and the sloping background observed at room temperature (Figure 4). The g = 4.3 line is narrower at 4K than at room temperature and shows a shoulder at 170 mT. The g = 2 signal for both NCS05-Med and NCS05-Ox keeps the same shape at low temperature but the g 4.3 /g 2 ratio is two times higher than at room temperature. For NCS05-Red, the g = 2 line (marked A in Figure 4) is wider (∆H pp ~60 mT) than in NCS05-Med and NCS05-Ox (∆H pp ~25 mT). These observations confirm a modification of the Wt.% Fe 3+ site repartition as a function of the redox state of the glass. As the intensity of the signal at g = 2 line disappears on the EPR spectra of NCS05-Red when temperature is lowered to 4K, this signal is assigned to the presence of Fe 3+ clusters, with Fe 3+ -Fe 3+ and/or Fe 3+ -Fe 2+ magnetic coupling becomes insignificant in comparison with the increase of the paramagnetic effect of isolated iron in regular sites at g = 4.3. [START_REF] Sakaguchi | [END_REF][15][16][17]28,[42][43][44][45][46]

Low-temperature Q-band EPR

To our knowledge, Q-band EPR spectra have not been investigated in silicate glasses despite a higher microwave frequency may help in assessing the rhombic distortion of the Fe 3+ site. Q-band (34 Ghz) EPR spectra have been investigated for NCS05-Med and NCS05-Red samples (Figure 6). The NCM05-Med signal is not constant before and after the g = 2 line, suggesting the presence of a broad signal originating from Fe 3+ -Fe 3+ interactions and overlapping with the isolated Fe 3+ signal at g = 2 (dashed line in Figure 6).

The signal proportion between g = 2 and g = 4.3 are inverted comparing to X-band. The dramatic loss of intensity of the g = 4.3 signal indicates that rhombic distortion is small relative to Q-band frequency. As a consequence the g = 2 signal of isolated paramagnetic species is expected to increase relative to X-band spectra, which is observed: the Q-band signal of the NCS05-Med sample at g = 2 is a relatively sharp and intense signal. In the NCS05-Med sample, about 80 to 90% of the integrated signal is at g = 2 whereas 10 to 20% is at g = 4.3. The remaining g = 4.3 weak line demonstrates the presence of a low amount of sites with a rhombic distortion. This confirms that Fe 3+ plays a network former role with a limited distribution of the polarization of the oxygens in polymerized glasses and that only a few sites show a rhombic distortion.

In the Q-band spectrum of NCS05-Red (Figure 6), the absence of g = 4.3 signal suggests a less distorted environment for Fe 3+ in reduced than in medium samples. The numerous Fe 3+ peaks between 1000 and 1300 mT may be related to a lower site distribution, in agreement with the splitting observed at g = 4.3 in X-band (inset of Figures 3 and4). The low-field signal (below 500 mT) confirms the presence of a few ppm of metallic Fe, already detected at X-band.

Evidence of non-random distribution of Fe 3+

As observed with optical absorption spectroscopy (Figure 3) and X-band EPR (Figures 7a, 7b and7c), NCMS shows an intermediate behavior between NCS and NMS. Surprisingly, the NCMS05-Med spectrum does not correspond to an average of the spectra of NCS05-Med and NMS05-Med, meaning that the environment of Fe 3+ in NCMS looks more like Fe 3+ in NCS than in NMS. This indicates that Fe 3+ sites are not randomly and homogeneously distributed into the glass matrix as they are sensing a more calcic than magnesian environment. A similar preferential coupling between transition elements and alkalis/alkaline earths has been observed for Cr 3+ ions in soda-lime glasses, where Cr 3+ shows a Wt.% preference for a Na-rich environment. [47]

Site distortion

The redox state of the samples investigated influences the EPR spectra (Figure 8). The almost linear evolution of I pp and ∆H pp with redox for a fixed total iron content (0.5 wt% of Fe 2 O 3 ) is similar to the variation obtained by changing the total iron content at a given redox. [START_REF] Elvers | [END_REF] This indicates a linear variation of the Fe 3+ concentration with the redox values in the glass elaboration conditions used. The quasi-linearity of the product I pp .∆H pp (Figure 8c) confirms the Lorentzian shape expected for the g = 4.3 signal of Fe 3+ in diluted glasses. [START_REF] Elvers | [END_REF] Increasing Fe 3+ concentration leads to a broadening of the EPR signal, which is indicative of a larger distribution including isolated Fe 3+ sites with a smaller average distortion. On the contrary, at low Fe 3+ -content, site distribution is smaller (discrete EPR transitions at g = 4.3 compared to oxidized sample without split signal) with a predominance of distorted sites (smaller ∆H pp ). However, the Q-band EPR signal at g = 4.3 (Figure 6) indicates that only a small amount of rhombically distorted sites are present in NCS05-Med while there is no resonance in NCS05-Red. Despite the average sites appear more distorted in NCS05-Red, NCS05-Med presents sites with a stronger rhombic distortion.

In Q-band, the integrated EPR signal at g = 4.3 participates in less than 10% of the total integrated EPR signal, while it represents 50-60% in X-band. The trend of the g = 2 signal to increase at Q-band relatively to X-band, in which the g = 4.3 signal is dominating, is in accordance with previous observations [42] on borate glasses. However, in the sodium-alkaline earth-silicate glasses investigated, the low intensity of the Q-band signal at g = 4.3 suggests a smaller proportion of rhombically distorted Fe 3+ sites than in lithium-borate glasses.

The nature of the alkaline earth cations has a smaller effect on I pp and ∆H pp than the redox.

Nevertheless, Mg-containing samples show weaker and broader g = 4.3 signals. Whatever the redox, the larger linewidth of the g = 4.3 signal (Figures 7b and8b) for NMS samples compared to NCS samples reflects a decrease of the rhombic distortion of Fe 3+ sites (i.e. lower value of λ = E/D). [START_REF] Elvers | [END_REF] A lower rhombic distortion is supported by the increase of the signal associated to axially distorted sites at g = 8 (Figure 7a) and g = 2 (Figure 7c). Due to smaller coordination numbers, 4-fold sites are indeed less subject to distortion. This effect is consistent with the increase of the concentration of tetrahedral sites in Mg-bearing glasses compared to Ca-bearing glasses, with lower average coordination number for Fe 3+ in soda-magnesia glasses (~4.5) than in soda-lime glasses (~5.0). [48] This is supported by our optical absorption data (Figure 3) suggesting an increase of the 25500 cm -1 contribution (attributed to 5-fold Fe 3+ ) and a decrease of the 26300 cm -1 contribution (attributed to tetrahedral Fe 3+ ).

Clustering vs isolated sites

Wt.% Clustering effects may be evaluated by the ratio between the intensity of the transitions linked to isolated and clustered Fe 3+ at g = 4.3 and g = 2, respectively (Figure 9). The reduced samples have the lowest I pp (g = 4.3)/I pp (g = 2) ratio, suggesting that they contain a higher proportion of Fe 3+ involved in clusters than more oxidized samples. This interpretation is corroborated by the antiferromagnetic origin of the g = 2 signal in reduced samples (Figure 5). According to previous articles that studied the effect of total Fe-content in sodium-silicate glasses, the reduced samples investigated here have a g = 2 signal shape similar to the one observed at higher iron content ([Fe 2 O 3 ] > 1 mol%). [17,18] In these high Fe-glasses, Fe 3+ -rich clusters have been shown by optical absorption and Mössbauer spectroscopic properties [49] and confirmed by neutron diffraction and numerical simulations. [50] The present study shows that, in reduced samples, residual Fe 3+ tends to segregate, with an EPR signature similar to that in concentrated glasses. In minerals, clusters involve [6] Fe 3+ cations. [51] By analogy, iron clusters in glasses would be favored by Fe 3+ located in 5-fold or 6-fold coordinated sites: this indicates that reduced glasses contain a higher proportion of [5] Fe 3+ and/or [6] Fe 3+ than the corresponding oxidized glasses. Indeed, the linkage between tetrahedral cation sites is unfavored energetically, as predicted by the Lowenstein exclusion rule [52] leading to the formation of oxygen triclusters. Whatever the redox, the ratio I pp (g = 4.3)/I pp (g = 2) decreases when Mg is added, suggesting that the NMS05 glass contains a higher proportion of Fe 3+ clusters than the NCS05 glass. This result indicates that smaller alkaline-earth ions promote Fe 3+ cluster formation, in agreement agrees with previous studies. [53]

Conclusions

Complementary data from X-and Q-band EPR at variable temperature, and optical absorption spectroscopy provide unique information on the structural behavior of Fe 3+ in sodium-alkaline earth silicate glasses as a function of redox (oxidized glasses with Fe 2+ /Fe tot ≈ 5%, medium glasses with Fe 2+ /Fe tot ≈ 25% and reduced glasses with Fe 2+ /Fe tot ≈ 99%) and glass composition. In medium and oxidized glasses, a majority of Fe 3+ occurs in tetrahedral sites with a minority of 5-fold coordinated sites. These sites are partitioned between isolated and clustered sites. There is a significant compositional dependence of the spectroscopic properties of sodium-alkaline earth silicate glasses.

Changing Ca to Mg increases the amount of tetrahedral Fe 3+ sites and decreases their distortion, which modifies glass optical transmission in the visible. It also reduces the amount of isolated rhombic Fe 3+ sites and promotes the formation of Fe-rich clusters. These compositional effects confirm the nonhomogeneous structure of silicate glasses.

The site distortion of the isolated sites has been evaluated by X-and Q-band EPR at helium and room temperature. The Fe-redox state has an important effect on the local environment of Fe 3+ . In medium and oxidized glasses, Fe 3+ sites are mostly isolated with a broad distribution centered on a modest rhombic distortion, but including a small proportion of rhombically distorted Fe 3+ sites. This indicates a high proportion of [4] Fe 3+ sites. Reduced glasses show a larger contribution of Fe-rich Wt.% clusters, with a possible presence of metallic Fe. An original feature is the presence of rhombically distorted, isolated Fe 3+ sites with a narrow site distribution, which is consistent with the presence of higher coordination numbers. The spectroscopic behavior of Fe 3+ in these reduced glasses indicates an original surrounding, which is hidden within a broader site distribution at higher Fe 3+ -content in the medium and oxidized glasses. Wt.%

Table 3: Fitting parameters and corresponding spin forbidden d-d transitions attribution for the absorbance peaks of Fe 3+

ions in the NCS05-Med glass.

Peak # Peak position (cm -1 )

Full width at half maximum (cm -1 )

Peak height (L.mol -1 .cm -1 ) Assignments 1 21200±200 2360±300 0.13±0.02 [4] Fe 3+ , 6 A 1 (S) → 4 T 2 (G) [START_REF] Sakaguchi | [END_REF] Fe 3+ , 6 A 1 (S) → 4 A 1 , 4 E(G) 2 22810±40 1450±70 0.46±0.02 [4] Fe 3+ , 6 A 1 (S) → 4 A 1 , 4 E(G) 3 23980±30 1370±100 0.38±0.02 [4] Fe 3+ , 6 A 1 (S) → 4 T 2 (D) 4 25600±30 1030±75 0.42±0.02 [START_REF] Sakaguchi | [END_REF] Fe 3+ , 6 A 1 (S) → 4 T 2 (D) 5 26340±10 910±10 1.71±0.02 [4] Fe 3+ , 6 A 1 (S) → 4 E(D) 6 27190±50 540±100 0.11±0.02 [5,6] Fe 3+ , 6 A 1(g) (S)

→ 4 E (g) (D) 7 36400±300 8050±500 900±30 OMCT Fe 3+ → O 2- 8 42400±300 16500±900 120±30 OMCT Fe 2+ → O 2-

Table 4: Fe 3+ spin forbidden d-d transitions attribution for the absorbance peaks (in cm -1 ) of 3 Fe-bearing minerals with different site symmetry and coordination number (CN). Band numbers in parenthesis correspond to the fit of the glass (Table 3).

Name CN Geometry 4 T 1(g) (G) 4 T 2(g) (G) 4 A 1(g) , 4 E (g) (G) 4 T 2(g) (D) Wt.% Wt.% Wt.%

Figure 1 :

 1 Figure 1: Optical absorption coefficient of the oxidized (NCS05-Ox / yellow), the intermediate redox (NCS05-Med / green) and the reduced (NCS05-Red / blue) soda-lime-silica glasses. The inset is a zoom on the Fe 3+ d-d transitions range.

Figure 2 :

 2 Figure 2: Resulting fit of Fe 3+ bands in the soda-lime oxidized glass (NCS05-Ox) after UV-edge removal. The Fe 3+ bands were fitted with Gaussian functions. The experimental spectrum and the fitted spectrum are shown as a blue line and a red dashed line, respectively.

Figure 3 :

 3 Figure 3: Optical absorption spectra for the three sodium-alkaline earth silicate glasses with a Fe 2+ /Fe tot ratio around 25%: NCS05-Med (red), NCMS05-Med (green) and NCM05-Med (blue). UV-edges have been removed using a Gaussian fit of the charge transfer band.

Figure 4 :

 4 Figure 4: Room temperature X-band EPR signals of a soda-lime-silica glass at three different redox: NCS05-Ox (yellow / R = 6%), NCS05-Med (green, R = 28%) and NCS05-Red (blue, R = 99%). The inset is a zoom on the g = 4.3 line.

Figure 5 :

 5 Figure 5: Low-temperature X-band EPR signals of a soda-lime-silica glass at three different redox: NCS05-Ox (yellow / R = 6%), NCS05-Med (green, R = 28%) and NCS05-Red (blue, R = 99%). The inset is a zoom on the g = 4.3 line.

Figure 6 :

 6 Figure 6: Q-band EPR signals for the samples NCS05-M (green, R = 28%) and NCS05-Red (blue, R = 99%) for the sodalime glass measured at 20K. The intensities are not normalized. The EPR signal of NCS05-Red has been magnified for clarity reasons.

Figure 7 :

 7 Figure 7: Effects on the X-band EPR signals when Ca 2+ is substituted by Mg 2+ in the glassy matrix for the three intermediate redox (NCS05-Med (red), NCMS05-Med(green) and NMS05-Med(blue)): (a) increase of the g = 8 line, (b) decrease and broadening of the g = 4.3 line and (c) increase of the g = 2 line.

Figure 8 :

 8 Figure 8: (a) g = 4.3 peak-to-peak intensity (I pp ), (b) g = 4.3 peak-to-peak linewidth (∆H pp ) and (c) the product of ∆H pp and I pp as a function of redox R = Fe 2+ /Fe tot for the three glass compositions under study: NCS, NCMS and NMS. The lines are a guide for the eyes.

Figure 9 :

 9 Figure 9: Ratio between the peak-to-peak intensity at g = 4.3 and the peak-to-peak intensity at g = 2 at 298 K as a function of the composition of the glassy matrix. The ratio I pp (g = 4.3)/I pp (g = 2) is correlated to the proportion of Fe 3+ in isolated distorted sites vs Fe 3+ in clusters.

Table 1 : Redox measurements: Fe 2+ /Fe tot (in %), of the 0.5 wt% Fe 2 O 3 doped silicates from wet chemical analysis (3% uncertainty).

 1

	Sample name	Oxidized (Ox)	Medium (Med)	Reduced (Red)
	NCS05	6	28	99
	NCMS05	5	25	99
	NMS05	6	24	99

Table 2 : Electron microprobe analysis (EMPA) measurements (in wt%) and nominal composition (converted in wt%), standard deviations of each element are in parenthesis.

 2

		SiO 2	Na 2 O	CaO	MgO	Fe 2 O 3	Total
	NCS05-Med	73.89 (0.73)	15.73 (0.37)	9.47 (0.23)	0.01 (0.01)	0.52 (0.02)	99.61 (0.56)
	NCS05-Ox	75.43 (0.50)	15.99 (0.40)	8.26 (0.31)	0.00 (0.00) 0.54 (0.004)	100.23 (0.43)
	NCS05-Red	74.32 (0.80)	15.61 (0.44)	9.35 (0.33)	0.00 (0.00)	0.51 (0.02)	99.79 (0.58)
	Nominal-NCS05	73.75	16.45	9.30	0.00	0.50	100.00
	NCMS05-Med	75.78 (1.18)	16.05 (0.51)	4.54 (0.23)	3.31 (0.31)	0.52 (0.03)	100.17 (0.84)
	NCMS05-Ox	77.32 (0.56)	15.89 (0.10)	4.43 (0.17)	2.84 (0.10) 0.52 (0.003)	100.99 (0.59)
	NCMS05-Red	74.50 (0.68)	16.37 (0.65)	4.97 (0.16)	3.67 (0.16)	0.55 (0.04)	100.05 (0.61)
	Nominal-NCMS05	74.73	16.67	4.71	3.39	0.50	100.00
	NMS05-Med	75.62 (0.96)	16.77 (0.43)	0.01 (0.01)	7.03 (0.21)	0.51 (0.03)	99.93 (0.88)
	NMS05-Ox	75.69 (0.47)	17.61 (0.28)	0.00 (0.00)	7.00 (0.16)	0.56 (0.01)	100.87 (0.46)
	NMS05-Red	75.45 (0.91)	17.04 (0.37)	0.02 (0.03)	7.19 (0.20)	0.52 (0.02)	100.19 (0.88)
	Nominal-NMS05	75.74	16.89	0.00	6.87	0.50	100.00

Acknowledgments

The authors would like to thank AGC for sample synthesis and technical support, Jürgen Von Bardeleben (INSP, University Pierre & Marie Curie) and Maxime Guillaumet (IMPMC, University Pierre & Marie Curie) for their help in data acquisition. This work was supported by ANRT, under the CIFRE Contract 2012/0640 and by AGC.


Thibaut Sylvestre
email: thibaut.sylvestre@univ-fcomte.fr

Jean-Charles Beugnot

Subwavelength-diameter optical fibers for Brillouin scattering

Keywords:

By confining light into subwavelength-diameter tapered optical fibers, we demonstrated a new type of Brillouin light scattering from surface and hybrid acoustic waves. This could be used to develop new optical sensors that are innovative and highly sensitive.

I. INTRODUCTION

Subwavelength-diameter optical fibers, also known as optical fiber micro and nanowires, are the tiny and cousins of standard optical fibers [START_REF] Tong | Subwavelength-diameter silica wires for low loss optical wave guiding[END_REF][START_REF] Brambilla | Optical fiber nanowires and microwires: a review[END_REF]. These hair-like slivers of glass, manufactured by tapering optical fibers down to a size hundred times smaller than a strand of hair, have a number of optical and mechanical properties that make them very attractive for both fundamental physics and technological applications. In addition to providing strong light confinement and enhanced nonlinear optical effects, optical microwires also exhibit a large evanescent field, enabling applications not currently possible with comparatively bulky optical fibers. Although microfibers have helped greatly to enhance the optical Kerr effect and stimulated Raman scattering for supercontinuum generation [START_REF] Foster | Nonlinear optics in photonic nanowires[END_REF], Brillouin scattering in these tiny waveguides has not been explored yet.

Brillouin scattering is one of the most important nonlinear optical processes in optical fibers resulting from the coherent interaction between light and acoustic waves. It has important applications ranging from optical sensors to telecommunications, and it also enables slow and fast light propagation. In standard optical fibers, light excites and interacts with shear and longitudinal bulk elastic waves, giving rise to well-known physical effects such as forward guided acoustic wave Brillouin scattering (GAWBS) and backward stimulated Brillouin scattering (SBS), respectively [START_REF] Shelby | Guided acoustic-wave Brillouin scattering[END_REF][START_REF] Ippen | Stimulated Brillouin scattering in optical fibers[END_REF].

In a subwavelength-diameter optical fiber, the situation changes dramatically. We report here the experimental observation of Brillouin light scattering from surface and hybrid acoustic waves [START_REF] Beugnot | Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fiber[END_REF]. To observe the surface waves, we injected a laser beam into a 1µm-diameter silica fiber wire. As the beam travels the fiber, it induces infinitesimal vibrations in the wire through the electrostrictive optical force, displacing the surface by a few nanometers. The distortions lead to surface acoustic waves that travel along the wire surface at 3400 m.s -1 . These acoustic phonons in turn affect the light propagation as a part of the light is backscattered with a slight shift of its carrier frequency due to the Doppler effect of in the range 5-6 GHz, far from the standard Brillouin frequency shift of standard optical fibers (11 GHz).

II. EXPERIMENT

To produce these silica microwires, colleagues at the Institut d'Optique in Paris heated and stretched standard singlemode optical fibers measuring 125 µm in diameter, down to a size of 1 µm over 8 cm. Figure 1(a) shows a scheme of the tapered optical fiber drawn from this technique. The remainder of the study was carried out in our lab at the Femto-ST Institute in Besançon. Figure 1(b) shows the experimental setup to detect the acoustic waves. As a pump laser, we used a narrowlinewidth continuous-wave distribute feedback laser (DFB) running at a wavelength of 1,550 nm. The laser output was split into two beams using a fiber coupler. One beam was amplified and injected in the optical microwire through an optical circulator, while the other beam served for detection. We then used a heterodyne detection in which the backscattered light from the microwire was mixed with the input coming from a second fiber coupler. The resulting beat note was then detected using a fast photodiode and Brillouin spectrum recorded using an electrical spectrum analyzer.

III. RESULTS

Figure 2 shows the experimental spectrum for an input power of 100 mW. We see the clear emergence of several frequency peaks with different weights in a radio-frequency range from 6 to 11 GHz. First, the high frequency at 10.86 GHz originates from standard Brillouin scattering in the untapered fiber sections. More importantly, two other peaks appear around 6 GHz. We identified them as resulting from Brillouin scattering from surface waves whereas the three other resonances around 9 GHz are clearly the signature of hybrid shear-longitudinal acoustic waves.

IV. SIMULATIONS

This observation was subsequently confirmed by a numerical simulation of the elastodynamics equation that includes the electrostrictive optical force [START_REF] Beugnot | Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fiber[END_REF][START_REF] Beugnot | Reduction and control of stimulated Brillouin scattering in polymer-coated chalcogenide optical microwires[END_REF]. Figure 3 shows the computed sound density generated by light in the silica microwire as a function of frequency and wire diameter. As can be seen, we retrieve most of the surface and hybrid acoustic resonances around 6 and 9 GHz, as those observed experimentally. Figure 3 also shows that, for a given microwire diameter, multiple hybrid waves with widely spaced frequencies can be simultaneously excited. This means that, in such small sub-wavelength waveguides, the light-sound interaction is fundamentally different from standard optical fibers. Brillouin spectrum is not simply the signature of a single bulk longitudinal sound wave. In such tiny waveguides, instead, waveguide boundary conditions induce a strong coupling of shear and longitudinal displacements, resulting in a much richer dynamics of light interaction with sound. We can thus the emergence of several anti-crossings in Fig. 3 due to the fact the hybrid waves are not orthogonal and thus strongly interact. The bottom figures show the associated transverse and axial displacements in the microwire for four of surface and hybrid modes. As can be seen, the surface acoustic mode A at 5.382 GHz exhibits a transversal displacement of a few picometres.

V. CONCLUSIONS This surface acoustic wave Brillouin scattering (SAWBS) had never been observed previously, for it only occurs when light is confined in a subwavelength-diameter waveguide. In standard optical fibers, light travels essentially in the core of the fiber (with a 10-micrometer diameter), and consequently does not generate surface waves. It thus may represent a landmark to foster the investigation of such surface waves in many different photonic platforms, such as photonic crystal fibers and integrated silicon or chalcogenide photonic chips.

Since the waves generated by the confinement of the light travel along the surface, they are sensitive to environmental factors such as temperature, pressure and ambient gas. This makes it possible to design highly sensitive and compact optical sensors for industry. These results also help improve our knowledge of the fundamental interaction between light and sound on an infinitesimally small scale. Surface Brillouin scattering also opens new opportunities in other domains such as nonlinear plasmonics. For instance, one could expect a strong interaction between surface waves and surface plasmons by depositing a thin layer of gold or silver on the microwire.

Fig. 1 .

 1 Fig. 1. (a) Scheme of the tapered silica microwire and the surface acoustic wave. (b) Experimental setup. DFB, distributed-feedback laser. FC, 10:90 fibre coupler. EDFA, Erbium-doped fibre amplifier. (c) Experimental

Fig. 2 .

 2 Fig. 2. Experimental beat note radio-frequency spectrum out of a 1µmdiameter silica microwire showing the Brillouin sidebands due to two surface acoustic waves and three hybrid shear-longitudinal waves. The highest frequency peak at 10.86 GHz are Brillouin sidebands in untapered fibre sections.

Fig. 3 .

 3 Fig. 3. Numerical simulations of the acoustic wave spectrum and mechanical displacements in a silica microwire generated by light-induced electrostriction. Top : Colour plot of sound density as a function of frequency and for a wire diameter varying from 0.6 to 3.5 µm. White arrows indicate the surface and hybrid acoustic waves, and the anti-crossing points due to the strong coupling between shear and longitufinal (P) waves. Bottom : Transverse (trans.) and axial displacements in the microwire associated with surface and hybrid acoustic modes labelled A,B,C,D.

ACKNOWLEDGMENT

The authors gratefully thank collaborators S. Lebrun and G. Pauliat from the Charles Fabry Laboratory (Institut D'Optique, Orsay, France) for providing the silica microwires, and Raja Hamad and Martin Rochette from McGill University for providing the chalcogenide microwires. These works have been supported by the Agence Nationale de la Recherche (ANR LABEX Action and OASIS projects).