Capillary focusing close to a topographic step: Shape and instability of confined liquid filaments
MMicrofluidics and Nanofluidics manuscript No. (will be inserted by the editor)
Capillary focusing close to a topographic step: Shape andinstability of confined liquid filaments
Michael Hein · Shahriar Afkhami · Ralf Seemann · Lou Kondic
Received: date / Accepted: date
Abstract
Step-emulsification is a microfluidic techniquefor droplet generation which relies on the abrupt de-crease of confinement of a liquid filament surroundedby a continuous phase. A striking feature of this ge-ometry is the transition between two distinct dropletbreakup regimes, the “step-regime” and “jet-regime”,at a critical capillary number. In the step-regime, smalland monodisperse droplets break off from the filamentdirectly at a topographic step, while in the jet-regimea jet protrudes into the larger channel region and largeplug-like droplets are produced. We characterize the
M. HeinSaarland UniversityExperimental Physics, Saarland University, 66123Saarbr¨ucken, GermanyTel.: +49-681-302 71704Fax: +49-681-302 71700E-mail: [email protected]. AfkhamiNew Jersey Institute of TechnologyDepartment of Mathematical Sciences, New Jersey Instituteof Technology, Newark NJ 07102, USATel.: +1-973-596-5719Fax: +1-973-660-6467E-mail: [email protected]. SeemannSaarland UniversityExperimental Physics, Saarland University, 66123Saarbr¨ucken, GermanyTel.: +49-681-302 71799Fax: +49-681-302 71700E-mail: [email protected]. KondicNew Jersey Institute of TechnologyDepartment of Mathematical Sciences, New Jersey Instituteof Technology, Newark NJ 07102, USATel.: +1-973-596-2996Fax: +1-973-660-6467E-mail: [email protected]
400 µm w ¥ Jet d (a)
400 µm w ¥ Step (b)
Fig. 1
Optical micrographs showing two regimes of dropletproduction. (a) “Jet-emulsification” for large Ca: Breakupoccurs downstream, (b) “Step-emulsification” for low Ca:Breakup occurs at the step breakup behavior as a function of the filament geome-try and the capillary number and present experimentalresults on the shape and evolution of the filament for awide range of capillary numbers in the jet-regime. Wecompare the experimental results with numerical sim-ulations. Assumptions based on the smallness of thedepth of the microfluidic channel allow to reduce thegoverning equations to the Hele-Shaw problem withsurface tension. The full nonlinear equations are thensolved numerically using a volume-of-fluid based algo-rithm. The computational framework also captures thetransition between both regimes, offering a deeper un-derstanding of the underlying breakup mechanism.
Keywords
Drops and Bubbles · Step-Emulsification · Capillary Focusing · Hele-Shaw Flow · Volume-of-Fluid
Hydrodynamic instabilities leading to droplet forma-tion by a decay of a liquid jet or filament have re-ceived considerable interest over the last two centuries(Rayleigh 1878), renewed especially since the emergenceof two-phase microfluidic systems. Hydrodynamic in- a r X i v : . [ phy s i c s . f l u - dyn ] S e p M. Hein et al. stabilities are employed to generate droplets on themicro-scale, a technique that is promising for many lab-on-a-chip applications, where droplets could serve asdiscrete vials for chemical reactions or bio-analysis. Ex-tensive reviews on experimental and theoretical studiesof droplet production units and droplet handling in mi-crofluidics have been written by Seemann et al. (2012)and W¨orner (2012). Typically three different dropletproduction units are being used: In a T-Junction (Thorsenet al. 2001), in which the dispersed phase enters a chan-nel filled with a continuous phase from a side channel,droplets are typically produced either by shearing orby a “Plug-and-Squeeze mechanism” (Garstecki et al.2006). In co-flow or cross-flow geometries, which havereceived considerable interest from an experimental aswell as from analytical perspective (Anna et al. 2003;Guillot et al. 2007, 2008), the dispersed phase flows par-allel to the continuous phase and decays into dropletseither directly at the inlet (“Dripping”), forms a jet thatdecays some distance downstream (“Jetting”) or formsa liquid jet, which is absolutely stable (“Co-Flow”). Allof these methods are widely used.The instability of a liquid filament confined in aquasi two-dimensional geometry triggered by a suddenexpansion of the channel has recently been described(Priest et al. 2006; Malloggi et al. 2010; Humphry et al.2009; Shui et al. 2011; Dangla et al. 2013). The systemconsists of a shallow terrace and a larger microfluidicreservoir downstream, as shown in Fig. 1. In the terrace,the non-wetting dispersed liquid flows as a straight fil-ament confined at the top and the bottom by the wallsand surrounded by a continuous phase. The main con-trol parameter is the capillary number, Ca, which rep-resents the effect of viscous relative to capillary forces.Two distinct droplet breakup mechanisms can be ob-served for droplet generation. At high Ca, a mecha-nism called “jet-emulsification” occurs, in which thedispersed phase creates a stable tongue in the terrace.This tongue narrows to a neck close to the topographicstep due to capillary focusing. From this neck, a jet pro-trudes, generating large and polydisperse droplets thatfill the whole reservoir channel (see Fig. 1a). Below acritical Ca, the so called “step-emulsification” occurs(see Fig. 1b). In the latter case, the inner stream formsa droplet immediately after it reaches the topographicstep, where the tip then becomes unstable, producingdroplets at high throughput with a monodispersity su-perior to the previously mentioned droplet productionmethods (Priest et al. 2006; Dangla et al. 2013). Re-cent efforts to describe this emulsification process haveresulted in models for predicting lower bounds for thegenerated droplet size in the step-regime (Dangla et al.2013) and estimating the width of the tongue at the step in the jet-regime (Malloggi et al. 2010). However, rig-orous study of the shape and instability of the confinedtongue and comprehensive comparison of experimentalresults with theoretical models have not been carriedout until now.In this article, we study the capillary focusing phe-nomena in a step-emulsification microfluidic device. Onone hand, the dynamic filament shape and stability inthe jet-regime is experimentally analyzed for a wideparameter range. On the other hand, a computationalframework is developed to solve the full Hele-Shaw (H-S) equations and to directly compute the shape of theliquid-liquid interface in the terrace, providing a betterunderstanding of the observed experiments. By directcomparison of the numerical results with experiments,we show that the shape of the liquid filament in the jet-regime can be computed as a function of Ca and theflow rate ratio between the dispersed and continuousphase, without resorting to any undetermined parame-ter.Here, a simple stability-criterion is used to showthat the experimentally observed transition betweenthe step- and jet-regime can also be predicted by thenumerical simulations; the description of this transitionbased on the direct numerical solution of the two-phaseH-S equations with surface tension has not been ad-dressed before.
To characterize the droplet breakup regimes in a step-emulsification geometry and the filament shape in thejet-regime, experiments are performed using a devicemicromachined into a polymethylmethacrylate (PMMA)-block with a terrace of width w = 400 µ m and depth b = 30 µ m (i.e. with aspect ratio w/b = 12 .
76) and adownstream reservoir of cross-sectional area of about1 × . The channel is sealed by a PMMA-sheet.An aqueous phase with 28 wt% glycerol is injected as adispersed phase into an oily continuous phase, IsoparM(ExxonMobil Chemical). For the chosen liquid system,the viscosity of the dispersed phase, µ , is matched withthe viscosity of the continuous phase, µ , and is ap-proximately 2 . γ , determined using thependant drop method, is 3 . ± . w ∞ ,far away from the neck, and the neck width, δ , directly hape and instability of confined liquid filaments 3 at the step, are automatically measured using numeri-cal image analysis (see Fig. 1a). Volumetric flow rates, Q and Q , of the dispersed and the continuous phase,respectively, are adjusted using computer controlled sy-ringe pumps. The corresponding Ca = U µ /γ is deter-mined from the average flow velocity of the dispersedphase, U . In addition to physical experiments, a computationalframework is developed to provide a deeper insight intothe effects governing the evolution of the liquid-liquidinterface in the jet-regime and to predict the transitionbetween the jet- and step-regime. The original govern-ing equations are reduced to the time-dependent two-phase Hele-Shaw equations with surface tension. Westress that, even in this case, the full problem is non-linear due to the curvature term in the surface tensionand therefore the full problem has to be solved numer-ically. Furthermore, although not considered here, ournumerical model allows to seamlessly vary the viscosityof both phases as well as the initial condition and thegeometry of the Hele-Shaw problem. We use a volume-of-fluid (VOF) based method to directly solve the gov-erning equations. Our numerical model, described indetail by Afkhami and Renardy (2013), has the distinc-tive feature of being capable of, accurately and robustly,modeling the surface tension force. In addition, the ac-curate interface reconstruction, the second-order cur-vature computation, and the use of adaptive mesh re-finement allow for resolved description of the interface,enabling us to investigate the complex features of in-terface profiles extracted from experiments, which wasnot possible in prior studies. The order of accuracy andconvergence properties of the numerical methods werepreviously studied by Afkhami and Renardy (2013).The computational framework consists of the clas-sical H-S model to simulate the shape of the interfacebetween the dispersed and the continuous phase in theshallow channel. The depth-averaged velocity field inboth phases is defined as u ( x, y, t ) = b µ {−∇ p ( x, y, t ) + F ( x, y, t ) } , (1)where µ is the viscosity of the considered phase definedas µ ( f ) = µ µ (1 − f ) µ + f µ , (2)where µ and µ refer to the viscosity of fluid 1 and 2,respectively, b is the depth of the H-S cell, and p ( x, y, t )is the local pressure. The VOF function, f ( x, y, t ), tracks y x Symmetry Line
Fluid 1 ( L, δ ) Fluid 2 ∂p∂y = 0(0 , − w ∞ ) Reservoir ∂p∂y = 0 ( L, w ) p = p (0 , ∂p∂x = − µQ b w ∞ ∂p∂x = − µQ b w ∞ Fig. 2
Schematic of the flow domain for the Hele-Shawmodel and the corresponding boundary conditions. The do-main is bounded by walls at | y | = w/
2. At x = 0, fluid 1occupies | y | ≤ w ∞ / w ∞ / < y ≤ w and − w < y < − w ∞ / w ∞ = w − w ∞ the motion of the interface. In a VOF method, the dis-crete form of the function f represents the volume frac-tion of a cell filled with, in this case, fluid 1. Away fromthe interface, f = 0 (inside fluid 2) or f = 1 (insidefluid 1); “interface cells” correspond to 0 < f <
1. Theevolution of f satisfies the advection equation ∂f∂t + u ( x, y, t ) · ∇ f = 0 . (3)In this formulation, surface tension enters as a singularbody force, F ( x, y, t ), centered at the interface betweentwo fluids (Afkhami and Renardy 2013). Figure 2 is aschematic of the flow domain, 0 ≤ x ≤ L , | y | ≤ w/ p , in the reservoir and, as a first approximationfor the outflow into a reservoir, an outflow pressureboundary condition at the exit, x = L , as p ( x, y, t ) = (cid:26) p + 2 Aγ/b < f ≤ p f = 0 (4)where the out-of-plane curvature is given by 2 /b . Thus2 γ/b is the Laplace pressure inside the dispersed phase1. Since the in-plane curvature at the step is typicallymuch smaller than the out-of-plane curvature, it canbe safely neglected. A pressure correction parameter, A ≤
1, is included for modeling the flow configura-tions for which the H-S approximation is expected to beless accurate, as discussed further below. The boundarycondition for f at the top and bottom walls is f = 0.At the outflow, the boundary condition for f is theinterface being perpendicular to the topographic step: ∂f /∂x = 0. This is justified experimentally in section4.1. On the solid boundaries (side walls), ∂p/∂y = 0. At M. Hein et al. the inlet, x = 0, the parallel flow solution holds, mean-ing that for both phases, the pressure gradient is pre-scribed as ∂p ( x, y, t ) i /∂x = − (12 µQ i ) / ( b w i ∞ ) where i = (1 , w i ∞ is the width occupied by the i th liquidat the inlet, and Q i is the flow rate of the i th phase. Asin the experiments, we chose dp /dx | x =0 = dp /dx | x =0 and fix b and w . The initial condition at t = 0 is a flatinterface defined by the initial distribution of f overthe domain. We keep L large enough so the results areunaffected by its value. δ/w ∞ , called the tongue shape factor, as afunction of time. As shown, at the beginning of thedroplet formation cycle (see Fig. 3a and regime 1 inFig. 3e), when the droplet is initially pushed into thereservoir, δ/w ∞ remains constant while the droplet in-flates, maintaining a nearly spherical shape. During thisinflation process, the liquid-liquid interface at the stepis perpendicular to the topographic step (see the in-set in Fig. 3a). The Laplace pressure that counteractsthe inflation of the droplet rapidly decreases as thedroplet radius increases and remains merely constant (a) t=0.2 s (b) t=2.73 s
400 µm (c) t=4.49 s (d) t=7.68 s d / w ¥ time (s) (e) regime 2 regime 3regime 1 Fig. 3
Temporal evolution of the tongue in the jet-regime.(a)-(d) Micrographs with the inset showing the interface di-rectly at the step and (e) the shape factor, δ/w ∞ , as a func-tion of time; Ca = 9 . × − and Q ∞ /Q ∞ = 3 when the droplet reaches the reservoir dimension. Addi-tional complexity may arise in the case of droplet colli-sion in the reservoir channel. When the droplet touchesthe walls of the reservoir, δ/w ∞ typically increases asthe droplet starts to travel downstream, pulling thetongue towards the step (see Fig. 3b-c and regime 2in Fig. 3e). Additionally, the droplet blocks the reser-voir, except for small regions along the corners of therectangular reservoir. Thus, an additional pressure con-tribution, that is expected to increase with the dropletlength (Labrot et al. 2009), is needed to push the con-tinuous phase at constant flow rate downstream in thereservoir. This pressure component is also expected todeform the interface in the shallow terrace. Thus δ/w ∞ reaches a maximum during regime 2 in Fig. 3e and thendecreases, showing the formation of a neck, that mayevolve to a stable width in regime 3 in Fig. 3e. In regime3, the neck elongates as the rear interface of the droplettravels downstream, until it bulges and finally ruptures(see Fig. 3d and regime 3 in Fig. 3e). After dropletbreakup, the process repeats starting in regime 1. Fromthe above description of the tongue evolution in the jet-regime at high Ca, it is obvious that only in regime 1 thepressure boundary condition at the topographic step isknown a priori, resembling an open outlet with a con-stant pressure, p . Thus the experimental data taken inregime 1 is used for further comparison with computa-tional results. In regime 2 and 3, however, the pressure hape and instability of confined liquid filaments 5 Fig. 4
Phase diagram demonstrating the transition of jet-regime (red triangles) to step-regime (black circles). Blueopen squares denote the transition regime, where droplet pro-duction alternates between step- and jet-regime. Blue solidsquares indicate the simulation results for A = 0 .
4, as dis-cussed in the text. For w ∞ /b ≤ at the topographic step is significantly influenced by theevolving droplet in the reservoir. We note that regime1 can be further expanded by increasing reservoir di-mensions.4.2 Transition from jet- to step-regimeA sudden transition to the step-regime occurs whenreducing the Ca below a certain threshold, as shownin Fig. 4. The transition between jet- and step-regimecan be qualitatively described as follows: At low Ca,when the width of the neck, δ , becomes comparable tothe height of the terrace, b , the dispersed phase formsa nearly cylindrical neck, which is prone to a surface-tension-driven instability. As the neck is no longer sta-ble, the filament breaks up rapidly and small dropletsare continuously produced, accompanied by a periodicretraction of the tongue. We have confirmed this crit-ical neck width experimentally by measuring δ in themetastable transition regime, where step- and jet-regimealternate (see shaded region in Fig. 4). Interestingly, de-spite the qualitative differences in tongue shape and sta-bility between step- and jet-regime (cf. Fig. 1), we notethat the tongue width far upstream from the step, w ∞ ,stays constant for a given Q ∞ /Q ∞ when varying Ca.In particular, w ∞ is unaffected by the transition fromthe jet- to step-regime, which allows to characterize thetransition as a function of the rescaled tongue width, w ∞ /b , and Ca. Figure 4 shows the emulsification phase y [ µ m ] x [µm]Experiment Simulation y [ µ m ] x [µm] (a)(b) w2 ¥ d /2w2 ¥ d /2 Fig. 5
Comparison of the tongue profiles in the terrace ob-tained from experiments (black solid line) and simulationswith A = 1 (red dashed line). Both experimental profiles areobtained in the jet-regime. (a) Ca = 0 .
019 and w ∞ /b = 7 . .
042 and w ∞ /b = 7 .
44. For experiments, Q ∞ /Q ∞ = 1 .
5. For simulations, Q ∞ /Q ∞ = 1 .
67 in (a)and Q ∞ /Q ∞ = 1 .
41 in (b) diagram in ( w ∞ /b, Ca) space, with an increased datarange compared to Priest et al. (2006), in which it isshown that the phase diagram is insensitive to the vis-cosity ratio of the dispersed phase to the continuousphase, µ /µ . We note that for w ∞ /b ≤
1, dropletbreakup occurs at the inlet of the dispersed phase. Thisregime is comparable to breakup in a T-junction and isnot considered here.4.3 Comparison of experiments with numericalsimulationsFigure 5 shows the direct comparison of the tongue pro-files obtained from experiments with the correspondingsimulations. As shown, the numerical simulations accu-rately reproduce the main features of the tongue shape,particularly the slight increase of the tongue width fol-lowed by the strong capillary focusing close to the topo-graphic step. We also note that the agreement betweenthe simulated results and experiments is best at higherCa values, shown in Fig. 5b, remote from the transitionto the step-regime, where the tongue width is larger andthe validity of the H-S approximation (assumed in the
M. Hein et al.
Fig. 6
Comparison of the experimentally determined shapefactor, δ/w ∞ , for Q ∞ /Q ∞ = 3 (black squares) and thesimulation results (red triangles) for Q ∞ /Q ∞ = 2 .
81 and A = 1 numerical scheme) is clearly fulfilled. To quantify thecomparison between the numerical and experimental re-sults, in Fig. 6, the shape factor, δ/w ∞ , measured fromthe obtained profiles, is plotted as a function of Ca. Theerror bars for the experimentally determined δ/w ∞ areworst case estimates based on the width of the inter-face after image segmentation, and the error bars for Caare Gaussian error estimates. The measured values for δ/w ∞ agree within experimental accuracy with the re-sults from the corresponding simulations, especially formedium to high Ca. Both the results from the experi-ment and the simulation show that the neck thicknessincreases with Ca, as also predicted analytically by Mal-loggi et al. (2010). However, for Ca (cid:46) .
02, close to thetransition, simulation predicts a stronger focusing, i.e. astronger decrease of δ/w ∞ with Ca, when compared tothe experimental data, as the neck becomes more andmore cylindrical in this regime, making the H-S ap-proximation less accurate. Thus the numerical modelunderestimates δ/w ∞ for small Ca. This underestima-tion can be compensated by introducing a pressure cor-rection, A , as described in Eq. 4, which increases δ byweakening the capillary focusing, at the cost of a re-duced bulging prior to the hydrodynamic focusing. Weapply this pressure correction and use the stability cri-terion, δ = b , as described in section 4.2, and employ thenumerical simulation to predict the transition from jet-to step-regime. Figure 4 shows the excellent agreementof the computationally predicted transition thresholdwhen compared to experimental data for A = 0 .
4. Asdiscussed previously, introducing A is needed when con-sidering the regime where the H-S approximation is notstrictly valid. A clear approach to improve the accuracy is to consider fully three-dimensional flow, as well as toconsider possible influence of the viscosity contrast andcapillary effects on the pressure correction factor. Weleave these non-trivial extensions for future work. In this article, the capillary focusing of a confined liq-uid filament at a topographic step is studied both ex-perimentally and numerically. A novel computationalframework, based on the Hele-Shaw approximation andthe volume-of-fluid approach, is shown to predict exper-imental results quantitatively. We find that by mod-eling the reservoir with an applied pressure boundarycondition, the computed shapes of the filament are inexcellent agreement with experiments for moderate tohigh capillary numbers, without resorting to any unde-termined parameters. Furthermore, we show that thenumerical model can accurately predict the width ofthe tongue remote from the step. We also character-ize the width of the tongue at the step as a functionof Ca and show that the computational results are invery good agreement with the experimental findingsin the jet-regime and close to the transition to thestep-regime. Direct computations also provide an ac-curate estimate of the transition between two distinctdroplet breakup mechanisms characterized experimen-tally. This work is expected to set ground for furthernumerical and analytical treatment of confined filamentshapes in similar geometries. The presented computa-tional framework solves the full Hele-Shaw equationswithout additional simplifications. Thus the model isnot limited to predicting two-phase flow behavior in thespecific geometry discussed in this paper and can eas-ily be adapted to similar problems. Additionally, know-ing the dependence of the breakup behaviour on fila-ment geometry and capillary number will also facilitateapplication-specific designs of microfluidic droplet pro-duction units.
Acknowledgements
Authors gratefully acknowledge Dr. Jean-Baptiste Fleury (Saarland University) for helpful discussionsand the DFG-GRK1276 for financial support. This work waspartially supported by the NSF Grant Nos. DMS-1320037(S.A.) and CBET-1235710 (L.K.).
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