Numerical Study of Surfactant Dynamics during Emulsification in a T

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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

Numerical study of surfactant dynamics during emulsification in a T-junction microchannel Antoine Riaud, Hao Zhang, Xue-Ying Wang, Kai Wang, and Guangsheng Luo Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00123 • Publication Date (Web): 30 Mar 2018 Downloaded from http://pubs.acs.org on March 30, 2018

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Numerical study of surfactant dynamics during emulsification in a T-junction microchannel Antoine Riaud,†,†,¶ Hao Zhang,‡,† Xueying Wang,‡ Kai Wang,∗,‡ and Guangsheng Luo∗,‡ †The two authors contributed equally to the work. ‡The State Key Lab of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 100084, China ¶Present address: INSERM UMR-S1147, CNRS SNC 5014, Paris Sorbonne Cité Université, Paris, France E-mail: [email protected]; [email protected]

Abstract Microchannel emulsification requires large amounts of surfactant to prevent coalescence and improve emulsions lifetime. However, most numerical studies have considered surfactant-free mixtures as models for droplet formation in microchannels, without taking into account the distribution of surfactant on the droplet surface. In this paper, we investigate the effects of non-uniform surfactant coverage on the microfluidic flow pattern using an extended Lattice-Boltzmann model. This numerical study, supported by micro-PIV experiments, reveals the likelihood of uneven distribution of surfactant during the droplet formation and the appearance of a stagnant cap. The Marangoni effect affects the droplet break-up by increasing the shear rate. According to our results, surfactant-free and surfactant-rich droplet formation processes are qualitatively different, such that both the capillary number and the Damköhler number should be

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considered when modeling the droplet generation in microfluidic devices. The limitations of traditional volume and pressure estimation methods for determining the dynamic interfacial tension are also discussed based on the stimulation results.

Keywords Surfactant, microfluidics, dynamic interfacial tension, lattice Boltzmann method, particle image velocimetry, stagnant cap, mass transfer, multiphase.

Introduction Dynamic interfacial tension is the variation of interfacial tension with time. An ubiquitous example is the adsorption of surfactant at a fluid interface at rest studied by Ward and Tordai in their seminal work. 1 Herein, the surfactant adsorption proceeds in two steps: (i) diffusion of the surfactant to the interface and (ii) adsorption and reorientation of the surfactant molecules (chemical kinetics). In microfluidics, dynamic interfacial tension phenomena are notoriously more complex 2,3 and have profound consequences on droplet volume, 4–7 pressure fluctuations, 8 emulsion stability 9–11 and inter-droplet mass transfer. 12 Droplet formation in microfluidic devices offers an especially interesting and rich playground to study surface phenomena. 13 As the droplet forms, it is exposed to a constant flow of the continuous phase which greatly enhances the surfactant mass transfer by convection. 5,14 Although dynamic interfacial tension in larger microchannels (hydraulic diameter L > 300µm) seems dominated by convection, 14 diffusion is accelerated in smaller microfluidic devices (L < 30µm) and ultimately surfactant adsorption rate is kinetically controlled. 11 Despite all these enhancement factors, the rate of droplet formation is so high that surfactant concentrations about two orders of magnitude above the critical micellar concentration (cmc) are necessary to overcome the dynamic interfacial tension. 11,14 In previous studies, 4–7,11,14 the dominant criterion G for dynamic interfacial tension was 2


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to compare the adsorption rate (either kinetically limited or mass-transfer limited) to the droplet formation time (or surface area generation rate) τf = L3 /QD = L/UD with L a typical lengthscale of the microchannel (e.g. the hydraulic diameter), QD the dispersed phase flow rate and UD the mean flow velocity at the dispersed phase inlet (similarly, UC and QC will refer to the mean flow velocity at the continuous phase inlet and the associated flow rate, respectively). If the adsorption is reaction limited, the characteristic surfactant kinetics are given by τkin =

Γ∗∞ Ikin

with Γ∗∞ the maximum interfacial coverage and Ikin the initial adsorption

rate. 11 In turn, this duration must be compared with the droplet characteristic formation time, which yields Gkin =

τkin τf

=

UD Γ∗∞ . Ikin L

The transport-limited case was extensively studied

by Wang et al. 14 by considering the creation of a boundary layer of thickness h∗m . This boundary layer develops due to the depletion of free surfactant monomers as they adsorb ∗

Γ∞ on the droplet surface. Two scenarios can be envisioned (i) diffusion limited h∗p = cCM 1/3C L 4 ∗ . with cCM C the critical micelle concentration and (ii) convection limited hcR = 2 3P eef f

In the latter, the effective Péclet number P eef f =

LUC Def f

accounts for an apparent increase

in surfactant diffusivity Def f (by up to two orders of magnitude 15 ) due to the additional monomers released by the micelles disaggregation. 16,17 Eventually, since the characteristic mass transfer time is given by τtr =

[min (h∗p ,h∗cR )]2 , Def f

time to the droplet formation time Gtr =

τtr . τf

we define the ratio of the mass transfer

In practice, Riechers et al. 11 measured Ikin '

1.1 × 10−3 mol/m2 s, Γ∗∞ ' 8 × 10−6 mol/m2 , cCM C ' 4 × 10−3 mol/m3 for perfluoropolyethercarboxylic acid (PFPE-CA) and suggested a typical diffusivity of 5 × 10−10 m2 /s for this surfactant (we will assume Def f = 100 × D = 5 × 10−8 m2 /s as the effective surfactant diffusivity). Consequently, for typical microfluidic experiments (L = 100 µm, UC = 5 cm/s, UD = 1.25 cm/s), Gkin ' 1 and Gtr ' 0.3 which suggests a significant yet incomplete adsorption of surfactant on the droplet surface. This estimate is a lower bound since some surfactant such as sodium dodecyl sulfate (SDS) can adsorb much faster. 14 Thus, the droplet surface can be either partially or completely covered with surfactant. While this variation of surfactant coverage over time has been widely considered, the

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possibility of a non-uniform coverage remains largely unexplored. It is often argued that the surface of the forming droplet is homogeneously covered with surfactant since these molecules adsorb quickly when the concentration exceeds by far the CMC. 4,6,7 Although this uniform coverage hypothesis is generally verified within a few percent of deviation as testified by the adsorption rate criterion G, it is not enough to neglect the Marangoni stress since the latter is greatly enhanced at high surface coverage. 18 According to Eggleton and Stebe, 18 two additional requirements are that (i) the surfactant desorption rate kd has to be much larger than the convection rate at the droplet surface (Bi =

kd a UC

>> 1) with a

the droplet radius and UC the continuous phase velocity and (ii) that the desorption must be fast enough to prevent large interfacial tension gradients dominating the shear stress B=

kd µC a σstat

>> 1 with µC the continuous phase viscosity and σstat the interfacial tension at

thermodynamic equilibrium. Although the desorption rate is difficult to evaluate directly, 19 it has been measured for some fluorinated surfactants such as perfluoropolyether-glycol 13 kd ' 6 × 10−3 s−1 (M ' 12500 g/mol) and PFPE-CA 11 kd ' 2 × 10−4 s−1 (M = 1634 g/mol). Consequently, for the same typical microfluidic experiment (L = 100 µm, σstat = 5 mN/m, µ = 1mPa.s, UC = 5 cm/s, UD = 1.25 cm/s), Bi ' 10−5 , B ' 10−7 suggesting an intense Marangoni effect. Such non-uniform surfactant coverage has been directly observed after the droplet formation by using a surfactant that turns fluorescent only after adsorption. 10 Further indirect observations with more usual surfactants are also given in the literature. 7,20 For instance, the build-up of surfactant at the droplet tip results in tip streaming, a phenomenon both observed in microchannel emulsification 20 and surfactant-covered droplets submitted to extensional flow. 21,22 Eventually, at the pinch-off stage, the dispersed phase detachment is slowed down by surfactants. 7 This neck strengthening is due to the Marangoni effect, also observed in the extensional flow simulations. 21 More recently, it was observed that the convective mass transfer of surfactant was happening as if the droplet interface was rigid. 14 Despite these remarkable experiments, no direct observations or mathematical models of the

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effects of non-uniform surfactant coverage on the droplet formation are available so far. Due to the various effects of surfactants and their high nonlinearity, non-uniform interfacial tension problems involving a mobile interface remain very challenging analytically. 23,24 This difficulty is alleviated for numerical simulations and a wealth of models of single droplets or bubbles covered with an insoluble 21,25–27 or soluble surfactant 18,28–36 are available. Nevertheless, the effects of surfactant non-uniform coverage on the droplet formation in microfluidic channels have yet to be investigated. In this study, we extend an existing LatticeBoltzmann model 37 to account for the dynamics of surfactant molecules during emulsification in microfluidic T-junctions. The 2D numerical results will be compared to experimental Micro-Particle Image Velocimetry (micro-PIV) measurements to investigate the influence of surfactant kinetics on the forming droplet flow pattern. Finally, we will study whether measuring the droplet size or the pressure fluctuations in microfluidic channels can provide some information about a possible non-uniformity of the droplet coverage.

Experimental setup and numerical method Micro-Particle Image Velocimetry The microchannels were similar to our previous study 8 except the depth was here 760 µm instead of 620 µm previously. They were drilled in polymethyl methacrylate (PMMA) with a cross section of 600 × 760 µm2 . We used a Dantec Dynamics (Denmark) micro-PIV. The time between laser pulses was set to 200 µs and the trigger rate was 7.4 Hz. Working fluids were water (dispersed phase) and octane with span 80 (continuous phase) seeded with 0.16 w% fluorescent melamine microparticles (density 1.6 g/cm3 , diameter 1 to 2 µm, labeled with Rhodamine B, Beijing Lifang Tiandi Technology Development Co.,Ltd., China). The flow in the median plane of the microchannel was magnified with a 5× microscope objective (NA = 0.25, depth of field 18 µm) and images were recorded with a 4 Mpixels camera (Flowsense E0, Dantec Dynamics, Denmark). Additional details can be found in Wang et al. 38 The 5


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water-octane interfacial tension without surfactant is 50.8 mN/m. The critical micellar concentration of Span 80 in octane is 0.03 w%. Above this threshold, the static interfacial tension ranges near 3.2 mN/m. 39 At 20o C, water and octane viscosities are approximately 1.0 mPa.s and 0.54 mPa.s, respectively.

Lattice-Boltzmann model The microfluidic emulsification was simulated with the Lattice-Boltzmann Method (LBM). It is an efficient numerical approach for multiphase flow undergoing large deformations in confined geometries. 40–42 LBM models the fluid as a statistical population of particles that stream between neighboring lattices and eventually collide. Macroscopic variables such as density, speed and pressure are then deduced from the particle distribution function. 43 In its multiphase implementation, intermolecular forces or energy constraints act on the particles and lead to phase separation. Thus, the interface is handled implicitly without need for tracking. Successful implementations of surfactants in the LBM framework include self-consistent free-energy based models 44–47 (see Van der Sman et al. 48 for a review article), force-based models with oriented particles 49 and interface-tracking hybrid methods. 25 Although these early models provide a remarkable insight in the phase transition of surfactants or complex membrane dynamics, 50 they do not set the interfacial tension and elasticity explicitly, but instead require a suitable free-energy functional or intermolecular force. The macroscopic interfacial tension and elasticity are then obtained by curve fitting. The scope of these empirical relations has been clarified in recent works 28 and these can now be reduced to a single universal constant. 51 We base the current model on an immiscible two-phase flow solver with an explicit surface stress tensor. 52 Due to the tensor representation, both the pressure difference across the interface and the Marangoni effect are accounted for in a controlled fashion. 53 The surfactant is then implemented 37 as a low-concentration solute that adsorbs to the fluid interface in a first-order kinetic. Prior any change, this combination of models was able to simulate 6


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the preferential accumulation of surfactant at the fluid interface rather than in the bulk and the mass transfer by diffusion and convection along the fluid interface. Therefore, simulating the effects of surfactant adsorption on the droplet formation required two further modifications: (i) relating the effects of solute distribution over interfacial tension and (ii) choosing an appropriate source term to model both the surfactant diffusion from the bulk to the interface and the subsequent adsorption. The LBM simulations were implemented in 2D since at small capillary number (squeezing flow regime), droplet emulsification resembles a 2D process where the dispersed phase is gradually squeezed by the continuous phase. Thus, qualitative flow features in 2D and 3D are similar. 54 However, 3D simulations would have increased the simulation time and memory requirements by up to 50 times. Since the resolution of the dynamics of the surfactant at the droplet surface depends on the number of lattices along each dimension of the microchannel, 2D simulations were more appropriate to accurately depict the complex phenomena happening at droplet interface. In order to improve the paper readability, we highlight here the key assumptions of the model and let the reader refer to the supporting information for further details regarding the implementation and validation of the code. In the following, we will measure distances in lattice units (l.u.) and time in time steps (t.s.) as it is common in Lattice-Boltzmann simulations. Most flow conditions are quantified with non-dimensional numbers such as the capillary number Ca, the Péclet number P e and the Damköhler number Da. In order to represent actual physical process, these dimensionless numbers must match experimental values. For convenience, the droplet volume Vd (originally in l.u.2 due to the 2D geometry) and the pressure in the microchannel P are made dimensionless quantities V˜d and P˜ defined as follows: V˜d = Vd /L2 and P˜ =

LP . σ0

Other values are converted to international units using

the Vashy-Buckingham Pi-theorem. 55 The Gibbs isotherm establishes a simple relation between the surfactant coverage Γ and the interfacial tension σ:

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σ = σ0 (1 + E0 ln(1 − Γ)),

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(1)

where Γ is the ratio of the surfactant coverage (Γ∗ in mol./m2 ) to the maximum packing allowed by the Gibbs isotherm Γ∗∞ , and σ0 is the clean interface interfacial tension. The elasticity E0 = RT Γ∗∞ /σ0 was not regarded as a relevant parameter since it ranges from 0.2 to 0.3 for most surfactants. 6,56 For instance, E0 ' 0.23 for span 80 at the water-octane interface. 56 Since this implementation of the coupling between flow pattern and surfactant coverage was new compared to our previous study, we first validated it against a commercial numerical solver (Comsol 4.2). The two methods yielded quantitatively similar flow pattern and surfactant coverage (more data available in the supporting information). As mentioned in the introduction, surfactant adsorption is complex and depends on the scale of the microchannel. For large microchannels, the mass transfer mostly depends on convection and a thin boundary layer is expected. In smaller microdevices, the adsorption is reaction limited and may be slowed down by some energy barrier. 57 Furthermore, actual emulsifiers operate much above the CMC so the micelles act as surfactant sources that maintain a constant monomer concentration near the CMC. Hence, both micelles and monomers transport should be accounted for in the model. 58 In order to keep a general viewpoint in this study, we gathered all the adsorption process, including bulk to surface transport and micelles disintegration, into an effective first-order kinetics (Eq. (2)). The surfactant is modeled only at the liquid interface in order to expand the range of adsorption rates and avoid redundancy with diffusive mass transfer from the bulk. Once adsorbed, the surfactant follows the usual mass conservation equation on a surface manifold 22 with a source term Rs accounting for the kinetic transport of surfactant from the bulk to the surface and given by equation (2). In an effort to validate this simplification, we checked that the droplet size and pressure fluctuations calculated with a model taking explicitly the bulk diffusion and convection into account — explicit mass-transfer model — faithfully reproduced those obtained using the current effective model (see supplementary information for models de8


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scription and validation). Therefore, although this effective model may not exactly reflect the actual ongoing process, it was favored over the explicit model since it provides a convenient unambiguous single parameter 0 < α < 1 to compare the surfactant adsorption rate and the droplet formation time, and hence simplifies the data analysis.

Rs =

1 τads

α(Γ0sat − Γ).

(2)

In this equation, Γ0sat is the surfactant coverage at thermodynamic equilibrium, Γ0sat < 1. For span 80, 56 the saturation coverage yields Γ0sat = 0.98. However, we have to set Γ0sat = 0.90 to maintain the numerical stability. τads is the adsorption time constant, and represents the global mass transfer and adsorption process. In order to observe the coupling between droplet growth and surfactant adsorption, we set τads = 3.03 × 103 t.s to the same order of magnitude as the droplet formation time. α is a non-dimensional parameter to tune the adsorption rate, 0 < α < 1. Comparing the LBM simulation to the micro-PIV experiments, the Pi-theorem based on channel width (L = 30 l.u.) and continuous phase inlet flow velocity (UC = 3.33 × 10−3 l.u./t.s.) indicates that one time step amounts to 0.3 µs and therefore τads = 1.0 ms. The parameter α then allows to slow down the adsorption by up to a hundred fold. The surfactant surface diffusivity is set to Ds = 3.33 × 10−3 l.u.2 /t.s., which leads to a Péclet number P e =

LUC DS

≈ 30. In spite of this moderate Péclet number, peak velocities

reached in the constrictions between the droplet and the channel wall lead locally to Péclet numbers near 1000. Even in regions of low velocity, we expect the Marangoni effect to dominate over diffusion. In this study, the emphasis is on the surfactant dynamics and the droplet formation process. Thus, the Damköhler number Da0 = UC τads /(αL) and the surfactant-free capillary number Ca0 = UC µC /σ0 with µC the continuous phase dynamic viscosity are the key parameters of the simulations. It is a considerable experimental challenge to tune Ca0 inde-

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pendently of Da0 since changing the flow rate 4 or the viscosity (which modifies the surfactant diffusivity 59 ) influences directly the Damköhler number. This prevents from discriminating the influence of the mass transfer rate relatively to the flow pattern. This difficulty is overcome in simulations by tuning the value of σ0 , which has no effect on the Damköhler number. In our computations, Ca0 started at 1.8 × 10−3 and was multiplied by 2 for each subsequent run, until it reached the value of 0.058. Reversely, we varied Da0 over two orders of magnitude by tuning 0 < α < 1 without changing the capillary number. In order to relate these idealized simulations to actual experiments where the flow velocity is varied, we define an isosurfactant dataset as a series of experiments keeping the same surfactant. In this case, Da0 increases proportionally to Ca0 . Numerically, the isosurfactant curves are obtained by linear interpolation over the Ca0 and Da0 dataset.

Results Simulation results During the simulations, we monitored the velocity field and surfactant coverage. At a given Ca0 , the droplet formation was observed with different surfactant kinetics. We compare these simulations in Fig. 1. Even though the snapshots have all been taken at the same simulation time, the formation process looks more advanced for faster surfactant kinetics. In addition, the flow pattern is significantly different: in the absence of surfactant (Fig. 1(a, e)), the emerging droplet is mixed by an intense recirculation eddy which progressively vanishes at intermediate adsorption rates (Fig. 1(c, g)). Beyond this point, the velocity field inside the droplet becomes more and more uniform, and the maximum speed magnitude decreases. We notice that in all these simulations, the surfactant preferentially concentrates at the droplet tip. Some surfactant (e.g Fig. 1(g)) may also accumulate at the upstream corner of the dispersed phase inlet due to the low convection at this particular point. This 10


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1

U/U max

0.8 0.6 = 0.0 Umax = 0.024

= 0.01 Umax = 0.024

= 0.1 Umax = 0.016

= 1.0 Umax = 0.014

0.4 0.2 0 !/ 1 !max

0.8 0.6 0.4 = 0.0 !max = 0.0

= 0.1 !max = 0.60

= 0.01 !max = 0.22

= 1.0 !max = 0.90

0.2 0

Figure 1: Simulation snapshots after 30 × 103 t.s with different surfactant kinetics and identical Ca0 = 1.8 × 10−3 . From left to right, the pictures correspond to progressive increase in surfactant kinetics (from 0 to 1). (a, b, c, d) show the velocity field (arrows) and speed magnitude (colors), droplet boundaries are indicated by the rainbow curve net. (e, f, g, h) indicate the surfactant coverage Γ, droplet boundaries are depicted by the white line. surfactant build-up may explain the experimental observation 7 that the droplet upstream surface shape switched from an arc to a chord when using surfactants. Indeed, the corner would have more surfactant and hence it may exhibit a low interfacial tension and a much larger curvature than the upstream side, which may appear relatively flat in comparison. The progressive immobilization of the dispersed phase was confirmed by our micro-PIV experiments as shown in Fig. 2. Although both droplets are formed in the squeezing regime, subtle differences appear in the flow field. Similarly to the simulations, the recirculation pattern present in the low-surfactant concentration droplet (visible in the inset of figure Fig. 2(a) and in the numerical model snapshot Fig. 1(a)) gradually weakens and vanishes at higher concentrations (see Fig. 2(g) and Fig. 1(b), respectively). The formation time is also shorter in the surfactant-rich case as testified by the reduced droplet size shown thereafter. We then measured the effect of surfactant kinetics on the droplet volume Vd in the simulations (see Fig. 3). First, looking at surfactant-free system, we notice that the curve is divided by a slope-break near Ca0 = 7 × 10−3 , likely to be the transition from the squeezing

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U/Umax 0

(a)

0.1

0.2

(b)

t = 400 ms 0.15 w% Umax = 3.8 mm/s

(d)

0.5

t = 592 ms 0.15 w% Umax = 11.9 mm/s

t = 405 ms 0.5 w% Umax = 4.0 mm/s

0.6

0.7

0.8

0.9

1

t = 783 ms 0.15 w% Umax = 15.5 mm/s

(f)

t = 613 ms 0.5 w% Umax = 11.4 mm/s

(h)

t = 405 ms 1.5 w% Umax = 3.9 mm/s

0.4

(c)

(e)

(g)

0.3

t = 798 ms 0.5 w% Umax = 14.6 mm/s

(i)

t = 587 ms 1.5 w% Umax = 12.3 mm/s

t = 759 ms 1.5 w% Umax = 15.1 mm/s

Figure 2: micro-PIV experimental results (QC = 100 µL/min, QD = 25 µL/min, Ca0 = 4.3 × 10−5 ). Droplet a different formation stage. (a, b, c) —with span 80 at 0.15 w%, (d, e, f ) —with span 80 at 0.5 w%, (g, h, i) —with span 80 at 1.5 w%. The insets in figures (a, b, c) are enlargements of the flow region in red showing the recirculation eddy. The droplet contour are guides for eyes.

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1.8 α=1 α=0.1 α=0.01 no surfactant isosurfactant

1.6

1.4

Vd/L2

1.2

1

0.8

0.6

0.4

0.2 −3 10

−2

−1

10 Ca0

10

Figure 3: Simulated non-dimensional droplet volume versus Ca0 at various surfactant adsorption rates. Lines are guides for eyes.

span80 1.5w% span80 0.5w% span80 0.15w% Isosurfactant

1.6

1.4

V˜d

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1.2

1

0.8

0.6 −3

−2

10

10

−1

10

Castat

Figure 4: Numerical (isosurfactant) and experimental (span80) non-dimensional droplet volume versus Castat at various surfactant concentrations. In experiments (3D) V˜d = Vd /L3 and in simulations (2D) V˜d = Vd /L2 . Lines are guides for eyes.

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regime to the dripping regime. 41 When adding surfactant, the droplet size is increasingly reduced for faster surfactant adsorption. We also notice that the slope-break is less obvious, as if the microdevice was always operating under the dripping flow regime. In order to facilitate comparison with experimental studies where the same surfactant is used under different flow conditions, we added an isosurfactant case. In this scenario, Ca0 is tuned by changing the inlet velocity while the surfactant adsorption rate remains unchanged. Therefore, the Da0 increases with Ca0 , and this trend was obtained by 2D-linear interpolation of the Ca0 , α, Vd dataset along the line α/αref = Caref /Ca0 . Hence, at low Ca0 , the surfactant adsorbs much faster than the droplet formation characteristic time, whereas as the Ca0 is increased, the formation frequency ramps up and the adsorption becomes relatively slower (decreasing the value of α). Interestingly, although a stark contrast between the different Da0 was obtained, isosurfactant curves are flatter and therefore likely to undergo smaller variations for different surfactant adsorption speeds. In figure Fig. 4, we compare this isosurfactant curve to experimental results obtained with the octane-water system. Since, due to numerical stability constraints, the interfacial tension in our model could only decrease by a factor of 2, whereas in experimental studies surfactant adsorption can decrease the interfacial tension by more than an order of magnitude, 4,5,7 we compared experimental and numerical results using the capillary number computed from the static interfacial tension σstat (obtained at thermodynamic equilibrium) instead of the dynamic interfacial tension. Then, the droplet sizes in simulations and experiment are in good agreement until Castat ' 0.01. Beyond this point, experimental droplet volumes drop more sharply until the onset of the jetting flow pattern. Our numerical model delays this flow transition until much higher capillary numbers. We believe that the accumulation of surfactant at the droplet tip may result in tip streaming, 21,22 a phenomenon not easily captured by our model due to the relatively small decrease of interfacial tension in the simulations. Nonetheless, the good qualitative agreement on the curve shape opens the prospect to establish a clear quantitative agreement once the numerical stability issue is

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overcome. 2.5

2

1.5 L∆P/σ0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1

0.5

0 −3 10

α=1 α=0.1 α=0.01 no surfactant isosurfactant −2

10 Ca0

−1

10

Figure 5: Continuous phase pressure fluctuations amplitude versus Ca0 at various surfactant adsorption rates. Lines are guides for eyes.

In addition to the droplet volume, various numerical 41,55 and experimental 60–62 studies have characterized the time-dependent pressure drop ∆P = Pin − Pout across the forming droplet (see Fig. 4 in the supporting information for the location of the points to measure Pin and Pout ). As the droplet grows asymmetrically in the continuous phase channel, it experiences an increasingly large pressure difference. Beyond the droplet size (given by the signal frequency), these pressure fluctuations also give a waveform and an amplitude, which may all contain useful data to estimate the channel flow pattern, and potentially dynamic interfacial tension. 8 A sample of these waveforms is available in the supporting information for Ca0 = 1.8 × 10−3 and 0 < α < 1. We report in Fig. 5 the maximum amplitude of the pressure fluctuations ∆P as a function of the capillary number. Several important features are present: the pressure fluctuations amplitude decreases beyond a given Ca0 , the transition threshold follows an opposite trend to the adsorption rate (early transition 15


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due to the adsorption of surfactant) and the isosurfactant curve starts at a small value and progressively increases until it reaches a maximum value just prior to the squeezing/dripping transition, and then decreases. Despite a good qualitative agreement between this figure and previous results, 60,61 it also highlights a limit of 2-D simulations: the transition from squeezing to dripping, which should occur near Ca0 ≈ 0.01 happens at Ca0 ≈ 0.03 for the surfactant-free system. This higher value might be related to some specificities of the 3-D droplet formation process, which includes the action of flow bypass around the droplet and spontaneous break-up prior to the complete squeezing. 63 Another discrepancy appears when comparing these results to the experiments of Wang and al. 8 where the authors measured the pressure fluctuations in microchannels containing surfactant. In this study, the pressure curves exhibit two inflexions: ∆P first grows, then decreases and increases again. The authors related this inflection to a switch of the surfactant mass-transfer mechanism from a quasi-diffusion mode (where the continuous phase stagnates behind the forming droplet) to a convection mode where the flow circulates around the droplet as shown in Fig. 1. This mass-transfer consideration is out of the scope of the current model since we assumed a constant adsorption kinetic given by the Damköhler number.

Volume and pressure-derived dynamic interfacial tension The rapid dynamics of microfluidics offer a powerful tool to investigate fast surfactantrelated phenomena. Since it is difficult to observe directly the surface coverage of forming droplets, most studies rely on indirect measurements such as the droplet size or the pressure fluctuations in microchannels to unveil the surfactant dynamics. In the following, we mimic these two experimental approaches to determinate a posteriori the interfacial tension of the forming droplet and thus the dynamic molecular coverage. The simplest approach to determine the dynamic interfacial tension is to use the droplet size. The basic assumption is that the droplet detaches when the shear rate overcomes the capillary forces 64–67 so that the final droplet size depends only on the (dynamic) capillary 16


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number at the break-up moment CaV = µC UC /σV . Thus, the droplet volume yields the terminal dynamic interfacial tension. In turn, this hypothesis of similar flow pattern requires neglecting the Marangoni effect by assuming a homogeneous distribution of surfactant over the droplet surface. Previous studies 4–6,14,68 provide a framework to estimate the dynamic interfacial tension under this uniform interfacial tension approximation (UITA). First, an empirical function Vˆd is devised to estimate the experimental droplet size Vd in surfactant-free conditions 14,a We construct a piece-wise cubic Hermite interpolation between the surfactant-free droplet volume and the capillary number:

Vˆd : Ca0 → Vd (Ca0 )|surfactant free .

(3)

Next, we estimate the CaV that would be required under normal conditions to obtain the same empirical droplet size Vd . Since we kept the fluid speed and viscosity unchanged, this capillary number yields the value of the volume-estimated interfacial tension σV = σ0 Ca0 /CaV : Vd (Ca0 ) Vˆd (CaV ) = . L2 L2 surfactant

(4)

Solutions of Eq. (4) are presented in Fig. 6. Similarly to previous works, smaller droplets (formed over a shorter period of time) exhibit a higher dynamic interfacial tension. However, this terminal dynamic interfacial tension does not necessarily reflect the surfactant coverage during the droplet growth: it is an average value over space measured at a single point in time. For instance, Fig. 6 indicates that at α = 1 and Ca0 = 1.8 × 10−3 , the volumederived dimensionless interfacial tension σV /σ0 is 0.1 and the coverage (computed from Eq. a

Due to wettability constraints (not considered in the simulation), surfactant-free droplet formation may be unstable, and the authors used surface-saturated droplets as the reference in some studies. 4,6 In UITA framework, the two hypotheses are very similar since under supersaturated conditions, the droplet surface is assumed to get immediately saturated and the interfacial tension should be a constant all along the droplet formation process, in an analog way to surfactant-clean droplets.

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1 α=1 α=0.1 0.8

α=0.01 no surfactant isosurfactant

σv/σ0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.6

0.4

0.2

0 −3 10

−2

10 Ca0

−1

10

Figure 6: Estimated dynamic interfacial tension versus capillary number from the volume data, based on the surfactant-free droplet taken as reference. The highest capillary number of α = 1 was not regressed because the corresponding volume was significantly lower from all measured values of surfactant-free droplets. Lines are guide for eyes.

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1) is Γ ' 0.978. This value is slightly higher than the Γ0sat = 0.9 set in the stimulations. Moreover, according to a series of snapshots of the droplet formation over time available in the supporting information, Γmax jumps from 0.91 to 1.13 at the breakup moment, leaving some uncertainty on the final coverage of the droplet. Despite this lack of accuracy, both methods agree to similar coverages within a few percent which tends to support previous mass transfer studies. 14 1

U/U max

0.8 0.6 = 0.0 !0 = !ref / 8 Umax = 0.010

= 0.01 !0 = !ref / 4 Umax = 0.013

= 0.1 !0 = !ref / 2 Umax = 0.012

= 1.0 !0 = !ref Umax = 0.014

0.4 0.2 0 "/ 1 "max

0.8 0.6 = 0.0

"max = 0.0

= 0.01 "max = 0.22

= 0.1 "max = 0.70

= 1.0 "max = 0.90

0.4 0.2 0

Figure 7: Simulation snapshots after 30 × 103 t.s with different surfactant kinetics and identical droplet volume Vd /L2 = 1.2. From left to right, the pictures correspond to progressive increase in surfactant kinetics (from 0 to 1). (a, b, c, d) show the velocity field (arrows) and speed magnitude (colors), droplet boundaries are indicated by the rainbow curve net. (e, f, g, h) indicate the surfactant density distribution, droplet boundaries are depicted by the white line.

In order to better understand the effect of the surfactant on the droplet volume, we compare in Fig. 7 several forming droplets at the same instant that will have the same terminal volume at break-up Vd /L2 = 1.2 despite different adsorption kinetics. This is achieved by tuning the capillary number Ca0 . In spite of the highly similar geometry of the forming droplets, we notice some differences in the flow patterns. Looking at the surfactantfree droplet, the speed magnitude evolves gradually in the whole droplet, indicating a viscous dissipation within the dispersed phase, whereas for surfactant-rich systems, the gradient is absent and the whole droplet moves at a uniform speed. If the droplet formation was exclusively controlled by an interfacial tension to viscous force balance, where a physical law

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would relate the droplet size to the capillary number, identical Vd should have the same flow pattern and viscous dissipation, which is not the case here. On the opposite, it appears that an intense Marangoni effect prevents the viscous dissipation within the droplet and makes it move like a solid. An alternative method 8 to determine the dynamic interfacial tension is to use the pressure fluctuations amplitude. Once again, according to the UITA, the droplet formation process is similar to the surfactant-free droplets ∆P if the pressure fluctuations have the same magnitude. We first compute a piece-wise cubic Hermite interpolation between the surfactant-free droplet pressure fluctuations and the capillary number. This interpolation relates the pressure fluctuations (and more importantly the associated droplet shape) to the capillary number for surfactant-free droplets:

∆Pˆ : Ca0 → ∆P (Ca0 )|surfactant free .

(5)

Similarly to Eq. (4), we solve the following dimensionless equation: L∆P (Ca0 ) L∆Pˆ (CaP ) = . σP σ0 surfactant rich

(6)

The pressure-estimated interfacial tension, shown in Fig. 8, is at odds with the monotonic trend of the volume-derived interfacial tension (Fig. 6). Indeed, σP starts by increasing slightly (region (a)), then decreases sharply (region (b)) and finally converges towards the surfactant-free value (region (c)). The increase in region (a) and the decay in region (b) are an expected consequence of a lower interfacial tension, which shifts the pressure curves in Fig. 5 to the left. Indeed, previous studies have shown that the transition from squeezing to dripping is marked by the uptake of shear forces over the capillary stress. 69 Consequently, such a transition indicates a weakening of the interfacial tension and is marked by a dramatic reduction of the pressure fluctuations amplitude. 41,60 We interpret the convergence in region (c) to the fast decay in pressure fluctuations amplitude (see Fig. 5) so that a wide range

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1.5 =1 =0.1 =0.01

0

no surfactant isosurfactant

1

/

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.5

(a) 3

(b)

(c) 2

10

10 Ca0

1

10

Figure 8: Estimated dynamic interfacial tension versus capillary number from the pressure data, based on the surfactant-free droplet taken as reference. Lines are guide for eyes. of ∆P correspond to a narrow range of dynamic interfacial tension. However, region (a) overshoot (σP > σ0 ) is hardly explainable from the uniform coverage standpoint.

Stagnant cap A more accurate approximation for multiphase flow with surfactant is the stagnant cap theory 70 where the droplet interface is divided into a mobile and a rigid region. The former is free of surfactant and the latter is completely saturated (see Fig. 9). This theory has been successfully applied to investigate the abnormal buoyant motion of surfactant-covered bubbles or oil droplets in water columns. 25,71–76 Such a stagnant cap is clearly observed in Fig. 1 and Fig. 7. In the following, we interpret how this stagnant cap affects the droplet size and the pressure fluctuations. According to several studies 63,65,66,69,77 on microfluidic emulsification, the droplets are formed in T-junctions microchannels by the continuous phase squeezing the dispersed phase

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Uniform Interfacial Tension Approximaton (UITA)

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Stagnant Cap Approximation

Stagnant cap

Higher shear rate

Lower interfacial tension

Figure 9: Two hypotheses proposed to explain the volume change of droplets for different surfactant coverage. In the first case (on the left), the interfacial tension is uniformly reduced by the surfactant on the whole droplet, and the shear rate keeps constant, whereas in the second case (on the right) a stagnant cap increases the drag force without large change in interfacial tension near the neck of the droplet. against the channel edge. Therefore, the droplet volume is the amount of fluid required to fill and obstruct the channel plus the volume of dispersed phase allowed to flow through during the squeezing time. In the simplest approximation, the squeezing time is the time required for the continuous phase to fill a certain squeezing volume. However, since the droplet does not completely obstruct the channel, some of the continuous phase can bypass the droplet and does not contribute to the squeezing. 77 When using surfactants, the droplet is rigidified which increases the hydrodynamic resistance of the bypass channels 7 and thus increases the fraction of continuous phase that contributes to the squeezing (see Fig. 1). Thus, the stagnant cap of the droplet reduces the squeezing time and the final volume of the droplet to a greater extent than the sole effect of a reduced interfacial tension. The stagnant cap theory also gives some insight about the overshoot of the pressureestimated dynamic interfacial tension. The initial increase of σP shown in Fig. 8 is only happening at low to moderate surfactant coverage (0.01 < α = 0.1) where a stagnant cap

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is well defined (see Fig. 1). Even for the highest coverage Γ = 0.6, the interfacial tension is expected to decrease only to 0.78σ0 . Therefore, the geometry of the droplet remains essentially the same. However, the stagnant cap prevents an efficient flow recirculation in the droplet. This immobilization increases the drag coefficient of the drop as shown in earlier studies on buoyant droplets in fluid columns. 25,71–76 It was also observed that this increased drag coefficient amplifies the pressure drop across bubbles flowing in microchannels at intermediate surfactant concentrations. 78 We believe the same phenomenon is responsible for the large amplitude of the pressure fluctuations and therefore for the overshoot of the pressure-estimated interfacial tension.

Conclusion Up to now, most studies of microchannel emulsifications have assumed a uniform surfactant coverage. Although this paradigm provides reliable results regarding the droplet size, some discrepancies between surfactant-free derived models and experimental pressure and droplet size measurements point to the limits of the model. In this paper, we used a two-dimensional numerical simulation to analyze the distribution of surfactant on growing droplets in Tjunctions microchannels. These simulations allow to flexibly tune the capillary and the Damköhler numbers. Mimicking experimental approaches, we showed that assuming a uniform interfacial tension over the droplet surface deviated from the average values of the dynamic interfacial tension. In addition, while droplet volume and pressure fluctuation amplitude are indicative of the total shear rate over the droplet surface, they should be combined into a new physical model to gain understanding on the droplet coverage and the extent of the stagnant cap. Our study highlights that comparatively to open media, the very high shear rate and the high droplet formation frequency of microfluidics lead to a stagnant cap when the surfactant adsorption kinetics are slower than the droplet formation rate. Under these conditions, the Marangoni effect hampers the flow circulation around the droplet and

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modifies the resulting force balance. Another effect of surfactant is to increase the contact angle of the dispersed phase. Although this effect was neglected in the current study by restricting the contact angle to 180o , a natural continuation to this work would therefore consider the effects of surfactant on the dynamic contact angle and consequently on the droplet size. Beyond these engineering issues, our study indicates that microfluidic emulsification allows reaching very high surface pressure in order to study the non-equilibrium thermodynamics of surfactants, including exotic two-dimensional phase transitions. 79–81

Acknowledgement The authors gratefully acknowledge the support of the National Natural Science Foundation of China (21776150, 91334201, U1302271) and the Fondation pour la Recherche Medicale (SPF20160936257).

Supporting Information Available Schematics of the experimental setup, description of the numerical model implementation and validation.

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Graphical TOC Entry With surfactant Surfactant-free

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Umax = 0.024 = 0.0

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Numerical Study of Surfactant Dynamics during Emulsification in a T

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