Scaling the drop size in coflow experiments

We perform extensive experiments with coflowing liquids in microfluidic devices and provide a closed expression for the drop size as a function of measurable parameters in the jetting regime that accounts for the experimental observations; this expression works irrespective of how the jets are produced, providing a powerful design tool for this type of experiments.


Introduction
The generation of emulsions is an area of active research due to its countless technological applications (see [1]- [5] for detailed reviews). Recent fabrication methods rely on microfluidics, as this technology provides great control over fluid flow and mixing of components. In many situations, the dispersed phase flows inside a coaxial coflow of the continuous phase; this provides several advantages with respect to using a quiescent bath [6]- [9]: (i) control of the drop size by appropriate tuning of the coflow properties; (ii) reduced coalescence between drops, in the absence of surfactants; and (iii) increased production frequency [10]- [17].
There are two major types of microfluidic devices that improve the drop or bubble generation process by making use of an outer, coaxial coflow: (i) those in which both streams flow through a small orifice, referred to as flow-focusing devices [15], [18]- [20] and (ii) those in which both streams flow in parallel, typically referred to as coflowing devices [10]- [14], [21]. The latter can also be classified in terms of the confinement provided by the outer bounding channel; there are situations where this confinement is significant [16,22,23], and situations where it is not [14,21,24,25]. Despite these differences, the focus of all these studies is on understanding the transition between the dripping and the jetting regimes [26]- [28]. Dripping is characterized by the fact that no long jets of the dispersed phase are formed. Thus, drops are generated right at the tip of the injection tube. By contrast, when jetting occurs, the dispersed phase forms long liquid jets and consequently, drops are emitted right at the tip of the liquid thread.
In the absence of confinement effects, two different types of jetting regimes have been identified [21]: the narrowing and widening regimes. These names simply reflect the shape adopted by the jet in either regime. The narrowing jets are formed when the viscous stresses on the interface due to the outer stream overcome surface tension confinement forces, or equivalently, when the capillary number based on the outer velocity, U o = Q o /D 2 o , and outer viscosity, µ o , is Ca o = µ o U o /σ O (1), with σ the interfacial tension between the two liquids. Since in these situations, the outer velocity is larger than the inner velocity, the jet stretches, thus narrowing downstream by a certain amount. The widening jets are produced in a different way; they result when the stresses due to the flow of the inner stream at the interface overcome surface tension confinement forces. This can happen when the Weber number of the inner fluid satisfies the condition: We i = 8ρ i Q 2 i /(π 2 σ D 3 i ) 1, with Q i and ρ i the inner-fluid flow rate and density, respectively, and D i the inner diameter of the injection tube. In this situation, the inner stream usually flows faster than the outer stream; consequently, these jets are decelerated as they move downstream, resulting in their widened shape.
In this paper, we extend the criterion needed to induce the formation of a widening jet and show that the condition We i > 1 is only applicable if the Reynolds number of the inner fluid, Re i , is also larger than one. In the opposite limit, when Re i < 1, the Weber number no longer reflects when a jet is formed. In this case, we find that the appropriate criterion is provided by the capillary number of the inner fluid; jetting occurs when Ca i > 1. More importantly, we provide here a general expression to estimate the drop size in either regime as a function of measurable parameters; this shows that despite the differences between these regimes, the drop size is governed by a unique scaling relationship. Our experiments confirm this prediction, which can thus be used to design coflow experiments aimed at obtaining droplets with a particular size distribution. This capability coupled to the possibility of multiplexing [29] could contribute to the widespread use of this methodology. The rest of the paper is structured as follows. The experimental setup is described in section 2. Section 3 is devoted to the analysis of the experimental results; we review the scaling of the drop size in the narrowing regime, provide the need to extend the current criterion to induce the formation of a widening jet, and derive and experimentally validate a simple equation to calculate the drop size in both narrowing and widening regimes. Finally, we conclude in section 4.

Experimental setup
Our experimental device is made of two coaxially aligned capillary tubes, as shown in figure 1. The inner capillary tube is cylindrical, with a tip tapered to an inner diameter D i that is varied between 40 and 60 µm and an outer diameter of approximately 80 µm. The outer capillary tube has a square cross section; coaxial alignment of the tubes is achieved by matching the outer diameter of the untapered portion of the inner capillary to the inner dimension of the square capillary, D o = 1 mm, as shown in figure 1(a). At this length scale, which is below the capillary length, the effects of gravity are negligible. Therefore, the orientation of the experimental device with respect to that of gravity is irrelevant. Nonetheless, all experiments were aligned horizontally. Although the flow in the square tube is not axisymmetric, since the tip is centered and D i D o , the local flow around the tip should be approximately axisymmetric. Both liquids are injected through syringe pumps (Harvard Apparatus PHD2000). For visualization and measurement purposes, we use a high-speed video camera Phantom V7.1, working between 2000 and 51 000 fps.  Example of the stretching regime [21], characterized by the ejection of a liquid jet with a diameter decreasing downstream. In the case illustrated in this figure, which shows a cone-jet transition, the coflowing device is operated under the tip-streaming regime, firstly described numerically by Suryo and Basaran [16]. The outer capillary number is Ca o = 4.32, Q i /(U o D 2 i ) < 3 × 10 −3 and the inner flow rate is decreased from bottom to top. Picture taken from Marin et al [29].
The liquids we employ are deionized water, glycerol and different polydimethylsiloxane (PDMS) oils, with viscosities varying between 1.5 and 100 cP; by interchanging the different liquids we can vary the viscosity ratio, µ i /µ o , from 0.1 to 20. The surface tension between the different liquids, σ , slightly varies around 40 mN m −1 .

Revision of the narrowing regime
It is well established that when the capillary number of the outer fluid in a coflow experiment exceeds a threshold value of order unity, a long liquid jet emerges from the injection tube [21,29]. If in addition, Q i /U o D 2 i 1, the diameter of the liquid jet is much smaller than that of the injection tube, as illustrated in figure 2. In this situation, drops with sizes down to one micron can be obtained [29].
Within this narrowing regime, if the inner-fluid flow rate is kept constant while U o increases, the diameter of the ejected jet decreases [21,29]. This results from the low Reynolds numbers in these experiments, which guarantee the effective diffusion of momentum across the whole section of the jet. As a result, the inner-liquid and outer-liquid velocities become equal some distance downstream of the injection tube and the jet diameter simply results from The thin jets observed in this narrowing regime are convectively unstable and consequently, the size of the drops obtained from their break-up, d d , can be deduced from the mass balance: the dimensionless wavenumber corresponding to the 5 maximum growth rate of sinusoidal capillary perturbations and π d j /k * its corresponding wavelength. Here, Oh = µ i / ρ i σ d j /2 is the Ohnesorge number based on the material properties of the inner fluid, expressing the relative importance of viscous and inertial timescales [30]. Since k * depends weakly on Oh for relatively large values of this parameter, as shown in the appendix (figure 12), it is sensible to write k * = k * t (µ i /µ o ), with k * t the wavenumber of maximum growth rate in the limit, firstly considered by Tomotika [31], of Oh → ∞. With these considerations, we obtain which has been confirmed experimentally for a wide range of viscosity ratios [29]. This equation can be obtained using an alternative way of reasoning. Indeed, the drop volume can be obtained from the mass balance π with f the drop formation frequency and T the corresponding period. We can approximate the period of drop formation by the time needed to elongate the jet a distance equal to the wavelength of maximum growth rate, where U p is the velocity of the jet at its most downstream position, as shown in figure 1(b). By combining this result with equation (3), we obtain [21,29] Since U p = U o for the narrowing jets considered so far, this equation (4) is identical to equation (2).

Revisiting the dripping to jetting transition in the widening regime
A different kind of jet, referred to as widening jet [21], is obtained if the inner momentum overcomes surface tension confinement forces: We i O(1). Generally, for these jets, U o Q i /D 2 i and thus, the jet is decelerated by the action of the shear stress exerted by the outer stream, leading to the observed jet widening, as shown in figure 1(b). We emphasize here that the condition for formation of these jets, If this is not fulfilled, the condition We i O(1) no longer predicts the dripping-to-jetting transition, as shown in figure 3(a), where we plot the values of the Weber number of the inner fluid for which we observe a jetting behavior; even for We i < 1, there are jets that form. In these cases, Re i < 1, as shown in figure 3(b). Interestingly, for these jets, the capillary number of the inner fluid is larger than one, Ca i = 4µ i Q i /(π D 2 i σ ) 1. We thus divide our data into two sets depending on whether the inner Reynolds number is smaller or larger than one. When Re i > 1, we observe jetting if We i > 1, as shown in figure 4(a). However, when Re i < 1, widening jets form when Ca i > 1, as shown in figure 4(b). As a result, widening jets can form either driven by inertial or viscous forces of the inner fluid; depending on Re i , they form when either We i or Ca i is larger than a threshold number, which is close to unity. Our results thus extend the dripping-to-jetting criteria [21] to cases where Re i < 1.
All the experiments considered in this study lie within the widening regime [21] in spite some of the values of We i are much smaller than unity. (b) Values of the inner Reynolds number, for those experiments in which Re i < 1. Thus, in order for widening jets to be generated, either We i 1 and Re i 1 or Ca i 1 and Re i 1. Numbers in the legend indicate inner/outer viscosities in centipoise.
. The values of the outer and inner viscosities are, respectively, µ o = 5 cP and µ i = 50 cP. (c) Drop size as a function of the outer flow rate for a fixed value of the inner flow rate, Q i = 10 ml h −1 and various inner/outer viscosities. Observe that the effect of increasing the outer flow rate is, in all cases, to decrease drop size.

Unified scaling for the drop size
In the widening-jet regime, we observe that the drop diameter decreases as the coflow velocity increases, as shown in figures 5(a) and (b). This trend is consistent with what has been observed previously for both the narrowing and widening regimes [21,29], and independent of the values of the inner and outer viscosities, as shown in figure 5(c).
We also observe that the drop size increases with the inner-fluid flow rate, for µ i = 1 cP and µ o = 10 cP, as shown by the images in figure 6, which correspond to two different values of Q i ; this is consistent with previous results too [21]. However, when µ i = 10 cP and µ o = 1 cP, we observe that the drop size is reduced if Q i is increased, as shown by the images in figure 7, which correspond to two different values of Q i . The viscosities of both inner and outer liquids thus play a relevant role in determining the drop-size dependence with Q i . Our experiments indicate that d d increases with Q i when the outer viscosity is sufficiently larger than that of water, irrespective of the viscosity ratio, as shown in figure 7(c), where we show data corresponding to µ i /µ o = 0.1, 1 and 10, all exhibiting the same behavior. By contrast, when the viscosity of the outer fluid is ∼ 1 cP, the drop size decreases with Q i , as also shown in figure 7(c).
To understand these observations, let us consider the drop formation period in equation (3). Similarly to bubble formation, this time corresponds to the time required to convect the inner fluid a distance λ at a velocity U p , t conv = λ/U p , plus the time required to collapse or pinch the liquid thread, t pinch : T = t conv + t pinch [17], where λ is taken as the distance traveled by the downstream location of the jet within two consecutive pinch-off events, as shown in figure 8. or In our experiments, the second term in the right-hand side of these equations is much smaller than unity. This implies that irrespective of the Ohnesorge number, the process of extending the liquid ligament a distance λ is much slower than the break-up time of the liquid thread: The scaling of d d thus depends on how λ and U p scale with the different control parameters in the problem. For U p , when the outer fluid viscosity is large compared to that of water, we find that U p U o , as shown in figures 9(a) and (b), where we plot U p /U o versus Q i and Q o , respectively. As a result, f ∝ λ/U o , as in the case of bubble formation in the presence of a liquid coflow [13,17] or for the case of drop formation in the narrowing regime [21,29]. However,  when the outer viscosity is similar to that of water, U p exhibits dependence on both U o and Q i , as shown in figure 9. This experimental observation can be qualitatively explained in terms of the viscous-stress balance on the jet surface. If we assume that the jet surface travels at a speed U s , the continuity of shear stresses at the interface demands that Based on this equation, we see that in the limit (µ i /µ o ) × (δ/D i ) 1, U s U o and consequently, U p U o , since in this case the differences between the interfacial and outer velocities are negligible. However, when the outer-fluid viscosity is not so large, then (µ i /µ o ) × (δ/D i ) O(1) and, due to the fact that . Under these circumstances, the inner stream drags the outer fluid creating a strong velocity gradient adjacent to the jet interface, as schematically shown in figure 10(b). Since the outer velocities in the neighborhood of the jet interface are larger than U o , U p is also larger than U o . As Q o increases, however, this difference decreases and U p approaches U o , as shown in figure 9(b). In addition, since U s grows with U i , so does U p , also consistent with our observations, as shown in figure 9(a).
Despite this qualitative agreement with our model, we have not been able to find a simple way of expressing U p as a function of the control parameters, other than solving the Navier-Stokes equations, which would have to be done numerically. Under confinement [22], the presence of the outer walls attenuates the growth of capillary waves and both the inner and outer velocity profiles are able to reach the parallel Poiseuille solution, simplifying the problem. In our experiments, this is not the case and the larger growth rate of the capillary waves disrupts the jet before the velocity profiles reach this analytic solution. Therefore, the velocities of the inner and outer streams evolve in the axial direction in a nontrivial manner and so does the jet tip velocity U p .
To obtain the appropriate scaling for λ, we take into account that the pinch-off process is driven by surface tension. Indeed, in order for surface tension to break a cylindrical piece of liquid into spherical drops, the wavelength of the growing, unstable mode, λ v , must satisfy the condition λ v > π d j , with d j the jet diameter upstream the location where the drops form [32]- [34] (see figure 8). In addition, t pinch t conv . As a result of these two ingredients, we can consider that right after a drop has been formed, the wavelength of the corrugations on the jet surface, shown in figure 8, is such that k = πd j /λ v > 1. Therefore, during the first instants of drop formation, surface tension smoothen all capillary waves. However, since the front of the jet is elongated at a velocity U p , the wavelength of the corrugations increases in time and thus, k decreases. When k 1, the liquid jet is close to being unstable and is prone to break into drops, however, the growth rates for k 1 are very small (see figure 11 for details) 4 . Consequently, k continues to decrease in time until the growth rate of capillary perturbations is significant, namely, until k k * . Since t pinch is so short compared to t conv , the pinch-off process only represents a small fraction of the period for drop formation, T , and breakup essentially takes place instantaneously as soon as λ π d j /k * . Therefore, the drop formation period can be approximated by the time needed to elongate the jet a distance equal to λ = π d j /k * . As a result: Since continuity demands that π/6 d 3 d = Q i f −1 , the drop diameter is given by: which is the same as equation (4) used to calculate the diameter of the drops generated in the narrowing regime. We thus arrive at the relevant conclusion that the drop size is given by the same equation either in the widening or narrowing regimes; the only difference resides in how k * , d j and U p depend on the control parameters. For the widening jets, the best approach to obtain k * would be to solve the dispersion relation corresponding to the capillary perturbations evolving in a geometry which is between a cylinder and a spherical droplet. Moreover, these capillary perturbations develop within coflowing streams with shear. Therefore, contrary to the simplified analysis in the appendix, which assumes that the geometry is cylindrical, that there is no relative motion between both streams and that perturbations decay at infinity in the radial direction, a rigorous dispersion relation describing wave evolution in the case of widening jets, should take into account all the real effects enumerated above. Nevertheless, this refinement in the calculation of k * would necessarily imply simplifications since, for instance, the precise geometry of the transition region between the jet and the drop is not known a priori. Thus, we adopt here the simplest approach and calculate k * through equation (A.1), which gives the dimensionless wavenumber corresponding to the maximum growth rate. Note that in the case of widening jets the values of Oh are not too large and thus the simplification k * k * t , assumed for the case of the narrowing jets, is not applicable, as shown in figure 12(a).
Finally, we need to estimate d j . While for the narrowing jets we can use equation (1), for the widening jets this is not possible, since the inner stream velocity never equals the outer stream velocity, U o . This results from the fact that for these jets, the flow is locally absolutely unstable [25,34], implying that at some axial location, the speed of the capillary disturbances becomes similar to the speed at which they are convected downstream. Therefore, the jet breaks before the inner velocity of the jet can become equal to the outer-fluid velocity. Based on this fact, we will estimate the value of d j from the condition U i = d j /t pinch , with U i = 4Q i /(π d 2 j ) the (approximate) propagation velocity of capillary disturbances and with d j /t pinch the characteristic velocity at which perturbations grow in time. By using that t pinch = 1/(i n * )(ρ i /µ i )(d j /2) 2 (see appendix), with (in * ) the maximum value of the growth rate, and the definition of Oh, we arrive at: Note that equation (11) needs to be solved iteratively since in * is also a function of d j through its dependence on Oh. This maximum growth rate depends on Oh, as shown in figure 12(b) for different viscosity ratios; it appreciably decreases as Oh increases. Equipped with the theoretical description of k * , U p and d j , we can critically test the drop size dependence predicted by equation (10). We thus plot the experimental dimensionless ratio d d /D i as a function of (6Q i d j /(D 3 i k * U p )) 1/3 , as shown in figure 13(a), where we measure U p directly from the experiments, and in figure 13(b), where we use U p = U o when µ o 5 cP. Remarkably, when doing so, all the data collapses onto the same mastercurve, irrespective of whether d d increased or decreased with Q i , and for a large number of µ i and µ o combinations. We obtain that the spread of the data is ±25% ( figure 13(a)) and ±35% (figure 13(b)) with Note that the wavelength of maximum amplification of capillary disturbances, k * , varies very slightly with Oh. Moreover, note that perturbations with wavenumbers k 1, possess growth rates close to zero. respect to the mean. In addition, there is a linear relation between these variables, consistent with our expectations. By performing a linear fit of the data, we obtain: for figures 13(a) and (b), respectively. The slopes of the linear fits are close to one and the ordinates are close to zero, in further agreement with equation (10). In spite of the relative errors in figure 13, we have shown that equations (12) and (13) can be used to approximately predict the drop size, even for the largest values of (6Q i d j /(D 3 i k * U p )) 1/3 , which correspond to drop sizes that are close to the outer geometrical dimension of the device. As a result, our conceptual description of the drop formation mechanism in coflowing liquids is correct, unifying the so-called narrowing and widening regimes in terms of the drop size.

Conclusions
We have studied in detail the process of drop formation from long, widening jets, in microfluidic coflowing devices [21]. By changing the values of the control parameters, which include innerfluid and outer-fluid viscosities and flow rates, we have extended the criterion for jetting to occur  i k * U p )) 1/3 . In both graphs, the slope of the linear regression fit to the experimental data is very close to 1, which is the experimental prediction given in equation (10). The relative errors, however, are ±30%. Since the maximum experimental error occurs in the measurement of the tip velocity but is only of the order of ∼ 10%, the dispersion is attributable to necessary simplifications in the way the wavelength of maximum growth rate and the tip velocity, U p , are calculated. Numbers in the legend indicate inner/outer viscosities in centipoise. as a function of the Reynolds number of the inner fluid. When Re i > 1, our results indicate that jetting occurs if We i > 1, consistent with previous experiments [21], while for Re i < 1, the correct measure to predict the transition from dripping to jetting is not We i , but rather the capillary number of the inner fluid; jetting occurs in this case when Ca i > 1.
By combining experiments and modeling, we have arrived at the conclusion that the physical idea underlying drop generation in both narrowing and the widening regimes [21] is the same: the period of drop formation is given by the time required to elongate the jet a distance equal to the wavelength of maximum growth rate. From a quantitative point of view, we have obtained that the size of the drops, d d , generated either in the narrowing or in the widening jetting regimes can be calculated, with relative errors of about ±30%, using a simple model relating d d to the dimensionless wavenumber corresponding to the maximum growth rate, k * , to the jet diameter, d j , and to the velocity of the most downstream point in the jet, U p . As a result, the only difference between the drop size generated in either the narrowing or widening regimes is the way k * , d j and U p depend on the control parameters.
For the narrowing jets, since the values of the Ohnesorge number are usually moderate or large, and due to the small variation of k * with Oh, we can approximate k * by k * t , which is the wavenumber corresponding to the maximum growth rate in the limit Oh → ∞ considered by Tomotika [31]. For the widening jets, this simplification in the determination of k * is not applicable since the values of Oh are not large enough.
Additionally, in the case of the narrowing jets, d j = (4Q i /π D 2 i ) 1/2 and U p = U o , while for widening jets, U p = U o Q o /D 2 o only if the viscosity of the outer fluid is large enough; otherwise it needs to be determined experimentally. For these jets, we have obtained the value of d j by using the fact that these jets break-up through an absolute instability, implying that the relevant capillary velocity can be approximated by the inner-fluid, average velocity; using this fact and the approximate pinch-off time, we have estimated the jet diameter.
Our results provide a general description of the drop formation mechanism in coflowing liquids for both the narrowing and widening regimes through a unique relation for the drop size. This understanding can aid future experimental approaches to the generation of emulsions using coflowing devices. deduced by Tomotika [31] and first solved numerically by Meister and Scheele [30]: