domain: spi nlin

text:
M Kalli

P Pico

L Chagot

L Kahouadji

S Shin

J Chergui

D Juric

O K Matar

P Angeli

Effect of surfactants during drop formation in a microfluidic channel: A

Keywords: Droplet formation, Flow-focusing microchannel, Surfactants, µPIV

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Introduction

Microfluidic channels are widely used to produce droplets with controlled size and low polydispersity index, which are important in applications such as emulsification, inkjet printing, cosmetics, and healthcare [START_REF] Lawrence | Microemulsion-based media as novel drug delivery systems[END_REF][START_REF] Yang | Drop-on-demand satellite-free drop formation for precision fluid delivery[END_REF][START_REF] Park | Microfluidic production of monodisperse emulsions for cosmetics[END_REF]. Surfactants are commonly added during drop formation to modify the interfacial properties and improve the stability of the emulsions. The presence of surfactant molecules significantly affects the non-linear drop formation process and thus the final size [START_REF] Chen | The dynamic adsorption of different surfactants on droplet formation in coaxial microfluidic devices[END_REF]. In the common dripping regime, drop formation depends on the balance between the interfacial tension and the drag forces, with the latter being the combination of shear and pressure forces [START_REF] Xu | Shear force induced monodisperse droplet formation in a microfluidic device by controlling wetting properties[END_REF]. As a result, a lower interfacial tension caused by surfactant addition will result in smaller and more rapidly-forming drops. As seen in previous work, the small characteristic timescales that are present in microfluidics can lead to interfacial tension values different from the equilibrium ones, which are often used in the calculation of capillary numbers Ca (Kalli & Angeli 2022) and in predictive equations for drop size. Additionally, shear stresses from high continuous phase velocities can redistribute surfactant molecules at the interface, which will cause interfacial tension gradients and thus Marangoni stresses [START_REF] Kovalchuk | Effect of surfactant dynamics on flow patterns inside drops moving in rectangular microfluidic channels[END_REF]. For surfactants that result in a large decrease in interfacial tension, the effect of interface retardation at low surfactant concentrations can be sufficiently large to change the velocity fields close to the interface and inside the drop, due to Marangoni stresses in the opposite direction to the flow. At high concentrations, surfactant exchange between the bulk and the interface is fast and interface remobilisation is seen at high velocities [START_REF] Martin | Mass transfer and interfacial properties in two-phase microchannel flows[END_REF]. Since both observations will greatly impact the final surfactant transport, it is important to study the behaviour of such surfactants at different concentrations, below and above the critical micelle concentration (CMC).

As first reported by [START_REF] Garstecki | Mechanism for flow-rate controlled breakup in confined geometries: A route to monodisperse emulsions[END_REF], drop formation in the dripping regime is typically divided into three stages called expansion, necking, and pinch-off. Following drop detachment, the interface pulls towards the inlet before recoiling in the downstream direction. At this stage, drop growth is mainly in the streamwise direction forming a parabolic interface profile and is dominated by interfacial tension. Once the pressure builds-up due to restriction of the inlet from the forming drop, the drag force overcomes the interfacial tension force and a reversal of the interface curvature is observed, which indicates the beginning of the necking stage. The neck width decreases until the drag force is large enough to cause drop detachment and thus pinch-off. In recent studies [START_REF] Wang | Determination of dynamic interfacial tension and its effect on droplet formation in the t-shaped microdispersion process[END_REF][START_REF] Xu | The dynamic effects of surfactants on droplet formation in coaxial microfluidic devices[END_REF], it was found that high dispersed phase flowrates enhance convection and surfactant mass transfer to the interface, which explain the reduced interfacial tension values at short formation times. Furthermore, it was found that absorption and mass transfer rates can compete with drop formation times (Kalli et al. 2022). Investigating the transport of surfactants during each phase of the drop formation process is vital in understanding the impact on the final drop size.

Improvements in experimental techniques allow the accurate visualisation of flows inside microchannels, which can be used to explore the effects of surfactant transport. [START_REF] Kiratzis | Effect of surfactant addition and viscosity of the continuous phase on flow fields and kinetics of drop formation in a flow-focusing microfluidic device[END_REF] used Ghost Particle Velocimetry (GPV) to study the effect of surfactant concentration, continuous phase velocity and viscosity on drop formation in rectangular channels. Results revealed smaller velocities at the interface when a surfactant of the same family but with a slower adsorption rate (lower CMC) was present, suggesting the presence of Marangoni stresses. In addition, larger internal circulation inside the drops was seen at higher continuous phase viscosity during the expansion stage. [START_REF] Roumpea | Experimental studies on droplet formation in a flow-focusing microchannel in the presence of surfactants[END_REF] used an innovative two-colour micro-Particle Image Velocimetry (µPIV) technique to study velocity fields in a flow-focusing microchannel in the presence of two cationic surfactants of the same family but different adsorption kinetics (C 12 TAB, dodecyltrimethylammonium bromide and C 16 TAB, hexadecyltrimethylammonium bromide). Addition of surfactant in the dispersed phase resulted in higher velocities at the tip and more uniform velocity fields inside the drop, which was attributed to surfactant accumulation at the tip that increases the interface rigidity in that area. A second observation was the decrease in the circulation inside the drop, which was again attributed to Marangoni effects that caused the interface to move in the opposite direction to the flow and opposed circulation. Similar effects were seen when a non-ionic surfactant (Span80) was added in the continuous oil phase, in a T-junction channel [START_REF] Riaud | Numerical study of surfactant dynamics during emulsification in a t-junction microchannel[END_REF]. A stagnant cap was observed in this case near the drop tip that split the forming drop interface into a mobile and a rigid region. The resulting Marangoni force in the opposite direction of the forming drop resisted the flow circulation. Studies on drop formation in a flow-focusing microchannel by [START_REF] Carrier | Time-resolved mixing and flow-field measurements during droplet formation in a flow-focusing junction[END_REF] revealed a flow inversion during pinchoff and weaker recirculation at the base of the drop compared to its front, upon addition of SDS (sodium dodecylsulfate) surfactant. Changes in circulation within surfactant-laden drops using µPIV were investigated in emulsions produced in a tapered microchannel leading to a geometrical constriction [START_REF] Leong | Investigating droplet internal flow in concentrated emulsion when flowing in microchannel using micro-piv[END_REF]. It was found that vortical structures forming when the drops passed the constriction, increase circulation up to 2.5 times compared to droplet mixing at the channel constriction.

Despite the work reviewed above, there are still important limitations to be addressed in order to fully understand the effect of surfactant on the drop formation process. As previously mentioned, it is essential to know the surfactant distribution at the interface as it can accumulate at the tip and cause Marangoni flows as the drop forms. The surfactant concentration at the interface is very challenging to obtain experimentally [START_REF] Dussaud | Spreading characteristics of an insoluble surfactant film on a thin liquid layer: comparison between theory and experiment[END_REF], especially during flow. Even if a fluorescent surfactant is used, the powerful intensity of the laser needed for visualisation can cause photobleaching [START_REF] Eggeling | Photobleaching of fluorescent dyes under conditions used for single-molecule detection: Evidence of two-step photolysis[END_REF], which is non-reversible. Another issue is self-quenching [START_REF] Aicolina | Fluorescence quenching studies of the self-association in ater of fluorescent surfactants. alkaryl-2-pyrazolines and alkyl-7hydroxycoumarins[END_REF][START_REF] Li | Optical fiber optofluidic laser based on surfactant solubilization of rhodamine b gain in an aqueous solution[END_REF], where the total emitted intensity can change upon temperature, pressure, or pH fluctuations. Additionally, most fluorescent surfactants are soluble in solvents such as chloroform and dimethylformamide, which can be toxic. Since most fluorescently-tagged surfactants are synthesised by adding a fluorescent group to the amphiphilic structure of the surfactant, they usually result in heavier molecules that exhibit low interfacial activity (i.e. small reduction in interfacial tension with concentration) or slow adsorption kinetics, especially when compared with the fast drop formation process. [START_REF] Dong | Laser induced fluorescence studies on the distribution of surfactants during drop/interface coalescence[END_REF] used Planar Laser Induced Fluroescence (PLIF) to capture the spatio-temporal distribution of a fluorescent surfactant during the coalescence of a mm size drop with a flat surface, using a laser sheet with 1 mm thickness. This dimension is an order of magnitude larger than typical microchannel length scales. On the other hand, the volume illumination used for microchannel LIF studies is not ideal when tracing molecules at a 3D interface.

The complex phenomena of drop formation in microchannels have also been investigated through numerical approaches. Computational Fluid Dynamics (CFD) simulations in particular are a powerful tool to overcome most of the above-mentioned experimental limitations and extract useful information related to the dynamics of the flow. Numerous CFD investigations based on Volume-of-Fluid (VOF) and Level-Set methods have been conducted to determine the role of operating conditions on the drop formation process in surfactant-free systems. These studies have revealed the interplay between capillarity, inertia, and viscosity for different microfluidic junction geometries and their effects on drop formation [START_REF] Li | Study on the mechanism of droplet formation in t-junction microchannel[END_REF][START_REF] Lan | Cfd simulation of droplet formation in microchannels by a modified level set method[END_REF][START_REF] Soh | Numerical investigation on the velocity fields during droplet formation in a microfluidic t-junction[END_REF][START_REF] Ngo | A numerical study on the dynamics of droplet formation in a microfluidic double t-junction[END_REF][START_REF] Kahouadji | Simulation of immiscible liquid-liquid flows in complex microchannel geometries using a front-tracking scheme[END_REF][START_REF] Filimonov | Toward computationally effective modeling and simulation of droplet formation in microchannel junctions[END_REF][START_REF] Van Der Graaf | Lattice boltzmann simulations of droplet formation in a t-shaped microchannel[END_REF][START_REF] Zhao | Simulation of liquid mixing inside micro-droplets by a lattice boltzmann method[END_REF]. Recent studies in the presence of surfactants have provided a detailed description of the heterogeneous distribution of surfactant concentration at the interface and the effect on drop formation time and drop size. [START_REF] Riaud | Numerical study of surfactant dynamics during emulsification in a t-junction microchannel[END_REF] performed numerical simulations inside a T-junction microchannel for a Span80-octane/water mixture and compared the results against µPIV experiments. The authors found that the interfacial distribution of surfactants on growing droplets is not uniform but surfactants tend to concentrate at the droplet tip. They also reported that surfactant adsorption is faster than drop formation at low velocities and low Ca. As Ca is increased, either by lowering the interfacial tension or by increasing the inlet velocity, the characteristic drop formation time is reduced and thus adsorption becomes slower than formation time. The authors showed that droplet volume increases with increasing dispersed phase flowrate and decreasing Ca; both parameters are therefore necessary to predict the drop size. [START_REF] Jin | The detachment of a viscous drop in a viscous solution in the presence of a soluble surfactant[END_REF] studied the effect of soluble surfactants on drop necking and used a nonlinear model to relate interfacial tension to surfactant concentration at the interface, by accounting for maximum surfactant packing at the interface. The analysis revealed that a family of necks formed, as a result of a primary and a secondary thinning rate of the neck, depending on the local accumulation of surfactants, and demonstrated that drop detachment behaviour can be used to quantify surfactant dynamics. [START_REF] Antonopoulou | Effect of surfactants on jet break-up in drop-on-demand inkjet printing[END_REF] performed experimental and numerical studies on the flow behaviour during inkjet printing and explained the effect different surfactants have on pinch-off during jetting.

This paper combines unique experimental and numerical approaches to investigate surfactant transport in the bulk, distribution at the interface and Marangoni phenomena during the fast drop formation process in a microfluidic channel. A non-ionic (CMC T X100 = 3.5mM) surfactant is used in the dispersed aqueous phase at concentrations below and above the CMC. A novel high-speed µPIV system is able to follow all stages of the fast drop formation process, which is a key improvement compared to previous works [START_REF] Riaud | Numerical study of surfactant dynamics during emulsification in a t-junction microchannel[END_REF][START_REF] Roumpea | Experimental studies on droplet formation in a flow-focusing microchannel in the presence of surfactants[END_REF]. The CFD simulations include the exchange of surfactant between the bulk of the drop and the interface, as opposed to previous works that considered only transport at the interface [START_REF] Antonopoulou | Effect of surfactants on jet break-up in drop-on-demand inkjet printing[END_REF][START_REF] Riaud | Numerical study of surfactant dynamics during emulsification in a t-junction microchannel[END_REF]. The effect of surfactant on the vortical structures and velocity fields are discussed for the different stages of drop formation. The CFD simulations provide information on surfactant concentrations at the interface that cannot be obtained experimentally. In addition, they reveal the vortical structures very close to the interface where measurements are not accurate because of the low concentration of the PIV tracer particles in this region (i.e. at the neck region just before pinch-off). They also enable investigations of the drop in three dimensions, which are very challenging to obtain experimentally. The experiments on the other hand provide data on the effect of the surfactant at high concentrations above the CMC, which is beyond the capabilities of the current model. The Marangoni stresses at the interface are calculated and presented as a function of time and space for the first time during the whole drop formation process.

Experimental Methods

Droplet generation

The experiments were performed in a glass flow-focusing microchannel from Dolomite Microfluidics (Part No. 3000436) already used in previous works [START_REF] Roumpea | Experimental studies on droplet formation in a flow-focusing microchannel in the presence of surfactants[END_REF][START_REF] Kovalchuk | Effect of surfactant on emulsification in microchannels[END_REF]. At the cross-junction the inlet dimensions are 195 µm × 190 µm (width × depth) and the dimensions of the main channel are 390 µm × 190 µm (width × depth). The continuous phase was introduced via the side channels and then the aqueous phase via the central channel of the junction (see figure 1a). For all configurations, silicone oil (Clearco, density: ρ c = 920 kg m -3 , viscosity: µ c = 4.6 mPa s at 20 • C) was used as the continuous phase and a mixture of 52% w/w glycerol and 48% w/w water (Sigma Aldrich, ⩾99.5%, density: ρ d = 1132 kg m -3 , viscosity µ d = 6.8 mPa s at 20 • C) with and without surfactant was selected as the dispersed phase. Based on the work of Kalli et al. (2022), the flowrates were selected to form drops in the dripping regime. Using syringe pumps (KDS Scientific, ± 5 ×10 -9 mL min -1 ), the continuous phase was introduced with a total flowrate Q c = 0.12 mL min -1 , while the dispersed phase had a flowrate of Q d = 0.01, 0.02 mL min -1 . To observe the impact of interfacial tension on the drop formation, several concentrations of the non-ionic surfactant Triton X-100 (TX100), (Acros organics, M w = 646.85 g/mol, ⩾95%) were dissolved in the dispersed phase at concentrations of (0.1, 0.2, 0.6, 1.0, 1.4, 2.1, 2.9, 4.3, 5.7, 8.6) × CMC. Figure 1(b) shows the equilibrium interfacial tension values (σ), normalised by the final equilibrium interfacial tension value at CMC = 3.5mM (σ Γ =Γ∞ = 2.9 mN/m) as measured using a Du Noüy ring attached to a Force K100 Tensiometer (Krüss GmbH).

Optical techniques

The experimental setup is shown in figure 2. Two different illumination modes were used for the micro Particle Image Velocimetry (µPIV) or the High-Speed Imaging (HSI). To allow full optical access for all measurements and avoid optical distortions, in particular close to the channel wall, both phases were selected to match the refractive index of the glass microchannel (n i = 1.39). For both modes, the images were taken with a 12-bit high-speed camera (Phantom v1212, 1280 × 800 pixels resolution). In the HSI experiments, the acquisition frequency was 10 000 Hz and the camera was equipped with a Nivatar ×20 zoom lens and an LED backlight for illumination. Results were averaged for at least 15 drops (drop size polydispersity ⩽ 1.2%). The experimental error is 3 µm per pixel for the spatial resolution and 0.1 ms for the time resolution. The high-speed µPIV experiments were performed by switching modes from the LED light (2) to the laser source (1) as shown in figure 2. The laser illumination was generated with a high-speed 60W pulsed Nd-Yag (Litron), which allows the observation in time of the droplet generation. The dispersed phase was seeded with 1 µm carboxylate-modified microspheres FluoSpheres, with an absorbance of 542 nm and an emission of 612 nm. For the PIV image pairs, the camera was equipped with a ×20 microscope lens (Mitutoyo Ltd) and focal depth of 1.6 µm. The lens was mounted with a high-pass filter (> 565 nm) to eliminate reflections from the channel walls. Moreover, to be able to follow the interface during the PIV measurements, a small amount of rhodamine B (with absorbance of 510 nm and emission of 610 nm) was added in the dispersed phase (0.05 ppm). The highspeed µPIV allows measurements in the channel centre during drop formation, which follow the whole process from the expansion t = t exp to the pinch-off stage t = t pin . For each measurement, 2 000 image pairs were acquired at a frequency of 3 000 Hz which represent ∼20 drop formation cycles. The measurements were performed in the entire channel width (y/l c ∈ [-1, 1], with y the transverse direction) and around the droplet emergence area (x/l c ∈ [-2, 2], with x the streamwise direction). The local coordinate system is defined in figure 1(a) where (x, y) = (0, 0) is the center of the cross junction. All velocity fields were computed with the free PIV software CPIV-IMFT developed at IMFT. It is based on an FFT cross-correlation method with peak-locking reduction schemes and parallelisation. In the processing, 16 × 16 pixels interrogation boxes and 50% overlap were used, yielding a grid resolution of 1.4 µm.

Numerical formulation, scaling, and validation

Numerical formulation and problem statement

As mentioned in §2, the microfluidic channel used in this study consists of a combination of both cross-junction and flow-focusing devices. Its computational representation and dimensions are illustrated in figure 1(c). The four branches of this device are identical with an oval cross-section shape of length l c = 390 µm and width w c = 190 µm. This cross-section is comprised of a rectangular piece of dimensions w c × (l c -w c ) connected to two semicircles of diameter w c . Figure 1(c) also shows the three-dimensional computational domain and its decomposition into 240 = 30 × 8 × 1 sub-domains used for the Message Passing Interface (MPI) computing protocol. The numerical construction of this device follows a similar approach as that used in [START_REF] Kahouadji | Simulation of immiscible liquid-liquid flows in complex microchannel geometries using a front-tracking scheme[END_REF]. This approach bypasses the typical obstacles in construction and meshing that arise when handling complex geometries by employing a static distance function for several primitive objects (e.g., cylinders, planes, and tori), and simple algebraic operations between them, such as "union" and "intersection".

The flow is considered incompressible and both the continuous and dispersed phases are assumed to be Newtonian. The following scaling is adopted to non-dimensionalise the equations governing the flow dynamics coupled to the surfactant transport in the bulk of the dispersed phase and at the interface:

x = x l c , ũ = u U sc , t = t l c /U sc , p = p µ c U sc /l c , σ = σ σ s , Γ = Γ Γ ∞ , C = C CM C , Cs = C s CM C . (3.1)
Here, the tildes designate dimensionless quantities, and x, t, u, and p are the spatial coordinate, time, velocity, and pressure, respectively; the interface surfactant concentration is represented by Γ , C and C s are the concentrations at the bulk and the bulk sub-phase, located immediately adjacent to the interface, respectively [START_REF] Shin | A hybrid interface tracking -level set technique for multiphase flow with soluble surfactant[END_REF], and σ is the local interfacial tension varying as function of Γ . The length scale l c , the average surface velocity of the continuous phase in the main portion of the channel,

U sc = Q c /A (A = l c w c + (π/4 -1) w 2 c
), the interfacial tension in a surfactant-free system, σ s , the saturation interfacial concentration, Γ ∞ , and the CM C, are used as the characteristic length, velocity, interfacial tension, and interfacial and bulk surfactant concentration scales, respectively; ρ c and µ c are the continuous phase density and dynamical viscosity, respectively. The dimensionless governing equations for the flow and surfactant transport are formulated in Eqs. (3.2)-(3.7):

∇ • ũ = 0, ρRe ∂ ũ ∂ t + ũ • ∇ũ = -∇p + ∇ • μ(∇ũ + ∇ũ T ) + 1 Ca à (σκn + ∇ s σ) δ x -xf d Ã, (3.2) ρ (x, t) = ρ d ρ c + 1 - ρ d ρ c H (x, t) , μ (x, t) = µ d µ c + 1 - µ c µ d H (x, t) , (3.3) ∂ Γ ∂ t + ∇ s • ( Γ ũt ) = 1 P e s ∇ 2 s Γ + Bi k Cs (1 -Γ ) -Γ , (3.4) ∂ C ∂ t + ũ • ∇ C = 1 P e c ∇ • (∇ C), (3.5) n • ∇ C| interf ace = -P e c DaBi k Cs (1 -Γ ) -Γ , (3.6) σ = max ϵ σ , 1 + β s ln 1 -Γ (3.7)
with ϵ σ set to 0.05. The dimensionless numbers shown in the above equations are defined by

Re = ρ c U sc l c µ c ; Ca = µ c U sc σ s ; P e c = U sc l c D c ; P e s = U sc l c D s ; Bi = k d l c U sc ; Da = Γ ∞ l c CM C ; k = k a CM C k d , (3.8)
where Ca and Re are the liquid capillary and Reynolds numbers, respectively; in which µ c is the continuous phase viscosity; P e c and P e s are the bulk and interfacial Peclet numbers that represent the interplay between convective and diffusive forces for the surfactant species at the bulk and interface, respectively, in which D c and D s are the (constant) surfactant diffusion coefficients in the bulk continuous phase and the plane of the interface, respectively. The Biot number, Bi, represents the competition between the time-scales of desorption and convection, characterised by k -1 d and l c /U sc , respectively. Finally, k represents the competition between adsorption and desorption time-scales, characterised by (k a CM C) -1 and k -1 d , respectively. Lastly, the Damkohler number, Da, provides a dimensionless measure of interfacial saturation with surfactant.

The continuity and momentum equations (Eq. (3.2)) are written in a three-dimensional Cartesian domain using a single-fluid formulation. H x, t represents a smoothed Heaviside function that takes the value of zero in the continuous phase and unity in the dispersed phase. The normal and tangential components of the interfacial tension force are expressed by the last two terms on the right-hand-side (RHS) of Eq. (3.2), respectively. The latter term representing the interfacial tension gradients across the interface, which give rise to Marangoni stresses; κ corresponds to the interface curvature and δ(x -xf ) is the three-dimensional Dirac delta function equal to unity at the interface (x = xf ) and zero elsewhere. In Eq. (3.4), ∇ s = (I -nn) • ∇ is the surface gradient operator, n is the outward-pointing unit normal to the interface, and I is the identity tensor; ũt = (ũ s • t) t corresponds to the tangential velocity, where ũs is the surface velocity, and t is the unit vector tangent to the interface. The last terms on the RHS of Eq. (3.4) and Eq. (3.7) represent the source term for surfactant exchange between the interface and the section of the bulk immediately adjacent to the interface.

As seen from figure 1(b) and Eq. (3.7), the dependence of σ on Γ is described by a nonlinear Langmuir equation of state in which β s = ℜT Γ ∞ /σ s modulates this dependency, as suggested by [START_REF] Muradoglu | Simulations of soluble surfactants in 3d multiphase flow[END_REF]; ℜ and T are the ideal gas constant and the temperature, respectively. The components of the interfacial tension force that induce Marangoni stresses can be written in terms of Γ by

1 Ca ∇ s σ • t ≡ τ Ca = -M a 1 (1 -Γ ) ∇ s Γ • t, (3.9)
where τ is the dimensionless Marangoni stress and M a ≡ β s /Ca represents a Marangoni parameter.

From the properties of TX100 and the operating conditions of the system, the relevant dimensionless numbers that characterise the system are given by Re = 2.35, Ca = 4.33 × 10 -3 , P e c = P e d = 2.35 × 10 5 , Bi = 8.41 × 10 -5 , Da = 1.04 × 10 -3 , k = 5.55 × 10 3 , β s = 0.10, and M a = 23.08. We define a capillary time-scale as τ cap = ρ c l 3 c /σ s = 1.31×10 -3 s, a velocity time-scale as τ vel = σ s /(ρ c l c ) = 2.99 × 10 -1 s, and a Marangoni time-scale as τ mar = µ c l c /∆σ = 6.16 × 10 -5 s. Here, it has been assumed that D c = D s = D, and Γ ∞ and K L = k/CM C were found by fitting the experimental data in figure 1(b) to the Langmuir-Szyszkowski equation [START_REF] Teipel | Adsorption behavior of nonionic surfactants studied by drop volume technique[END_REF], as in previous work (Kalli et al. 2022). Γ was calculated using the Langmuir equation of state with the measured experimental values of σ for each C (see Eq.(3.7) and plotted in a dimensionless form 1. Finally, the diffusion coefficient, D, was estimated using the Wilke-Chang correlation [START_REF] Wilke | Correlation of diffusion coefficients in dilute solutions[END_REF].

Fully three-dimensional direct numerical simulations were performed within the context of the Level Contour Reconstruction Method (LCRM) for the interface advection, as previously detailed in [START_REF] Shin | Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity[END_REF] and [START_REF] Shin | Accurate representation of surface tension using the level contour reconstruction method[END_REF][START_REF] Shin | A solver for massively parallel direct numerical simulation of three-dimensional multiphase flows[END_REF][START_REF] Shin | A hybrid interface tracking -level set technique for multiphase flow with soluble surfactant[END_REF]. This method handles the interface and the forces arising from surface tension through a hybrid fronttracking/level-set technique in conjunction with surfactant transport at the interface and bulk. At the inlet branches, an analytical solution following a Poiseuille profile is set for the velocity, together with a Neumann condition for the pressure (∂p/∂n = 0). The full derivation of this velocity profile (Eq.(3.10) in dimensionless form) in the cross-section shown in figure 1(c) (right) can be found in §A. At the outlet, a Neumann condition is specified for both velocity and pressure. The walls are treated as no-slip boundaries. For the surfactant-laden cases, Neumann conditions are set for surfactant concentration on all boundaries, with the exception of the disperse phase inlet. For this inlet, C is specified as a constant value according to the experimental setup ( C = 0.06 and C = 0.21), and Γ as calculated in figure 1

(b). ũ(x, ỹ) =          Ã (ỹ -ỹ0 ) 2 R2 -1 if |x -x0 | ⩽ L Ã (x -x0 -L) 2 R2 + (ỹ -ỹ0 ) 2 R2 -1 otherwise, (3.10) where A = -Q 8 3 RL+ π 2 R 2 , Q
represents the flow rate of the phase on each inlet (continuous for the two lateral branches and disperse for the central branch), x 0 and y 0 are the coordinates of the centre points of the inlet cross sections, and R = w c /2 and L = (l c -w c )/2, as highlighted in figure 1(c). Each variable has been non-dimensionalised according to the scaling parameters described previously, with à = -QUsc

8 3 R L+ π 2 R2 .

Validation

The numerical simulation in an identical geometry was validated experimentally with a surfactant-free system in previous work [START_REF] Kahouadji | Simulation of immiscible liquid-liquid flows in complex microchannel geometries using a front-tracking scheme[END_REF]. Further validation with experimental results for surfactant-laden cases is made in this section. As seen in previous work (Kalli et al., 2022), a semi-empirical model can be developed to relate the dimensionless drop diameter to the Capillary number and thus the interfacial tension, with the latter being proportional to the drop size in a microfluidic system. As a result, the addition of surfactant is expected to result in smaller drops with shorter formation times, due to the lower interfacial tension. Figure 3(a) shows the effect of surfactant addition on the drop diameter for Q c = 0.12 mL/min and 0.01 mL/min ⩽ Q d ⩽ 0.02 mL/min. As C increases from 0 to 3, the dimensionless drop diameter d/l c decreases for both flowrates, due to the lower interfacial tension. Larger drops are observed at higher Q d as expected.

The effect of surfactant addition on the drop formation time for Q c = 0.12 mL/min and 0.01 mL/min ⩽ Q d ⩽ 0.02 mL/min is shown in figure 3(b). As C increases from 0 to 8.6, the dimensionless formation time t decreases for both flowrates, due to lower interfacial tension. Shorter formation times are observed at larger Q d , as opposed to drop diameter. The above results are compared with the numerical simulations and good agreement is found, with an average error of 0.5% for d/l c and 8.6% for t. The error between the high-speed imaging (HSI) and the µPIV experiments was below 1%.

Results and discussion

The vortical structures and the velocity fields obtained from the CFD simulations and the high-speed µPIV experiments in surfactant-free and surfactant-laden systems, as well as the concentration profiles in the bulk and at the interface are discussed here, for all stages of the drop formation process. The results for the surfactant-free case will be discussed first to show the spatio-temporal evolution of the interface, as well as the vortical structures during each stage. In addition the surfactant concentrations in the bulk and at the interface for the various stages of drop formation will be discussed based on the CFD simulations.

Surfactant-free case

The spatio-temporal evolution of the interface during the complete drop formation process was first simulated for the surfactant-free case. Figure 4 shows the changes in the shape of the interface during expansion (a), necking (b), and pinch-off (c) for Q d = 0.02 mL/min and Q c = 0.12 mL/min. A different output time-step is used for each stage equal to 1 ms, 0.5 ms, and 0.1 ms, respectively, with the longest one used for the expansion stage and the shortest for pinch-off. Specifically, the evolution of the interface is slow during expansion, becomes moderately rapid during necking, and very rapid just before pinchoff. In previous work, expansion times were also found to be longer compared to necking ones, for both surfactant-free and surfactant-laden cases in the dripping regime (Kalli & Angeli 2022). A similar trend was seen by [START_REF] Kahouadji | Simulation of immiscible liquid-liquid flows in complex microchannel geometries using a front-tracking scheme[END_REF] for pancake-shaped drops.

The vortical structures inside the drop are shown in figure 4(d) for the expansion and necking stages and in figure 4(e) for pinch-off. The velocity fields from the experiments with and without surfactant are shown in figure 5, which agree very well with the simulation results. Initially for the surfactant-free case (at C = 0 and t = 0.10, figure 4d), two vortical structures are seen close to the front of the drop which grow larger and move closer to the interface later on (at t = 0.50). This is in agreement with the experimental results, where large internal circulation patterns are observed in expansion at t = 0.10 and t = 0.50 (figures 5a-b respectively). The circulation pattern is more pronounced in the second stage of expansion ( t = 0.50, figure 5b), where two circulation zones are clearly seen, one on each side of the growing drop and high velocities in the middle, reaching a maximum of |U max |/U sc = 0.45. From the shape of the interface (black dotted line) it is clear that originally the drop grows in the radial direction, as the continuous phase obstructs the flow of the dispersed phase. The high velocities on the sides of the dispersed phase during expansion suggest that the liquid is mainly transported through the sides of the drop. This is in agreement with observations reported previously by several authors [START_REF] Pirbodaghi | Investigating the fluid dynamics of rapid processes within microfluidic devices using brightfield microscopy[END_REF][START_REF] Roumpea | Experimental studies on droplet formation in a flow-focusing microchannel in the presence of surfactants[END_REF]. Previous works using Ghost Particle Velocimetry in rectangular flow-focusing microchannels reported a stagnation point at the base of the growing drop, during the second stage of expansion, but no significant circulation in the surfactant-free case. Even though a stagnation point can be seen in this work during the second stage of expansion (Figure 5b), the presence of the strong circulation in this study can be attributed to the differences in microchannel geometries (oval in the present study and rectangular in the study of [START_REF] Kiratzis | Effect of surfactant addition and viscosity of the continuous phase on flow fields and kinetics of drop formation in a flow-focusing microfluidic device[END_REF]) and higher flowrates used. The predominant reason for this difference is due to the presence of corner flow in the gutters of the rectangular channel, which is absent in this work with an oval flow-focusing geometry. This can also reverse the direction of the flow fields after the drop has formed [START_REF] Kovalchuk | Effect of surfactant dynamics on flow patterns inside drops moving in rectangular microfluidic channels[END_REF][START_REF] Mießner | µpiv measurement of the 3d velocity distribution of taylor droplets moving in a square horizontal channel[END_REF].

As the drop grows, it obstructs the inlet of the oil phase, which now pushes the interface perpendicular to the drop and changes its curvature. This marks the beginning of the necking stage, with the neck defined as the bridge between the growing drop and the dispersed phase remaining attached to the inlet. The curvature of the interface becomes negative and the pressure build-up from the continuous phase causes the flow to be higher in the neck compared to the rest of the dispersed phase (with |U max |/U sc = 0.25), while the velocities near the interface decrease. In addition, the vortical structures decrease in size ( t = 0.67 in figures 4d and 5c). This indicates that the transport of the dispersed phase from the sides is significantly weaker than in the expansion stage. As the process approaches pinch-off (see figure 4e, left panel), large vortical structures form in the region of the dispersed phase that remains attached to the inlet. Immediately after the pinchoff, the vortical structures grow large in the middle of the cross-junction (figure 4e, right panel). This vortex dipole just before and at pinch-off was also observed during the formation of surfactant-free plugs [START_REF] Kahouadji | Simulation of immiscible liquid-liquid flows in complex microchannel geometries using a front-tracking scheme[END_REF]. Another agreement between the two works is the stagnation point present just at the location of pinch-off. A small satellite drop is also seen at the point where the neck broke. The distance between four subsequent drops is equal, as shown in figure 4(f), indicating that the drop formation process is periodic with exactly the same formation time of 13.5 ms in each cycle.

Surfactant-laden cases

Expansion and necking stages

Upon addition of the TX100 surfactant at C = 0.06, the velocity fields change significantly from their surfactant-free counterparts throughout all stages of the drop formation process. At the start (see figure 5d at t = 0.08), the average velocities appear higher and more uniform at the back of the forming drop and a less intense circulation pattern can be seen in the centre. The two circulation zones still appear in the second phase of the expansion stage (see figure 5e at t = 0.33), however, their velocities are lower than in the surfactant-free case. During the necking stage at TX100 concentrations below CMC (figure 5f), higher velocities are observed near the neck, with maximum values of |U max |/U sc = 0.45. The reduced interfacial tension due to the presence of surfactant accelerates the neck thinning and results in high velocities (figure 5f at t = 0.41), as also observed by [START_REF] Kiratzis | Effect of surfactant addition and viscosity of the continuous phase on flow fields and kinetics of drop formation in a flow-focusing microfluidic device[END_REF]. The circulation pattern completely disappears during necking at low TX100 concentrations, which agreed with the CFD simulations. For concentrations above the CMC, during the first expansion phase (see figure 5g at t = 0.05), velocities are high inside the whole of the growing drop, reaching a maximum As discussed by several authors, high dispersed phase flowrates can enhance surfactant mass transfer possibly due to increased convection, especially in the TX100 case which is convective mass transfer controlled at low concentrations [START_REF] Wang | Determination of dynamic interfacial tension and its effect on droplet formation in the t-shaped microdispersion process[END_REF][START_REF] Wang | Mass-transfer-controlled dynamic interfacial tension in microfluidic emulsification processes[END_REF]. In this work, high velocities near the sides of the forming drops are observed at Q d = 0.02 mL/min in both the simulations and experiments, which can enhance surfactant mass transport to the interface, as opposed to Q d = 0.01 mL/min. This could explain the lower dynamic interfacial tension measured at Q d = 0.02 mL/min compared to the one in Q d = 0.01 mL/min in our previous work (Kalli et al. 2022), even though drops formed faster and surfactant had less time to transfer to the interface (see figure 3b).

Pinch-off

Pinch-off is further discussed in this section, using both the numerical and experimental results. The effect of TX100 surfactant on drop shape and velocity fields at pinch-off from HSI and µPIV experiments, as well as CFD simulations at Q d = 0.02 mL/min as shown in figure 6. As can be seen in figures 6(a), (d) and (g), smaller droplets are formed upon TX100 addition and satellite drops are seen after pinch-off. It should be noted that the shadow from the edge of the channel observed when using HSI does not exist in the µPIV images, which can mislead the comparison of the droplet sizes in figures 6(c and f). In the case of no surfactant present, high velocities at the rear of the formed drop and circulation patterns in the dispersed phase remaining attached to inlet are seen in figure 6(b-c) (see also figure 4). The higher velocities at the rear of the drop as opposed to its front are also observed in the surfactant-free simulations of [START_REF] Zhao | Simulation of liquid mixing inside micro-droplets by a lattice boltzmann method[END_REF]. Similar observations were reported from CFD simulations of mass transfer in a flow-focusing microchannel by [START_REF] Lan | Cfd simulation of droplet formation in microchannels by a modified level set method[END_REF].

When surfactant is added at concentrations below CMC, smaller droplets form due to the lower interfacial tension and a weaker circulation pattern is seen at the remaining dispersed phase. The velocities inside the formed droplet are higher compared to the surfactant-free case (figure 6e-f at C = 0.06). A stagnation point is also observed in the CFD output just at pinch-off for both surfactant-free and surfactant-laden cases. At concentrations above the CMC, the experimental µPIV results show that the circulation pattern completely disappears and uniform velocity fields are observed in the formed droplet, with |U max |/U sc = 0.5 (figure 6h at C = 2.89). As was discussed above, the model does not account for micelle formation, and no numerical results are shown for concentrations above CMC.

Concentration distribution

Figure 7 shows the relative bulk concentration of the TX100 surfactant (a), C, the vortical structures (b), and the interfacial concentration (c), ( Γ ), during the drop formation process at concentrations below CMC ( C = 0.06). The concentration profile of the surfactant in the bulk follows the circulation patterns as seen in the expansion stage ( t = 0.39, top of figures 7a-b). During the necking stage, the circulation is weaker and the surfactant transfers downstream, following the drop growth and into the main channel ( t = 0.78). Just before pinch-off, the two circulation zones in the drop disappear and the concentration of surfactant is high at the rear of the drop ( t = 0.98). Large circulation patterns appear at the front of the phase remaining attached to the inlet, which grow larger after drop detachment ( t = 0.99). Interestingly, the small satellite drop appears to have no surfactant. As also observed in the µPIV experiments, the circulation zones in all stages are smaller compared to the surfactant-free case. This is possibly due to Marangoni forces acting in the opposite direction to the circulation when surfactant is present.

As previously discussed, information on surfactant concentration at the interface is paramount in understanding the effect of TX100 on the drop formation process. The spatio-temporal evolution of the relative interfacial concentration ( Γ ) is shown in figure 7(c). As can be seen the interfacial concentration of the surfactant is high (and equal to the equilibrium value) at the left part of the channel, and before the cross-junction where the interface forms. At the interface initially there is very little surfactant concentration, apart from the tip of the drop. Even at the tip, however, the concentration is below the equilibrium value. This concentration profile remains at all stages of the formation process, including expansion (top), necking (middle) and pinchoff (bottom). The difference in surfactant concentration along the interface will cause Marangoni stresses that can affect the drop formation time and the velocity fields inside the drop; this will be discussed further in §4.2.4.

Marangoni stresses

Marangoni stresses are usually discussed in works for drop-coalescence [START_REF] Constante-Amores | Role of surfactant-induced marangoni stresses in drop-interface coalescence[END_REF][START_REF] Dong | Laser induced fluorescence studies on the distribution of surfactants during drop/interface coalescence[END_REF], rising bubbles [START_REF] Bastani | Cfd simulation of bubble in flow field: investigation of dynamic interfacial behaviour in presence of surfactant molecules[END_REF], liquid bridges [START_REF] Kovalchuk | Kinetics of liquid bridges and formation of satellite droplets: Difference between micellar and bi-layer forming solutions[END_REF][START_REF] Kamat | Role of marangoni stress during breakup of surfactant-covered liquid threads: reduced rates of thinning and microthread cascades[END_REF]) and jet formation [START_REF] Antonopoulou | Effect of surfactants on jet break-up in drop-on-demand inkjet printing[END_REF]), but less frequently during the fast process of drop formation. In this section, Marangoni stresses at the interface will be calculated and presented as a function of time and space for the first time during the whole drop formation process. Additionally, the competition between Marangoni stresses and the interfacial tension reduction caused by the surfactant absorption will be explored in all dimensions of the forming drop.

Figure 8 shows the shape of the interface and the dimensionless interfacial concentration of surfactant during drop formation, with the magnitude of Γ shown in colour in (a) and (b) for top-(x-y plane) and side-(x-z plane) views, respectively. The corresponding local interfacial tension (σ/σ s ) and the Marangoni stress (∇ (σ/σ s )•t) are shown in figures 8(c-d) and (e-f), respectively. The 3 profiles represent the expansion stage at t = 0.39, the initial part of the necking stage at t = 0.78 and the final part of necking before pinch-off at t = 0.98, for Q d = 0.02 mL/min and Q c = 0.12 mL/min. The concentration gradient at the interface decreases as the drop grows, with the concentration at the drop tip reaching a maximum of Γ = 0.58 at t = 0.39, which decreases to Γ = 0.48 at t = 0.78 and Γ = 0.35 at t = 0.98 (as also seen in figure 7(c)). This causes a weaker interfacial tension gradient for the top-view as observed from the slope in figure 8(c) which decreases as the drop forms. The Marangoni stresses on the interface are small and in the opposite direction to the flow, as seen from the negative peaks in figure 8(e) with ∇ (σ/σ s ) = -2.4, -0.7 and -0.3 at t = 0.39, 0.78 and 0.98 respectively (0.4 < x/l c < 1.2). The positive peak in the direction of the forming drop, around x/l c = 0 in the range of 0.0 < x/l c < 0.2 with ∇ (σ/σ s ) • t ∼ 1.5 exists for all three stages, but is far from the neck region to contribute to pinch-off.

The flow of the continuous phase from both the top and bottom inlets (Q c = 0.12 mL/min) is high enough to sweep surfactant molecules away from the neck towards both sides of the neck and results in an essentially surfactant-free interface at the neck region. The depletion of surfactant in the neck area between 0.2 < x/l c < 0.9 (as also seen in figure 7c) and the flat concentration profile of Γ show that there are no Marangoni stresses to affect the process of pinch-off. The shorter drop formation times observed when surfactants are present are caused by the lower interfacial tension compared to the surfactant-free case. The effect of this dominates over Marangoni stresses as the high continuous flowrates cleans the interface during the whole formation process (see Figure 8. The maximum in Γ at all times close to the drop tip causes local Marangoni stresses in that area whose strength decreases with time. These stresses slow down the velocity inside the drop and weaken the circulation intensity (as discussed in §4.2) rather than delay pinch-off. This is opposite to the case of jet formation where Marangoni stresses delay break-up and result in longer times to pinch-off (Kalli & Angeli 2022).

The influence of the continuous phase flowrate on the surfactant distribution at the interface is obvious when the top-and side-views are compared. The surfactant concentration distribution at the interface is less uniform in the side-than in the topview, while the neck area has some surfactant, apart from the last part of the necking stage (Figure 8b). This results in a lower interfacial tension overall, which will contribute to the decrease in drop formation time. Marangoni stresses now appear with peaks close to the neck region between 0.1 < x/l c < 0.3 at t = 0.78 and 0.1 < x/l c < 0.2 at t = 0.98. Nonetheless their magnitude is small (∇ (σ/σ s ) • t < 1) and are not close to the thinning part of the neck (0.4 < x/l c < 0.6 at t = 0.98) so should not to contribute to pinch-off. Antonopoulou et al. (2021) compared experimental results with numerical simulations of a weak (β s = 0.1, low CMC, identical to this study) and a strong (β s = 1, high CMC) surfactant during jet formation and break-up. It was found that the distribution of the weak surfactant at the interface was less uniform as opposed to the strong surfactant. Even though surfactant accumulation near the neck follows the opposite trend between the formation of drops (low Γ at drop pinch-off) and jets (high Γ at jet break-up), there is agreement on the fact that surfactant concentration gradients and thus Marangoni stresses decrease with time during the formation process, as seen in this work.

(c) (d) 0 0.2 0.4 0.6 0.8 1 1.2 -3 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 1.2 -3 -2 -1 0 1 2 (e) (f)

Fully formed drops

In order to investigate surfactant transport in the whole drop volume, the vortical structures and surfactant concentrations of the formed drops as obtained by the CFD simulations are shown in figure 9 for surfactant-laden (c-d) cases after pinch-off (at the end of the channel, 10 × d). The vortical structures for surfactant-free drops are shown in figures 8 (a-b) for comparison. The difference in the shape of the formed drop for top-(a),(c) and side-(b),(d) views is due to the channel dimensions, as explained in §2 and §3. In the surfactant-free case, the vortical structures are completely symmetrical from the top-view, while a strong vortex dipole is seen from the side-view near the rear boundaries of the drop. The opposite is seen in the surfactant-laden drop; a strong vortex dipole is forming in the centre of the top-view, while no circulation is seen in the side-view.

Both bulk and interfacial concentrations of the formed drops are plotted in figure 9, in their dimensionless form as C and Γ , respectively. For C, the surfactant concentration is high in the centre and low around the periphery of the drop in both the top-and side-views. The variations in Γ are small, as can be seen from the range of the scale in the figure . Here it should be noted that adsorption is significantly faster than desorption, as also expected from the large K L = 1585.00 m 3 /mol value. As mentioned in §3, Bi = 8.41 × 10 -5 is very small, which can be considered negligible when compared to k = 5.55 × 10 3 , denoting negligible desorption as opposed to adsorption of surfactant at the interface. When compared to the concentrations during drop formation, it is obvious that the surfactant absorbs very fast after pinch-off. As seen in figure 7, the bulk concentration reaches the maximum value inside the drop throughout all stages ( C = 0.06) and the interface is not saturated ( Γ < 1). This is not the case for fully formed drops, where the bulk concentration is lower than the maximum value. However, there are some differences in the interfacial concentration of the surfactant. From the top-view, Γ is seen to be higher at the rear of the drop. On the other hand, the interfacial concentration of the surfactant is larger near the rear edges of the drop from side-view. This is due to advection of surfactant at the back of the drop as it travels through the channel.

Conclusions

This paper presents a thorough study of the formation process of surfactant-laden drops inside a flow-focusing microchannel. High-speed imaging experiments (HSI) and novel micro-Particle Image Velocimetry (µPIV) experiments were used in conjunction with numerical CFD simulations, to obtain velocity and surfactant concentration profiles and understand the effects of surfactants on the different stages of the drop formation process. Surfactant concentrations below and above the critical micelle concentration (CMC) were considered. The numerical simulation utilised a three-dimensional multiphase solver which accounted for the transport of soluble surfactants in the bulk and at the interface and was adapted to the context of the hybrid Front-Tracking/Level-set method.

A low viscosity silicone oil (4.6 mPa s) was used as the continuous phase and a mixture of 48% w/w water and 52% w/w glycerol was the dispersed phase. A non-ionic surfactant (CMC T X100 = 3.5mM) was added in the aqueous phase, at several concentrations below and above CMC. Numerical simulations for the drop formation in the microfluidic channel showed good agreement with experiments on drop size, formation time and vortical structures during drop formation.

Results revealed smaller dimensionless drop size (d/l c ) and shorter formation times ( t) at higher surfactant concentrations ( C) in the drop bulk due to the lower interfacial tension forces. During the drop expansion and necking stages, velocities at the back of the forming drop were higher at higher C values while the circulation patterns completely disappeared at concentrations above the CMC. At the pinch-off point, the velocities were higher at the rear of the drop as opposed to the front, while circulation patterns in the dispersed phase remaining attached to the inlet were weaker when surfactant was added compared to the cases with no surfactant. The effect of the dispersed phase flowrate was also investigated and it was found that an increase in flowrate increased circulation intensity in the drop during the expansion stage and thus enhanced surfactant mass transport to the interface.

The surfactant concentrations in the bulk and at the interface of the formed drop were also explored. Marangoni stresses at the drop interface were calculated for all stages of drop formation and found to progressively weaken with formation time. It was concluded that the high continuous phase flowrate plays an important role in sweeping surfactants away from the neck region and contributed towards pinch-off mainly. The Marangoni stresses did not seem to affect the neck pinch-off but contribute mainly to the velocity profile inside the drop, as opposed to what is seen in jet formation and break-up.

In future work, a new experimental setup will be developed using µLIF techniques to trace both time and spatial evolutions of the surfactant concentration at the interface and compare with numerical results. Additionally, the numerical simulations will be expanded to model concentrations above CMC by accounting for micelle formation. Replacing the value of c 1 on Eq.(A 2) and the cartesian version of Eq.(A 5), we obtain the inlet velocity profile for the cross-section considered in this study, illustrated in figure 10.

u(x, y) =        A (y -y 0 ) 2 R 2 -1 if |x -x 0 | ⩽ L A (x -x 0 -L) 2 R 2 + (y -y 0 ) 2 R 2 -1 otherwise, ( A 7)
where

A = -Q 8 3 RL + π 2 R 2
.

Figure 1 .

 1 Figure 1. (a) Numerical configuration, and an illustrative snapshot of a top-view of the interface (top), and the corresponding high-speed image from experiments (bottom); (b) interfacial tension isotherm for TX100 surfactant showing the variation of equilibrium interfacial tension , σ. Black points represent σ (normalised by the surfactant-free value of interfacial tension, σs) with the semi-log variation of the surfactant bulk concentration C (normalised by the CMC). Red points represent the interfacial surfactant concentration, Γ (normalised by its saturation value, Γ∞) obtained using Eq. (3.7). The blue dashed line represents the fitting of σ/σs = 1 + 0.1 ln(1 -Γ/Γ∞); (c) geometrical details of the microfluidic channel decomposed into 240 = 30 × 8 × 1 subdomains. Each subdomain contains a equal and structured mesh resolution of 32 3 , for a total of 960 × 256 × 32 cells.

Figure 2 .

 2 Figure 2. Experimental setup of the HSI and µPIV techniques.

Figure 3 .

 3 Figure 3. Effect of TX100 concentration on dimensionless drop (a) diameter (d/lc) and (b) formation time ( t)).

Figure 4 .

 4 Figure 4. Numerical results for the surfactant-free case generated with Q d = 0.02 mL/min and Qc = 0.12 mL/min. Panels (a)-(c) depict the spatio-temporal evolution of the interface with a time difference between the curves is 1 ms, 0.5 ms, and 0.1 ms, respectively; (d) highlights the vortical structures for three different times that can also be compared with experimental PIV of figures 5(a,b,c) (shown in red lines); (e) shows the development of a neck followed by pinch-off which is accompanied by satellite formation; (f) shows top-views of the periodic drop formation in the channel with a period of 13.5 ms.

Figure 5 .

 5 Figure 5. Effect of TX100 concentration on velocity fields from PIV measurements during expansion and necking stages at Q d = 0.02 mL/min for (a) -(c) C = 0, (d) -(f) C = 0.06 and (g) -(i) C = 2.89.

Figure 6 .

 6 Figure 6. Effect of TX100 concentration on velocity fields from HSI experiments (a), (d) and (g), PIV measurements (b), (e) and (h) and CFD (c), (f) and (i) at the pinch-off point for 0.02 mL/min for (a) -(c) C = 0, (d) -(f) C = 0.06 and (g) -(i) C = 2.89.

Figure 7 .

 7 Figure 7. Spatio-temporal evolution of (a) the relative bulk surfactant concentration C, (b) the vortical structures in the channel reference frame and (c) the relative interfacial surfactant concentration Γ , respectively. The snapshots from top to bottom correspond to times t = 0.39, 0.78, 0.98, and 0.99, respectively, and C = 0.06.

Figure 8 .

 8 Figure 8. Quantitative measures highlighting (a)-(b) the interface shape and the interfacial surfactant distribution with the magnitude of Γ/Γ∞ shown in colour, (c)-(d) the corresponding local interfacial tension σ/σs, and (e)-(f) the Marangoni stress ∇ (σ/σs) • t, for top-and side-views, respectively.

Figure 9 .

 9 Figure 9. Top-and side-views of resulting drop for the surfactant-free case (a-b) and surfactant-laden case (c-d) accompanied with their vortical structures in the moving reference frame. Both cases are for Q d = 0.02 mL/min, Qc = 0.12 mL/min and C = 0.06. These conditions correspond to the non-dimensional parameters: Re = 2.35, Ca = 4.33 × 10 -3 , P ec = P e d = 2.35 × 10 5 , Bi = 8.41 × 10 -5 and k = 5.55 × 10 3 .

Figure 10 .

 10 Figure 10. Normalised inlet velocity profile. The contours are equally spaced with steps of 0.05 and (x0, y0) = (0, 0).

Table 1 .

 1 Physical Properties of TX100 surfactant. the secondary axis of figure 1(b)). Using the literature value of TX100 desorption in water (k d ) (Gassin et al. 2012), k a was calculated using k a = K L k d and shown in Table

	Physical properties	TX100
	Γ∞ (mol/m 2 )	1.42 × 10 -6
	KL	(m 3 /mol)	1585.00
	k d	(1/s)	0.0065
	ka (m 3 /mol s)	10.3
	D	(m 2 /s)	5 × 10 -11

on

† Email address for correspondence: p.angeli@ucl.ac.uk, o.matar@imperial.ac.uk

Acknowledgements. This work is supported by the Engineering and Physical Sciences Research Council, United Kingdom, through the EPSRC MEMPHIS (EP/K003976/1) and PREMIERE (EP/T000414/1) Programme Grants. M.K. would like to acknowledge the EPSRC Doctoral Training Programme (EP/ R513143/1) for her studentship. O.K.M. acknowledges funding from PETRONAS and the Royal Academy of Engineering for a Research Chair in Multiphase Fluid Dynamics. P.P. and L.K. acknowledge HPC facilities provided by the Research Computing Service (RCS) of Imperial College London for the computing time. D.J. and J.C. acknowledge support through HPC/AI computing time at the Institut du Developpement et des Ressources en Informatique Scientifique (IDRIS) of the Centre National de la Recherche Scientifique (CNRS), coordinated by GENCI (Grand Equipement National de Calcul Intensif) Grant 2022 A0122B06721. The numerical simulations were performed with code BLUE (Shin et al. (2017(Shin et al. ( , 2018))) and the visualisations have been generated using ParaView.

We confirm that there is no conflict of interest.

Appendix A. Inlet velocity profile

The cross-section of the three inlets is comprised of two main areas: two semi-circles of radius R = w c /2 and a rectangular portion of dimensions w c × (l c -w c ). The coordinates of centre point of the cross-section are defined as x 0 and y 0 , and the distance in the x-axis from the edge of the rectangular portion to the centre line as L = (l c -w c )/2. We begin with a general parabolic function consistent with a Poiseuille profile for the velocity at the rectangular portion (f 1 (x, y)):

where a 1 , b 1 and c 1 are constants. Applying a no-slip condition at the channel walls (f 1 (x, ±R) = 0) and considering symmetry across the x-axis, we obtain a 1 = 0 and b

We define the parabolic profile for the circular sections in cylindrical coordinates (f 2 (r, θ)):

where x -x 0 = L + sin θ and y -y 0 = r cos θ. Applying a continuity condition at the limit between the rectangular and circular sections, ∂f1 ∂y x-x0=L

, as well as the no-slip condition at the walls,

Considering symmetry across the θ-axis:

These profiles must satisfy the exact entry flow rate of each branch. Therefore, we refer to the definition of flow rate Q in order to find constant c 1


Gautier Meyer

Ralf Schweins

Tristan Youngs

Jean-Franc ¸ois Dufrêche

Isabelle Billard
email: isabelle.billard@grenoble-inp.fr

Marie Plazanet
email: marie.plazanet@univ-grenoble-alpes.fr

How Temperature Rise Can Induce Phase Separation in Aqueous Biphasic Solutions

Ionic liquid-based acidic aqueous biphasic solutions (AcABS) recently offered a breakthrough in the field of metal recycling. The particular mixture of tributyltetradecylphosphonium chloride ([P 4,4,4,14 ]Cl), acid and water presents the unusual characteristic of a Lower Solution Critical Temperature (LCST), leading to phase separation upon a temperature rise of typically a few tens of degrees. We address here the microscopic mechanisms driving the phase separation. Using Small Angle Neutron Scattering, we characterized the spherical micelle formation in the binary ionic liquid/water solution and the micelle aggregation upon addition of acid, due to the screening of electrostatic repulsion. The increase of both acid concentration and/or temperature eventually leads to the micelle flocculation and phase separation. This last step is achieved through chloride ions adsorption at the surface of the micelle. This exothermic adsorption compensates the entropic cost, leading to the counter-intuitive behavior and may be generalised to numbers of molecular systems with LCST.

Recycling is one of the biggest tasks of our present society and a very challenging problem. In particular, metallic wastes evolve at a very high rate, due to the exponential growth of technological objects including all kinds of metals, the mixing of which is completely different from what is traditionally found in ores. For this purpose, liquidliquid extraction (LLE) is an efficient and major way to separate components. The principle is that each chemical, when mixed in the solution, is solvated in a preferential phase of the well chosen two-phase mixture. Recycling should however be performed without the use of polluting solvents, driving the search for green chemistry in this area [START_REF] Ventura | Ionic-Liquid-Mediated Extraction and Separation Processes for Bioactive Compounds: Past, Present, and Future Trends[END_REF][START_REF] Duan | Comprehensive Evaluation of Deep Eutectic Solvents in Extraction of Bioactive Natural Products[END_REF][START_REF] Hilali | Green Extraction of Essential Oils, Polyphenols, and Pectins from Orange Peel Employing Solar Energy: Toward a Zero-Waste Biorefinery[END_REF] . In this context, a recent breakthrough was achieved in using aqueous biphasic solutions [START_REF] Gras | Ionic-Liquid-Based Acidic Aqueous Biphasic Systems for Simultaneous Leaching and Extraction of Metallic Ions[END_REF] . Composed by more than ca. 60% water, ionic liquid (IL) and acid, such a solution (further denoted by AcABS for acidic aqueous biphasic system) avoids the use of carcinogenic, mutagenic and reprotoxic (CMR) solvent while enabling to have large quantities of metallic ions in solution without hydrolysis (depending on ions, for ex. up to 40 g/L for Fe (III) in HCl/H 2 O solutions [START_REF] Sinoimeri | Uncommon biphasic behaviour induced by very high metal ion concentrations in HCl/H2O/[P44414]Cl and HCl/H2O/PEG-600 systems[END_REF] ). Eventually, according to their complexation state, metallic species migrate toward their preferred phase, enabling a very efficient partition [START_REF] Gras | Ionic-Liquid-Based Acidic Aqueous Biphasic Systems for Simultaneous Leaching and Extraction of Metallic Ions[END_REF][START_REF] Schaeffer | A HNO3-Responsive Aqueous Biphasic System for Metal Separation: Application towards CeIV Recovery[END_REF] .

Such AcABS are thermoresponsive systems. Among them, the mixture of water, tributyltetradecylphosphonium chloride ([P 4,4,4,14 ]Cl) and acid, also presents the uncommon characteristic of a Lower Critical Solution Temperature (LCST), expressing the property of the solution to separate into two immiscible aqueous phases upon heating. This behavior is opposite to the usual demixion mechanism for which phase separation disappears at high temperature because of the predominance of the entropy of mixing. A more subtle mechanism underneath the phase separation therefore needs to be addressed in order to fully exploit such a phenomenon for extraction purposes, such as metal recycling.

In this letter, we present a combination of structural investigation at the microscopic scale and chloride ion titration, enabling to draw a complete picture of the mechanisms driving the phase separation upon heating. Our system is illustrated by the figure 1. The biphasic region increases with temperature. Interestingly, the system also forms a thermoresponsive ABS in the presence of salt instead of acid, as for example NaCl substituting HCl.

The phase diagram presents moreover the same characteristics in terms of inverse thermal response in the presence of acid or salt (for ex. HCl or NaCl). Since all the macroscopic observations are similar, the phase transition in presence of acid or salt is expected to be based on the same mechanisms. We therefore used one system or the other depending on the technical requirements of the experiment performed. Moreover, within the precision of our measurements, the binodal is not affected by deuteration of water or acid, also enabling the use of partial deuteration, a great advantage for small angle neutron scattering experiments. Eventually, the nature of the acid has a large influence. In presence of sulfuric acid H 2 SO 4 , the thermal response is similar although the biphasic region is increased. The effect is even more striking if nitric acid HNO 3 is used, since a single drop turns the IL/water solution in a biphasic one [START_REF] Mogilireddy | Understanding the fundamentals of acid-induced ionic liquid-based aqueous biphasic system[END_REF] . Such a drastic effect makes however difficult the characterization of the phase transition in presence of HNO 3 and will not be discussed in this work. [P 4,4,4,14 ]Cl is a compound close to a classical ionic surfactant, for which self assembling in water is expected, similarly to many ionic liquids [START_REF] Triolo | Nanoscale Segregation in Room Temperature Ionic Liquids[END_REF][START_REF] Greaves | Solvent nanostructure, the solvophobic effect and amphiphile self-assembly in ionic liquids[END_REF][START_REF] Bowers | Aggregation Behavior of Aqueous Solutions of Ionic Liquids[END_REF] . Such an organization of charged units is expected to be influenced by the ionic strength of the solution, modified by the presence of acid, and consequently in the different regions of the phase diagram, since acid concentration also varies. Characterizing the structural organizations of the IL in the different phases should therefore lead us to the microscopic mechanisms of the phase separation. Small angle neutron scattering (SANS) provides a unique tool to probe such a structural organization in solution. The contrast, in our systems, is given by the difference in scattering length between the hydrogenated IL and deuterated water and acid. We will be able to extract both the form of the objects adopted by the self-organized ionic liquids (form factor) simultaneously with the interactions between these objects leading to the structure factor.

The first step into the structural investigations was performed on binary IL/water mixtures for various IL contents.

The experimental data together with their fitted curves, at 25 o C, are shown in figure 2. The best fit is obtained for a model of spherical micelles formed by the IL, dispersed in suspension because of the repulsive electrostatic interaction modeled by a hard sphere form factor calculated with effective radius (see SI for details). For rather dilute conditions (IL %wt < 20), the micelle radius has little variations around 19 Å, i.e. roughly the length of a single cation. Considering the charged head and hydrophobic tails of the IL cation, we reasonably assume that the heads of the cations form the external layer of the micelles, surrounded by water and counter ions (Cl -), as schematically represented in the insert of figure 2. Moreover, charge compensation of the outer surface of the micelle by the chloride anions (counter ions of the IL, unique anions of the solution) is poor since few ions are present to participate to the electrical double layer (EDL) around the micelle. This leads to a residual electrostatic repulsion between the micelles, giving rise to the correlation peak around 0.1 Å-1 . Note that the peak consistently moves toward higher values of Q, i.e smaller distances, when the IL concentration increases.

Increasing the IL concentration does not induce any phase transition toward hard spheres crystal or lamellar phase as observed for many surfactants [START_REF] Israelachvili | Intermolecular and Surface Forces; Intermolecular and Surface Forces[END_REF] . For IL content larger than 65%wt., the microemulsion is inverted and water droplets of radius ∼ 5 Å are surrounded by IL (see figure S1).

Eventually, measurements were also performed for temperatures increasing up to 55 o C. The IL/water solution is homogeneous over this whole temperature range, while it undergoes a phase separation in the presence of acid. No significant changes were indeed observed on the measured intensity as shown in figure S4. Only a slight increase of S(0) can be extracted and also plotted in figure S8, which is directly related to the osmotic compressibility [START_REF] Cousin | Small angle neutron scattering[END_REF] . Such a trend indicates an increase of the attractive interactions with temperature, however not strong enough to enable the micelle flocculation. the decrease of the Debye layer (EDL) size when the concentration of the microions increases [START_REF] Barrat | Basic Concepts for Simple and Complex Liquids[END_REF][START_REF] Durand-Vidal | Electrolytes at Interfaces[END_REF] . The more concentrated the microions are, the closer they come to the charged surfaces. This electrostatic screening eventually strongly reduces the electrostatic interaction between the micelles. The screening becomes more efficient and the attractive interactions between micelles originating e.g. from Van der Waals interactions are no longer overwhelmed by the repulsive interactions. The effect is similar for both acids. However, we emphasize that the structural transformation cannot be rationalized to the ionic strength of the solution. First, the ionic strength is ill-defined in case of sulfuric acid because of its bi-acid nature ; second, if HCl and H 2 SO 4 samples exhibit similar behavior of the structure and transition temperature with acid concentration, it is very different with HNO 3 that induces the separation at very low concentration. Indeed, the concept of ionic strength is especially important around the Debye-H ückel regime for which ion interactions are mainly coulombic. In our case, the ion concentration is too high and specific ion interactions play a role [START_REF] Smith | The Electrostatic Screening Length in Concentrated Electrolytes Increases with Concentration[END_REF][START_REF] Anousheh | Ionic structure and decay length in highly concentrated confined electrolytes[END_REF][START_REF] Adar | Screening length for finite-size ions in concentrated electrolytes[END_REF] . Other effects have to be taken into account such as ion size or polarization, or more sophisticated screening approaches due to the high concentration of charges in the solution are necessary [START_REF] Parsons | Hofmeister effects: interplay of hydration, nonelectrostatic potentials, and ion size[END_REF][START_REF] Duignan | Toward a First-Principles Framework for Predicting Collective Properties of Electrolytes[END_REF][START_REF] Dubouis | Anion Specific Effects Drive the Formation of Li-Salt Based Aqueous Biphasic Systems[END_REF] .

Technically, a unique model for fitting the whole set of data, including the high acid contents, could not be found. This strongly suggests that different micellar structures (cylinders, double shell, bicontinuous phase or mixtures of different objects) appear in the solution upon increase of the acidity or ionic strength, as observed in other systems [START_REF] Aswal | Role of different counterions and size of micelle in concentration dependence micellar structure of ionic surfactants[END_REF][START_REF] Sanchez-Fernandez | Counterion binding alters surfactant self-assembly in deep eutectic solvents[END_REF][START_REF] Danov | Analytical modeling of micelle growth. 4. Molecular thermodynamics of wormlike micelles from ionic surfactants: Theory vs. experiment[END_REF] . We therefore concentrate here on the low acid content as a model system, giving a clear picture of the process that can be, omitting the form factor details, applied to any region of the phase diagram. The best fit is still obtained for a hard sphere form factor, but now combined with a sticky hard sphere structure factor in order to model the attraction between the micelles. The effect of temperature is similar to the addition of acid to the solution, the effect being illustrated by the corresponding top and bottom panels of the figure 3. The screening of electrostatic repulsion between micelles increases and so the stickiness of the structure factor. This temperature behavior is represented in figure 4.

The stickiness of the potential, directly related to the attractive interaction between the micelles, is therefore the relevant parameter for the description of the phase separation. The potential represented by the sticky hard-sphere model is akin to the potential of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [START_REF] Barrat | Basic Concepts for Simple and Complex Liquids[END_REF][START_REF] Verwey | Theory of the Stability of Lyotropic Colloids[END_REF][START_REF] Dufrêche | Charged-stabilized colloidal suspensions: counterion condensation and phase diagrams[END_REF] . Inter-micelle potential reflects the attractive Van Der Waals (VdW) force and the repulsive electrostatic force. Within the DLVO description, the two contributions are assumed to be additive and the electrostatic term is calculated from the linearized Poisson-Boltzmann equation. The resulting (DLVO) potential presents a barrier that depends on acid content and temperature. When the temperature is high enough so that the thermal energy k B T is close to the height of the barrier, the probability for the micelles to cross the energy barrier and be attracted to each other by Van der Waals interactions increases. In other terms, at high temperature, the micelles eventually flocculate and the phase separation is observed.

The upper phase is formed by the lowest density phase, i.e. the ionic liquid rich phase. Eventually, the effect is confirmed by the trend of the osmotic compressibility S(0) with temperature, with values way above unity, indicating an increased interaction between the particles and increasing even more with temperature (see SI figure S8).

Although the general trend of screening of electrostatic interactions with the addition of salt or acid is expected, its temperature dependence is less intuitive. The overall increase of the attraction is not believed to be due to stronger Van der Waals interactions that are poorly temperature dependent and mostly size dependent. It therefore has to be a decrease in the repulsive interaction, meaning a variation in the charge distribution around the micelle or a change of EDL composition. The long distance repulsion between the micelle is typically (in k B T units) proportional to the Bjerrum length of the solution. The latter is weakly temperature dependent. Thus the increased attraction at high temperature is not related to dielectric constant changes but rather to the (effective) charge of the micelles, i.e. to the ion adsorption at the surface. In order to address this last point in solutions where the zeta- The adsorption of Cl -in the EDL, following the interpretation of the structural data, leads to a decrease in the repulsive interactions, easing micelles flocculation. Assuming a Van't Hoff behavior for this adsorption process and two possibilities only for the Cl -(free or bound), we get the enthalpy of adsorption of chloride ions on the micelles to be around 12 kJ/mol. This enthalpy of adsorption relates the interaction potential between anions and the head of the IL cations on the micelle surface. We emphasize that this potential energy is actually a free energy since it is averaged over the configurations of the system. Consequently, this interaction is globally attractive and increases with temperature.

A complete picture of the microscopic mechanisms leading to the phase separation induced by temperature can now be drawn. Most probably because of the increasing disorder of the short aliphatic chains around the phosphonium ions of the ionic liquid [START_REF] Kang | Molecular insight into the lower critical solution temperature transition of aqueous alkyl phosphonium benzene sulfonates[END_REF] , chloride ions are attracted to the surface of the micelle, increasing the screening of electrostatic repulsion that maintains the micelles in suspension. Once the effective energy barrier of the DLVO potential is low enough, the micelles are able to reversibly aggregate and form a biphasic system. The phase separation is thus a kind of liquid-gas separation of a fluid of micelles.

At high temperature, if the magnitude of the micelle configurational entropy increases, it is no longer sufficient to compensate for the enthalpy gain, which also increases due to the adsorption of ions. This enthalpic gain compensates the entropic cost of the phase separation upon temperature increase. The phase separation mechanism is different here from the one described by the Flory-Huggins solution theory. Although the free energy gained by the chloride adsorption may account for an entropic contribution due to the water molecules released from the ion hydration shell, an enthalpic mechanisms drives the transition, not the entropy overpassing a weak interaction between the components.

Such a phase separation upon rise of temperature is unusual but not unique, as found in other systems presenting LCST. Although more commonly encountered in polymeric systems [START_REF] Lee | Lower Critical Solution Temperature (LCST) Phase Behavior of Poly(ethylene oxide) in Ionic Liquids[END_REF][START_REF] Bruce | Nonadditive Ion Effects Drive Both Collapse and Swelling of Thermoresponsive Polymers in Water[END_REF] , other mixtures of small molecules [START_REF] Plazanet | Freezing on heating of liquid solutions[END_REF] , including ionic liquids exhibit such behavior [START_REF] Schaeffer | A HNO3-Responsive Aqueous Biphasic System for Metal Separation: Application towards CeIV Recovery[END_REF][START_REF] Kang | Molecular insight into the lower critical solution temperature transition of aqueous alkyl phosphonium benzene sulfonates[END_REF][START_REF] Fukumoto | LCST-Type Phase Changes of a Mixture of Water and Ionic Liquids Derived from Amino Acids[END_REF][START_REF] Wu | Assembly Pattern of Supramolecular Hydrogel Induced by Lower Critical Solution Temperature Behavior of Low-Molecular-Weight Gelator[END_REF][START_REF] Wang | Microemulsion-like aggregation behaviour of an LCST-type ionic liquid in water[END_REF][START_REF] Kohno | Temperature-responsive ionic liquid/water interfaces: relation between hydrophilicity of ions and dynamic phase change[END_REF] . The LCST mechanism is however not fully identified in all these systems. If the hydrophilic/hydrophobic balance of the ionic liquid plays a role in the self assembling of IL cations in water, anions also play a role [START_REF] Kang | Molecular insight into the lower critical solution temperature transition of aqueous alkyl phosphonium benzene sulfonates[END_REF][START_REF] Kohno | Temperature-responsive ionic liquid/water interfaces: relation between hydrophilicity of ions and dynamic phase change[END_REF] . Moreover, droplet size extracted from zeta-sizer measurements can be misleading on the growth of micelles in such concentrated solutions. Such measurements indeed also indicate an increase in micelle size with temperature in our systems, while direct characterization from SANS prove an increasing attraction between micelles of constant radius. Another approach was proposed by Schaeffer et al. based on molecular dynamics, where the adsorption of molecular HNO 3 at the micelle surface is proposed as mechanism driving the phase separation [START_REF] Schaeffer | A HNO3-Responsive Aqueous Biphasic System for Metal Separation: Application towards CeIV Recovery[END_REF] . Here also, an adsorption on the cation's head is at the origin of the LCST. If the adsorbed specie is specific to each system, we believe that the adsorption mechanisms is at the origin of a large number of LCST in charged molecular solutions.

The methodology presented here based on a microscopic structural investigation coupled with ionic species titration, would therefore apply to many systems of soft matter, providing a bridging description from the nanoscale to macroscopic properties.

Supplementary Information: samples preparation, experimental protocol describing the chloride titration; experimental conditions for the SANS measurements and data analysis, additional parameters extracted from the fits including osmotic compressibility.

How Temperature Rise Can Induce Phase Separation in

Aqueous Biphasic Solutions: Supplementary Information

Gautier Meyer, † Ralf Schweins, ‡ Tristan Youngs, ¶ Jean-Franc ¸ois Dufrêche, § Isabelle Billard, * ,∥ and Marie Plazanet Selective deuteration makes SANS a very powerful technique to characterize aggregates in solution through various parameters such as their shape, size or volume fraction. The hydrogen/deuterium scattering length density (SLD) contrast is used to maximize the SLD contrast between the IL cation and the surrounding solvent, reducing simultaneously the background due to the incoherent neutron scattering of hydrogen. In the present case, we used hydrogenated IL and deuterated solvents. The scattered neutron intensity is measured as a function of the wave vector transfer momentum q = 4π sin( θ 2 ) λ . The q-range covered respectively on NIMROD and D11 are 0.05 Å -5 Å and 0.0018 Å -0.45 Å. Such an extended q-range enables to determine molecular structural information on the shorter scale (high q), while contrariwise the supramolecular organization is probed at a larger scale at smaller q.

Eventually, the lowest q-range was measured only for some selected samples and did not provide any additional information and is therefore not presented here.

NIMROD data were reduced with the program GUDRUN. D11 data reduction was performed with the LAMP software (https://www.ill.eu/fr/users/support-labs-infrastructure/software-scientific-tools/lamp), while the dedicated software SASView was used for model fitting, enabling the combination of different models for both structure and form factors. For the various samples and conditions, the best fit to the measured intensity was obtained in combining the hardsphere form factor with a structure factor of the form either hard sphere with electrostatic interactions (meaning a hard sphere structure factor with an effective radius different than the one used for the form factor), or a sticky hard sphere potential.

The intensity scattered by a spherical object in a solution is given by ? :

I(q) = V f V . ( 3V(∆ρ). sin(qR) -qR cos(qR)) (qR) 3 ) 2 + B
where V f is the volume fraction, V is the volume of the scatterer, R is the sphere radius, B is the background level and ∆ρ is the scattering length density difference between the scatterer and the solvent.

Polydispersity of the spheres was accounted for in the modeling process. The distribution of radii is considered to follow a gaussian profile, and the polydispersity PD parameter is given by: PD = σ/x , where σ is the standard deviation of the Gaussian distribution and x is its center value.

Complement on the SANS analysis of binary IL/water solutions

The data were fitted using a hard sphere model for the form factor and hard sphere with electrostatic interactions,

Complement on the SANS analysis of higher acidic concentration AcABS

The attractive (sticky) potential is modeled as a square well potential, which depth and width are represented in the figure ??. The spherical form factor alongside the sticky hardsphere structure factor fits nicely all the data obtained for the lowest acid concentrations and those at room temperature for all the acid concentrations. The polydispersity or the sphere radius (form factor) is also taken into account. The data measured on the intermediate acid concentrations could also be fitted with the same model, although the agreement was slightly degraded at low q for the highest temperature. The stickiness of the potential, showing similar behavior as the lowest concentration, at least up to the phase separation, is represented in the figure ??. An increase of the attractive potential between the micelles (which results from the sum of the Van Der Waals and the electrostatic repulsion) leads to the micelles flocculation. The depth of this potential increases with the temperature, in good agreement with the attractive potential interpretation that leads to the flocculation observed at higher temperature. A higher concentration of acid leads to an increase of the potential, that is attributed to the screening of the electrostatic repulsion and not to an increase of the Van der Waals forces, since the size of the micelles remains constant. We did not find a unified model to fit all the samples we have characterized. Upon increasing the acid concentration, we got very good fitting agreement using models with coexisting shapes (spheres and cylinders for example) and bicontinuous phases. The intensity plateau expected at very low q was not reached at higher temperature, indicating the presence of very long objects or structural effects on a very large scale. We present here the results obtained for both acids with a cylindrical form factor. The radius of the cylinders was found to be roughly equal to the radius of the spherical micelles at lower concentration, and their length is given in figure ??.

Osmotic compressibility extracted from SANS analysis of IL/water and ABS solutions

The osmotic compressibility is the limit of the structure factor at q=0. In the case of non interacting particles, S(0)=1.

It is larger than 1 for attractive interactions, and smaller than 1 for repulsive interactions. In order to get this value, the model fitting the experimental data was extrapolated at q=0.

Figure 1 :

 1 Figure 1: DCl based samples representation on the ternary phase diagram of DCl/[P 4,4,4,14 ]Cl/D 2 O ABS at 25 o C, recalculated from 7 (see SI for more details). 100% of DCl solution corresponds to 37% DCl in water. The points S 1 , S 2 and S 3 give the compositions of the solutions studied by SANS (see SI for more details and the corresponding compositions in 100% DCl). The schematic representation of the cation P[ 4,4,4,14 ]Cl of the ionic liquid is also shown.

Figure 2 :

 2 Figure 2: Normalized SANS data (symbols) and fits (solid lines) using a spherical form/hard-sphere structure factors for IL/water mixtures. The insert is a schematic representation of the micelle surrounded by chloride ions.

Figure 3 :

 3 Figure 3: Left panels : measurements on hydrochloric acid solutions; right panels : sulfuric acid solutions. Top : normalized data and fitting results of the SANS measurements performed on ABS solutions at 25 o C for three acid contents (S 1 , S 2 and S 3 for DCl samples); bottom : data at different temperatures from the monophasic to the biphasic state (upper phase), for transitions at 38 and 36 o C for DCl and D 2 SO 4 solutions respectively at the lowest acid content. The schematic representation of the micelle now contains several types of ions (different colors) in the surrounding electrical double layer.

Figure 4 :

 4 Figure 4: Attractive potential intensity (in k B T) as a function of T for the lower concentration samples. Phase separation temperature is represented by the dashed line of the corresponding color.

Figure 5 :

 5 Figure 5: Free [Cl -] in the solution as a function of temperature, assuming that the activity is equal to the concentration (left axis). Corresponding ratio of Cl -to cations in the EDL of the micelle (right axis) assuming all anions are either free or bound.

SANS methods .

 methods SANS measurements on IL/water solutions have been carried out on the diffractometer NIM-ROD at the ISIS neutron source (Oxford, UK) and on the small-angle neutron scattering instrument D11 at the Institut Laue-Langevin (ILL, Grenoble, France). SANS experiments on the D11 instruments were performed under the following conditions. Neutrons have been measured using a 3He MWPC (CERCA) detector with 256 x 256 pixels of 3.75 mm x 3.75 mm size. Three instrument configurations have been employed, all with a neutron wavelength of 6 Å (FWHM 9%), and sample detector distances of 34 m, 8 m and 1.4 m (with collimation distances of 34 m, 8 m and 20.5 m respectively). A fourth setting has been used for time-resolved experiments, with a sample detector distance of 4 m and a collimation distance of 20.5 m. The neutron footprint on the sample was 16 mm in diameter. The cuvettes had an internal size of 45 mm x 45 mm and were made by Thuet (France). All data were put on absolute scale making use of a 1 mm H 2 O measurement, serving as secondary calibration standard, and having a differential scattering cross section of 0.983 cm -1 .

  leading to an effective radius for the structure factor. The relevant parameters, radius (form factor, figure ??) and effective radius (structure factor, figure ??) are plotted in the following figures. The polydispersity is taken into account, as represented in the figure ??.

Figure 1 :Figure 2 :

 12 Figure 1: IL/water : radius of the micelle form factor vs IL content

Figure 3 :Figure 4 :

 34 Figure 3: IL/water : polydispersity of the radius of the micelle as a function of IL content

Figure 5 :

 5 Figure 5: Temperature evolution of the stickiness of the potential for the intermediate acidic concentrations. The data are fitted with the same model for all samples : a hard sphere form factor (radius around 18 Å) and sticky hard spheres structure factor. The dotted lines indicate the transition temperature.

Figure 6 :

 6 Figure 6: Polydispersity of the sphere form factor radius as a function of temperature for the lower and intermediate acid concentration samples.

Figure 7 :

 7 Figure 7: Length of cylinders found in the form factor for the two highest concentration samples as a function of temperature. The dotted lines indicate the transition temperature.

Figure 8 :

 8 Figure 8: S(0), or osmotic compressibility as a function of temperature for binary 20% IL in water solution and two ABS solutions in DCl and D 2 SO 4

  * ,⊥ †Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France ‡Institut Laue Langevin, 38042 Grenoble, France ¶ISIS Rutheford Laboratory, UK §Institut de Chimie Séparative de Marcoule, France ∥Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, Grenoble INP, LEPMI, Grenoble, France⊥Univ. Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France E-mail: isabelle.billard@grenoble-inp.fr; marie.plazanet@univ-grenoble-alpes.fr since the binodal of the NaCl solution is shifted to the left with respect to the HCl solution ? . A composition of 2.6 / 14.2 / 83.2 wt% of NaCl/IL/H 2 O was used instead, in order to be in a similar region of the phase diagram with respect to the binodal curve.

Acknowledgement

This work is supported by the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02). We also thank the Institut Laue-Langevin (Grenoble, France) for the allocation of neutron beam time. The neutron data set is available at DOI: 10.5291/ILL-DATA.6-02-604.

Supporting Information Available Materials and methods

Samples.Tributyltetradecylphosphonium chloride was purchased from IoLiTec guaranting a purity higher than 95%. Several mineral acids were used, in their deuterated form, together with D 2 O to optimize the contrast in neutron scattering (see methods). Deuterated water (99,90% deuterium), D 2 SO 4 (98 wt% in D 2 O, > 99% deuterium) and DNO 3 (65-70 wt% in D 2 O, > 99% deuterium) were purchased from Eurisotop and DCl (35 wt% in D 2 O, > 99% deuterated) from Sigma-Aldrich.

The ABS samples phase transition temperatures were determined using the cloud point method, where the monophasic samples temperature is increased until turbidity is observed.

The phase separation in all of our deuterated samples was observed at the same temperature than in the hydrogen based samples for the same molar compositions. Thus, the binodal curves of deuterated samples were extrapolated from the hydrogenous ones available in the literature ? . We chose in the Figure 1 The electrode used for the free chloride titration has an upper bound of 1M of Cl-, which is much lower than the quantities of Cl-in an ABS using HCl that would have a phase transitions under the convenient temperature of 50C°. For this reason, we used NaCl-based ternary solutions which is more efficient at inducing the phase separation than HCl with respect to the molar quantities added to the solution.

The composition of the NaCl/P 4,4,4,14 Cl/H 2 O solution was not identical to the HCl/P 4,4,4,14 Cl/H 2 O investigated