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Article Cite This: ACS Omega 2019, 4, 9800−9806

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Viscous Liquid Threads with Inner Fluid Flow Inside Microchannels Jaap Molenaar,† Willem G. N. van Heugten,‡ and Cees J. M. van Rijn*,‡ †

Mathematical and Statistical Methods, Wageningen University, Droevendaalsesteeg 1, 6708PB Wageningen, The Netherlands Micro Fluidics and Nano Technology, ORC, Wageningen University, Stippeneng 4, 6708WE Wageningen, The Netherlands

‡

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S Supporting Information *

ABSTRACT: Forming droplets are often accompanied by an interconnecting liquid thread. It is postulated that this phenomenon can only exist as long as a pressure gradient exists within the thread, for instance, when a viscous liquid is conveyed via the liquid thread to the forming droplet. We have built a microfluidic setup to form and sustain a liquid thread, which after a length L ends in a droplet. To prevent the droplet from moving up too fast due to buoyancy, we force the droplet to shift along a tilted ceiling, which can be positioned at three different angles. This enables us to keep the gradual lengthening of the liquid thread under control. Based on the Navier−Stokes equation, we are able to predict the axial shape of such a liquid thread as a function of fluid mass density, initial thread radius, initial fluid velocity at the nozzle, fluid viscosity, and surface tension. Although an explicit solution of the governing differential equations is not known, we managed to find an explicit approximating expression for the shape function, which shows excellent agreement with both the measured and the numerically calculated shape functions. An intriguing phenomenon observed in the experiments is the breakup of the thread. This breakup always occurs close to the droplet. Using our approximating solution, we derive a relation that connects, for any time in the development of the thread, its length and the pressure gradient stemming from, among other effects, the shear at the interface of the liquid thread due to motion of the inner liquid. For relatively short thread lengths, this relation is linear on a log−log scale, due to the fact that in this regime, viscosity effects are dominant. However, if the thread length increases, this relation starts to deviate from linear behavior, due to surface tension effects. We show from the experimental results that the thread starts to show unstable behavior as soon as these capillary effects come into play. We show how to predict the thread length at which the capillary instability sets in for any liquid thread system. It is found that the predicted maximum dimensionless thread length is given by Lmax,pred ≈ 12Ca with Ca the capillary number.

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limitations on thread length. In this paper, we first point out how to build a fluidic microchannel system for making stable liquid threads with a length slowly increasing in time (Figure 1). Next, we show how the thread shape is accurately described by a differential equation containing three dimensionless numbers: one representing internal viscosity effects, one representing surface tension effects, and one related to the viscous dissipation in the outer fluid. Eventually, we show how an upper bound on thread length can be predicted.

INTRODUCTION In many liquid thread and droplet formation processes, a general pattern is observed: droplets appear and coupled liquid threads quickly disappear. Fluid threads seem to be intrinsically unstable, and under the influence of the interfacial tension force, they tend to be readily transformed into droplets.1,2 One might anticipate that any sufficiently extended fluid element will always break up and convert into droplets. However, experiments and calculations have shown that this is not always the case.3−5 For example, Eggers has derived that a strongly stretched viscous element will never break up, as long as inertial forces can be neglected.6 Anomalous breakup processes with unexpectedly slender and stable viscous threads have also been reported.7−10 In recent years, microfluidic devices11 are progressively used to form droplets from liquid threads with techniques based on flow focusing,12−14 co-flow,15 crossflow,16,17 and microchannels.18−21 In most of these devices, a liquid thread is created to transport liquid toward a forming droplet, enabling the growth of the droplet, until a maximum is reached followed by pinching, and then the droplet formation process restarts. The breakup of the liquid thread normally appears downstream at a point close to the droplet. Liquid threads can be stabilized by regulating the inner fluid flow.22 Our experiments show that this indeed holds but only within © 2019 American Chemical Society

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EXPERIMENTAL SECTION

We used as inner fluid silicone oil (viscosity η = 373 mPa s and density ρ = 972 kg/m3) in aqueous 1% Tween 20 (interfacial tension γ = 5.0 mN/m) in a microfluidic device (10 × 10 × 120 mm3), tilted 7 or 20° or not tilted from the horizontal position. Inner fluid flows from a nozzle in the stationary outer fluid from the bottom of the device. The generated droplet floats to the tilted ceiling of the device still attached to the Received: March 22, 2019 Accepted: May 14, 2019 Published: June 4, 2019 9800

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Figure 1. Snapshots taken at times 24.90, 20.00, 14.90, 9.90, 4.90, and 0.1 s before breakup of a liquid thread developing in time. For a movie, see the Supporting Information. These pictures are taken from experiment no. 1, for which the parameters are given in Table 1. (a) The droplet touches the ceiling and starts to shift along it under the influence of buoyancy. (b−e) Both thread length and droplet volume grow in time. (f) Eventually, an instability occurs leading to breakup close to the droplet. Blue arrows (―▶) indicate the location where the initial liquid thread radius has doubled in size and defines the length L of the liquid thread as measured from the nozzle exit. Scale bars are 2000 μm.

∂ (R2(∂zv)) i1y ρ∂tv + ρv∂zv = −γ ∂zjjj zzz + 3η z +Q R2 kR{

nozzle via a liquid thread (see Figure 1 and the corresponding movie). Part of the buoyancy force is counteracted by the ceiling when the droplet reaches the ceiling. We focus on liquid threads in the viscous regime with a very small Reynolds number (Re = ρvR/η < 0.01) and a very small Weber number (We = ρv2R/γ < 0.1), with v being the (average) velocity and R being the initial thread radius. The corresponding capillary numbers Ca  We/Re are ≈ 1−10. We did experiments like the one in Figure 1 for six different experimental conditions. In Table 1, we summarize the parameter settings for these experiments. In this table, we also give the values of the governing dimensionless parameters α and β to be specified later on.

here, z is the coordinate along the axis of symmetry of the thread, R(z) is its radius, and ν(z) is the axial velocity at position z. These equations are obtained by averaging the velocity in the radial coordinate and further assuming that the radius of curvature at R(z) is equal to the radius of the thread at R(z). Equation 1 expresses the conservation of mass, and eq 2 expresses the conservation of impulse. In eq 2, we recognize terms representing inertia (left hand terms with ρ), surface tension (term with γ), and viscosity (term with η). In addition, there is factor Q, which represents all forces other than the ones related to inertia, surface tension, and viscosity. These other forces contributing to Q may comprise the following: (i) the pressure difference between the nozzle exit and droplet, (ii) the buoyancy force experienced by the droplet, and (iii) the friction force exerted by the surrounding fluid on the thread interface. The Laplace pressure inside the droplet can be derived from its (time-dependent) radius, but the pressure at the nozzle is not measured, so unknown. Effect (ii) is discussed in the Experimental Section and can be considered small. The friction and associated dissipation (iii) between the moving thread surface and the outer fluid contribution will create an additional pressure gradient between the nozzle and droplet. However, it is difficult to write down a simple expression for this contribution. In view of these uncertainties, we cannot model the remaining force Q explicitly. However, it is one of the key notions of this paper that we do not need such an explicit modeling to still arrive at insights explaining and predicting why and when the thread will break up. In section Fitting Experimental Thread Shapes, we show that Q can be used as a fitting parameter to accurately fit the measured thread shapes using the model contained in eqs 1 and 2. In section Start of Breakup, we are even able to show that the model in eqs 1 and 2 predicts a unique relationship between Q and thread length L (in eq 16). Thus, simply

Table 1. Typical Parameter Values for the Six Experimental Microfluidic Setups Considered in This Papera exp. nr.

angle ceiling (deg)

R0 (μm)

v0 (mm/s)

α

β

1 2 3 4 5 6

7 7 no ceiling no ceiling 20 20

120 120 70 70 70 70

70.2 58.1 45.5 33.8 27.7 23.6

4.35 6.35 17.8 32.2 48.0 65.7

136 164 359 483 590 690

a

R0 and v0 are the thread radius and the velocity at the nozzle, respectively. The dimensionless parameters α and β are defined in eq 5b. For all experiments, it holds that mass density ρ = 972 kg/m3, surface tension coefficient γ = 5 mN/m, and viscosity η = 370 mPa s.

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MODELING THREAD SHAPE The dynamical behavior of the liquid thread is appropriately described by equations, derived by Eggers and Dupont,23,24 for slender, axisymmetric configurations. We use them in a form adapted to our experimental configuration ∂t(R2) + ∂z(vR2) = 0

(2)

(1) 9801

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measuring L already specifies the value of Q. The derived relationship between Q and L provides us with the insight why and how the breakup of the thread happens. Thus, without modeling Q itself, it is still possible to study the (in)stability of this rather special experimental system. An important observation is that thread shape changes quite slowly in comparison with the velocity of the fluid within the thread. This implies that a quasi-steady-state approach is likely to be applicable. This also follows from a comparison of time scales. As seen from the v0 values in Table 1, the fluid velocity is on the order of 10 mm/s. The snapshots in Figure 1 show that the thread length changes on the order of 0.1 mm/s. This difference in time scales leads us to set the time derivative in eq 1 equal to zero. We then immediately find that v(z) = v0R 02/R2(z)

Figure 2. Experimental and fitted thread shapes R*(z*), made dimensionless according to eq 4. The dotted experimental curves correspond with the threads depicted in Figure 1a−e and are snapshots taken at times 24.90, 20.00, 14.90, 9.90, and 4.90 s before breakup. The solid lines are calculated using eq 5a with the value of Q* chosen to fit the experimental curves. The dimensional Q values are 37.62, 28.37, 17.20, 9.67, and 4.73 kN/m3, respectively.

as approximation for its solution. In Figure 3, we show for a number of quite different cases that the agreement between

(3)

here, R0 is the radius of the thread at the nozzle exit where z = 0, and v0 is the velocity at that point. Substituting eq 3 in eq 2, we obtain in the quasi-steady-state approach a second-order differential equation for R(z). It is convenient to rewrite this equation in a dimensionless form by introducing the scaled variables z* ≡ z /R 0 , t * ≡ v0 t /R 0 , R *(z*) ≡ R(z)/R 0

Figure 3. Comparison of R*(z*) curves calculated by solving eq 5a numerically (dotted lines) with R*(z*) curves calculated using approximation eq 6. The curves correspond to thread shapes shortly before breakup of the six different experiments listed in Table 1. The calculation of the parameters c and d in eq 6 to obtain optimal fits is explained in the text.

(4)

Applying these scalings and the transformation y(z*) = 2 ln(R*(z*)), eq 2 can be written as β e−y∂z * z *y − e−2y ∂z *y − α e−0.5y ∂z *y = Q *

R*(z*) curves calculated by solving eq 5a and R*(z*) curves calculated from eq 6 is nearly perfect, provided that the parameters c and d are chosen appropriately. To determine c and d, we substitute expression eq 2 in eq 5a and evaluate this equation for z = 0 and L. The first and second derivatives of expression eq 6 are given by

(5a)

The three dimensionless parameters are defined as Q* ≡

QR 0 ρv02

,α≡

3η γ 1 3 , and β ≡ ≡ ≡ 2 2We Re ρv0R 0 2ρv0 R 0

2

Thus, α (or inverse Weber number 1/2We) represents surface tension effects and β (or inverse Reynolds number 3/Re) represents viscosity effects. Since confusion is hardly possible, we omit in the following the * index for dimensionless variables, unless stated otherwise. The initial values for eq 5a are y(0) = 0 and ∂zy(0) = 2∂zR(0). Since very close to the nozzle the thread hardly tapers, we may effectively take ∂zy(0) = 0.

= 2cd(1 + 2dz 2)edz

2

2

(7)

Setting z = 0, we find that ∂zy(0) = 0, ∂zzy(0) = 2cd. Substituting these relations in eqs 5a/5b, we conclude that 1 cd = WeQ (8) 4 The next step is to evaluate eq 5a at the thread endpoint z = L. Since the thread transforms in a smooth way into the droplet (cf. Figure 1), the exact position L is not fully fixed. In practice, we use as its definition that it is the position where the thread radius has doubled its size, so where the dimensional R equals 2R0. By the way, this specific choice is hardly relevant for the results of the paper. This choice implies that (in dimensionless variables)

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FITTING EXPERIMENTAL THREAD SHAPES To check the reliability of the model in eqs 1−5a,5b, we compare measured R(z) curves with solutions of eq 5a. In that procedure, Q is used as the fitting parameter: we numerically integrate eq 5a and select the Q-value that yields the best fit. In this way, we may estimate Q as a function of α and β and thread length L. As an example, we show in Figure 2 the measured R(z) curves of the developing thread in Figure 1 at five consecutive times together with the corresponding fitted solutions of eq 5a. We observe that the fits are quite good. This confirms that the quasi-steady-state assumption seems indeed reliable. In the caption of Figure 2, we also give the dimensional Q-values that lead to the best fits. Notice that Q decreases if the thread becomes longer. Since eq 5a is strongly nonlinear, an explicit solution is not known. However, in the following, we will use the expression

R(L) = 2 ↔ y(L) = 2 ln(2)

(9)

2 2 1 i1 y βcd(1 + 2dL2)edL − jjj + α zzzcdL edL = Q (10) 2 k8 { Substituting eq 2 in eq 10 and introducing the notation x  dL2, we obtain

Evaluating eq 5a at z = L and using eq 9, we arrive at

1 1 yz ReL i1 zz − jjj + (11) 2 2We { 3 k8 By solving this implicit equation numerically, we may easily calculate x and thus d, given the parameters L, We, and Re. 2 e −x − x =

2

y(z) = c(edz − 1)

2

∂zy(z) = 2cdz edz , ∂zzy(z) = 2cd edz + 4cd 2z 2 edz

(5b)

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contains Re and not We, we conclude that the shape of short threads is completely determined by viscosity. If L increases, we find from eq 11 that x increases. For elongating threads, we arrive at a regime where log(x/(ex − 1)) ≪ 0. This causes the downward bending of the curves in Figure 4. As seen in Table 1, we have in all experiments We < 0.1. Consequently, we may approximate eq 11 by

Parameter c then directly follows from eq 8. The calculation of Q in terms of the other parameters is dealt with in section Start of Breakup. It is interesting to remark that for small values of z, so close to the nozzle, approximation eq 6 behaves quadratically in z 3Q 2 z 2Re

y(z) ≈ cdz 2 =

(12)

4 e−x − 2x ≈ 1 −

The corresponding dimensional thread profile behaves similarly R(z) ≈ R 0 ecdz

2

/2

≈ R0 +

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QR 0 2 z 12v0η

START OF BREAKUP Approximation of eq 6 allows us to derive a direct relation between Q and thread length L. Evaluating eq 6 at z = L and using eq 9, we find that

x≈

(14)

Q≈

Thus 2 ln(2) ex − 1

(15)

12 ln(2) x ReL2 e x − 1

(16)

If α, β, and L are specified, we first calculate x from eq 11 and then Q from eq 16. Taking logarithms, we find that i x log Q = log(12 ln(2)/Re) − 2 log L + logjjj x ke −

(19)

2 ln(2) −L /6Ca e WeL

(20)

From this, we see that for long threads, Q decays exponentially. However, Figure 5 shows that this regime is never reached in practice. For large x values (x > 1) from eq 18, we find that the Q values will be mainly determined by the capillary number Ca and thus by the ratio of surface tension and viscosity. A physical meaning of x is thus that it describes the transition of a viscous-driven regime (x ≪ 1) to a capillary-driven regime (x > 1). Alternatively, one can say that in the viscous-driven (with high Q values) regime, the liquid threads are rather stable, whereas the liquid threads become unstable when the capillary regime is approached (with Q values below 1 or x values above 1). To check the reliability of Q(L) in eqs 16 and 17, we plot in Figure 5 the experimental (Q, L) data for lengthening threads for all our six experimental situations specified in Table 1, together with the Q(L) curves calculated from eq 17. We observe that the theoretical curves perfectly fit the experimental data. Furthermore, we see that all threads break up as soon as the Q(L) curve starts to bend down, when surface tension effects become significant. To analyze the breakup in more detail, we rewrite eq 11 as

Combining eqs 8 and 15, we arrive at Q = 2cdβ =

1 L 6Ca

In this regime, eq 17 reads as

2

c=

(18)

The capillary number Ca measures the balance between the viscosity forces and the surface tension forces. For elongating threads, Q becomes smaller and smaller. For increasing x, we shift to a regime where

(13)

y(L) = 2 ln(2) = c(edL − 1)

1 L 3Ca

yz zz 1{

(17)

In Figure 4, we plot log(Q) versus log(L) for the six experimental conditions summarized in Table 1.

L=

12 (1 + 2x − 4 e−x) Re(1 + 4/We)

(21)

Note in Figure 1f that the defined length L matches with the actual place of breakup. In Table 2, we give for each of the six experiments the observed thread length Lmax at which the steady-state behavior is lost and the thread surface starts to oscillate. During experiments (see Figure 6), we observed that breakup occurs when small perturbations along the liquid thread induce a fast-growing instability close to the droplet. The final pinch-off is always close to the droplet. This instability can be absolute25 or convective.26,27 It should be realized that the exact start of the breakup process is somewhat difficult to detect, but always a reproducible maximum length of the liquid thread is found when the pinch-off takes place. The liquid thread begins undulating radially close to the droplet, possibly because the Laplace pressure of the large droplet is much smaller than the Laplace pressure of the liquid thread close to the droplet, causing the fluid system to become

Figure 4. Q as a function of L (both dimensionless) according to relation (eq 17), for the six different experiments summarized in Table 1. For short threads, the curves are straight lines with the height being fully determined by the dimensionless Reynolds number, so by viscosity. For longer L values, the curves tend to bend downward, due to the increasing influence of surface tension effects, represented by the Weber number. After bending, the shape of the curves is nearly completely determined by the quotient Re/We, i.e., the capillary number Ca.

We observe that all of the Q(L) curves in Figure 4 show similar behavior. For short threads, the curves start as straight lines with slope −2. In this regime, the factor log(x/ex − 1) ≈ 0. From eq 11, we indeed find x ≈ 0.6 for small L. According to eq 17, the slope of the straight line in Figure 4 is for small L determined by the factor 12 ln(2)/Re. Since this factor only 9803

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Figure 5. Predicted Q(L) curves (solid lines) according to eq 17 and experimental results (colored dots) for six different experiments as summarized in Table 1. For each experimentally found thread length L, a corresponding value of Q is derived by solving eq 5a. At a specific value of L (denoted Lmax), breakup sets in near the droplet. We observe that theory and experiment agree quite well and also that the thread tends to break up (at Lmax) as soon as the Q(L) curve starts to deviate from a straight line, i.e., when surface tension effects become appreciable.

Table 2. Values of Lmax, being the Length of the Thread at Which the Instability Sets ina We

exp. nr. 1 2 3 4 5 6

1.1 7.9 2.8 1.6 1.0 7.6

× × × × × ×

10−1 10−2 10−2 10−2 10−2 10−3

Re 2.2 1.8 8.4 6.2 5.1 4.3

× × × × × ×

Ca = We/Re

x fitted

Lmax,pred eq 22

Lmax observed

5.19 4.3 3.37 2.5 2.05 1.75

2.3 1.8 1.8 2 1.8 1.9

62.3 51.6 40.4 30 24.6 21

79 51.3 40 34.1 23.8 21.9

10−2 10−2 10−3 10−3 10−3 10−3

The x values are calculated by fitting eq 21 to the data. The predicted Lmax values follow from eq 22.

a

Figure 6. Snapshots depicting one period of oscillation in the neck. The thinnest part of the neck (marked arrows) pulsates at this rate. The time between two snapshots is 22 ms, and one period is 170 ms. Velocity v0 is 23.6 mm/s. The scale bar is 500 μm.

inner fluid as stated in eq 2. The contribution of buoyancy in Q is small. Although buoyancy is needed to elongate the liquid thread in a controlled way, most of the buoyancy force is counteracted by the ceiling when the droplet reaches the ceiling. Moreover, the mass density difference between the inner fluid (silicone oil, ρ = 972 kg/m3) and the outer fluid (water, ρ = 998 kg/m3) at room temperature is small. Also, any contribution of buoyancy will be negative, whereas we find that the sign of Q is positive. We find that the main contribution to Q is caused by the friction and associated dissipation between the moving thread surface and the outer fluid, as we will verify below. In the first order, we will neglect the tapering of the thread profile and thus approximate the thread surface by a cylinder and also assume the flow to be laminar in the outer fluid. As a boundary condition for the tangential force balance at the interface, we use

unstable at a fixed location near the droplet. In our setup, the observed vibrations are close and also more distant to the droplet. It is at present unclear if the oscillations are following an absolute or convective instability. We now compare the fitting of the theoretical prediction eq 21 to the actual measured Lmax values. To that end, we substitute in eq 21 per experiment We, Re, and Lmax and calculate x. The resulting x values in Table 2 suggest that the unstable behavior starts if x is in the range 1.8−2.0. If we substitute, e.g., the value x = 1.9 in eq 21, we obtain that Lmax,pred ≈

12 4.0 ≈ 12Ca Re(1 + 4/We)

(22)

In this approximation, the maximum thread length Lmax is thus fully determined by the capillary number only. As mentioned above, Q represents all forces other than the ones related to inertia, surface tension, and viscosity of the

n ·σ ·t = ηouter γouter ̇ 9804

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here, σ is the stress tensor and n and t are the unit vectors orthogonal and tangential to the thread surface, respectively. For σ, the common Newtonian stress tensor is used, with viscosity η of the thread inner fluid. On the right-hand side, we have γ̇ as the shear rate or derivative of the longitudinal velocity in the surrounding water with respect to the radial coordinate, taken at the thread surface and viscosity ηouter of the surrounding water. The moving thread surface drags the water with it and sets it into motion. This way energy is transferred, which is dissipated in the water due to internal friction. Thus, the water induces a slowing down force on the thread. If one uses expression eq 23 as a boundary condition in the derivation given in ref 24, this results in a Q term given by Q=

and is described by the Boussinesq−Scriven constitutive model.28 As a result, a model of the interface must also account for these surface viscosities and gradients in surface tension. The interface may therefore display the properties of a two-dimensional fluid, as originally proposed by Boussinesq.28 The interfacial tension will also considerably vary during liquid thread formation due to the time required for adsorption and transport of surfactant molecules to the liquid thread interface,28−30 giving rise to Marangoni effects. Such effects are known to be considerable around or below the critical micellar concentration (CMC), in the case of Tween 20 of around 0.008%. In this paper, we have not taken into account the possible effect of interfacial viscosity and the contribution of Marangoni effects; the inner fluid viscosity is rather large, η = 373 mPa s, and also the Tween 20 surfactant concentration of 1% is far above the CMC.

2 ηouter γouter ̇ R0

(24)

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We have to check whether the predicted magnitude for Q corresponds with eq 24. We have measured therefore velocity profiles by tracking particles in the outer fluid. Note that we model the thread surface by a cylinder with a fixed radius and thus that Q is uniform along the liquid thread. To get a good indication for the value of Q and the outer fluid shear rate γ̇, all measurements were performed half way the liquid thread. In Figure 7, an example of the measured velocity data is given just a few moments before the actual breakup of the liquid thread.

CONCLUDING REMARKS We have shown that, based on the approximated axisymmetric Eggers−Dupont eq 2, it is possible to predict the axial shape and length of a liquid thread between a nozzle and a growing droplet. Thread shapes can be calculated as solutions of differential equation eq 5a containing a pressure gradient term Q that can be regarded mainly as a compensation term for the presence of remaining forces acting on the liquid thread. Q can be obtained in two ways: (1) fitting the solutions of eq 5a to the measured thread shape and (2) calculating Q from the theoretically derived relation eq 16 between Q and thread length L. Figure 5 shows excellent agreement between both approaches. The theoretical Q(L) curve eq 16 (Figure 4) reveals that initially, when the thread is still short, Q(L) only depends on the dimensionless number β, representing viscosity effects. However, the more the thread elongates, the more the surface tension effects come into play, causing the Q(L) curve to be determined by the capillary number. At some moment in time, the thread surface starts to oscillate, leading to a breakup quite close to the droplet. Our analysis predicts how and when this unstable behavior will set in. This is most clearly seen from Figures 4 and 5, where it is shown that log(Q) as a function of log(L) is a straight line for small L. In this regime, the whole thread dynamics is driven by viscosity. For longer threads, surface tension becomes important, indicated by a deviation from the straight line. The experimental data in Figure 5 show that it is exactly at this length that the thread starts to break up. Since we were able to derive an explicit expression for Q(L), we may predict for which L breakup will happen. Equation 22 predicts that the maximum thread length is directly proportional to the capillary number. Table 2 shows that the predictive power of this formula is quite good.

Figure 7. Plot of the outer fluid velocity profile. The r = 0 value starts at the interface between the inner and outer fluid. R0 = 70 μm, v0 = 27.7 mm/s. The fitted red line corresponds to a shear rate value of γ̇ = 360 s−1.

Clearly, the Q values will be dependent on the initial inner velocity v0 and liquid thread length L. Higher v0 values will lead to more energy dissipation due to the moving interface and thus to higher Q values, whereas longer thread lengths L will result in larger and broader dissipation areas in the outer fluid and thus corresponding to smaller values for the shear rate. A shear rate value of γ̇ = 360 s−1 (Figure 7) gives according to eq 24 a Q value of ca. 10 kN/m2, which in the order of magnitude matches with the derived Q data a few moments before breakup (Q = 5−10 kN/m2) as given in Figure 2. From another shear rate fit, we find that γ̇outer may vary between 350 and 1500 s−1. The properties of an interface stabilized by surfactants and other components will alter the dynamic behavior of the liquid thread. In-plane shear and dilatational friction may arise at the interface from the molecular components sliding over each other. This phenomenon, known as dynamic interfacial viscosity, exists in concentrated layers of insoluble lipid films

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsomega.9b00796. Lengthening and breakup of the liquid thread of Figure 1 (AVI)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 9805

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ORCID

(22) Javadi, A.; Eggers, J.; Bonn, D.; Habibi, M.; Ribe, N. M. Delayed capillary breakup of falling viscous jets. Phys. Rev. Lett. 2013, 110, No. 144501. (23) Eggers, J. Drop formation - An overview. Z. Angew. Math. Mech. 2005, 85, 400−410. (24) Eggers, J.; Dupont, T. F. Drop formation in a one-dimensional approximation of the Navier-Stokes equation. J. Fluid Mech. 1994, 262, 205−221. (25) Utada, A. S.; Fernandez-Nieves, A.; Gordillo, J. M.; Weitz, D. A. Absolute instability of a liquid jet in a coflowing stream. Phys. Rev. Lett. 2008, 100, No. 014502. (26) Driessen, T.; Jeurissen, R.; Wijshoff, H.; Toschi, F.; Lohse, D. Stability of viscous long liquid filaments. Phys. Fluids 2013, 25, No. 062109. (27) Leib, S. J.; Goldstein, M. E. Convective and absolute instability of a viscous liquid jet. Phys. Fluids 1986, 29, 952. (28) Gounley, J.; Boedec, G.; Jaeger, M.; Leonetti, M. Influence of surface viscosity on droplets in shear flow. J. Fluid Mech. 2016, 791, 464−494. (29) Schroën, K.; Ferrando, M.; de Lamo-Castellví, S.; Sahin, S.; Gü e ll, C. Linking Findings in Microfluidics to Membrane Emulsification Process Design: The Importance of Wettability and Component Interactions with Interfaces. Membranes 2016, 6, 26. (30) van Rijn, C. J. M.; van Heugten, W. G. N. Droplet Formation by Confined Liquid Threads inside Microchannels. Langmuir 2017, 33, 10035−10040.

Cees J. M. van Rijn: 0000-0001-8380-3834 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Dutch research program MicroNed for financial support.

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REFERENCES

(1) Rayleigh, L. On the instability of jets. Proc. London Math. Soc. 1878, s1−s10, 4−13. (2) Eggers, J. Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 1993, 71, 3458. (3) Stone, H. A. Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 1994, 26, 65−102. (4) Stone, H. A.; Leal, L. G. Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 1989, 198, 399−427. (5) Gunawan, A. Y.; Molenaar, J.; van de Ven, A. A. F. Break-up of a set of liquid threads under influence of surface tension. J. Eng. Math. 2004, 50, 25−49. (6) Eggers, J.; Fontelos, M. A. Isolated inertialess drops cannot break up. J. Fluid Mech. 2005, 530, 177−180. (7) Wilkes, E. D.; Phillips, S. D.; Basaran, O. A. Computational and experimental analysis of dynamics of drop formation. Phys. Fluids 1999, 11, 3577−3598. (8) Kowalewski, T. A. On the separation of droplets from a liquid jet. Fluid Dyn. Res. 1996, 17, 121−145. (9) Shi, X. D.; Brenner, M. P.; Nagel, S. R. A cascade of structure in a drop falling from a faucet. Science 1994, 265, 219−222. (10) Doshi, P.; Cohen, I.; Zhang, W. W.; Siegel, M.; Howell, P.; Basaran, O. A.; Nagel, S. R. Persistence of memory in drop breakup: the breakdown of universality. Science 2003, 302, 1185−1188. (11) Thorsen, T.; Roberts, R. W.; Arnold, F. H.; Quake, S. R. Dynamic pattern formation in a vesicle-generating microfluidic device. Phys. Rev. Lett. 2001, 86, 4163. (12) Anna, S. L.; Bontoux, N.; Stone, H. A. Formation of dispersions using “flow focusing” in microchannels. Appl. Phys. Lett. 2003, 82, 364−366. (13) Utada, A. S.; Lorenceau, E.; Link, D. R.; Kaplan, P. D.; Stone, H. A.; Weitz, D. A. Monodisperse double emulsions generated from a microcapillary device. Science 2005, 308, 537−541. (14) Shah, R. K.; Shum, H.; Rowat, A.; Lee, D.; Agresti, J.; Utada, A. S.; Chu, L.; Kim, J.; Fernandez-Nieves, A.; Martinez, C.; Weitz, D. A. Designer emulsions using microfluidics. Mater. Today 2008, 11, 18− 27. (15) Utada, A. S.; Fernandez-Nieves, A.; Stone, H. A.; Weitz, D. A. Dripping to jetting transitions in coflowing liquid streams. Phys. Rev. Lett. 2007, 99, No. 094502. (16) van Rijn, C. J. M. Micro Engineered Membrane Technology, 1st ed.; Elsevier Publishers: 2004; pp 1−384. (17) Atencia, J.; Beebe, D. J. Controlled microfluidic interfaces. Nature 2005, 437, 648−655. (18) Kawakatsu, T.; Kikuchi, Y.; Nakajima, M. Regular-sized cell creation in microchannel emulsification by visual microprocessing method. J. Am. Oil Chem. Soc. 1997, 74, 317−321. (19) Kobayashi, I.; Nakajima, M.; Chun, K.; Kikuchi, Y.; Fujita, H. Silicon array of elongated through-holes for monodisperse emulsion droplets. AIChE J. 2002, 48, 1639−1644. (20) van Rijn, C. J. M.; van Heugten, W. G. N. Droplet Formation by Confined Liquid Threads inside Microchannels. Langmuir 2017, 33, 10035−10040. (21) Kobayashi, I.; Mukataka, S.; Nakajima, M. CFD simulation and analysis of emulsion droplet formation from straight-through microchannels. Langmuir 2004, 20, 9868−9877. 9806

DOI: 10.1021/acsomega.9b00796 ACS Omega 2019, 4, 9800−9806