Pinch-off of microfluidic droplets with oscillatory velocity of inner phase flow
Pinch-off of microfluidic droplets with oscillatory velocity of inner phase flow
OPEN When one liquid is introduced into another immiscible one, it ultimately fragments due to hydrodynamic instability. In contrast to neck pinch-off without external actuation, the viscous twofluid system subjected to an oscillatory flow demonstrates higher efficiency in breaking fluid threads. However, the underlying dynamics of this process is less well understood. Here we show that the neck-thinning rate is accelerated by the amplitude of oscillation. By simply evaluating the momentum transfer from external actuation, we derive a dimensionless pre-factor to quantify the accelerated pinch-off. Our data ascribes the acceleration to the non-negligible inner fluid inertia, which neutralizes the inner phase viscous stress that retards the pinch-off. Moreover, we characterize an equivalent neckthinning behavior between an actuated system and its unactuated counterpart with decreased viscosity ratio. Finally, we demonstrate that oscillation is capable of modulating satellite droplet formation by shifting the pinch-off location. Our study would be useful for manipulating fluids at microscale by external forcing.
In many natural systems, coordination between dynamic and geometrical parameters plays a crucial role in
retaining the reliability and efficiency for system performance. For example, a beating heart expands and
contracts periodically to pump blood through vessels1, thereby maintaining one?s life; most male frogs distend vocal
sacs to intensify the calls, thereby attracting females and reinforcing the success in reproduction2. In principle, the
reliability and efficiency usually arise from the periodic deformation in system configuration coordinated with
dynamic parameters such as pressure gradient. Technologically, human beings benefit from this coordination in
developing new techniques, for instance the invention of steam engine3 by which high-pressure steam expands
to perform mechanical work. Recently, scientists also use the coordination strategy to control droplet generation
by acoustically actuating4,5 or mechanically vibrating6?8 fluid systems. Subjected to an oscillatory flow, the
bulbous drops pulsate with the rarefaction and compression of fluid pressure, followed by a sudden breakup due to
a fast drainage in the fluid neck8. The resultant micro-droplets have various potential applications, ranging from
material synthesis9, chemical reactions and mixing10 to biological assays11, drug encapsulation and delivery12
and cell cultures13. In these applications, highly-controllable droplet generation with high throughput is usually
required. Note that the formation of satellite droplets is unfavourable in the interest of controllable generation of
droplets, and the production frequency is often limited by neck thinning velocity. It is thus essential to have deep
understanding toward the dynamics of droplet breakup to meet the demand of uniformity and high throughput.
The dynamics of Newtonian threads rupture without external forcing have been well established for both
liquid-in-air and two-fluid systems14. In the vicinity of pinch-off, the neck profile is self-similar and the minimum
neck radius Rmin obeys scaling laws of Rmin ? (tc ? t)?, with exponent ? determined by the asymptotic force
balances under various circumstances14; tc is the critical time at neck pinch-off and t is the time. Unlike pinch-off
in air, viscous effect of the surrounding liquid is non-negligible in viscous two-fluid systems15. Stokes flow
dominates two-fluid pinch-off when Rmin < ?i?o/(?i?)15, where ?, ? and ? represent respectively the dynamic viscosity,
volumetric density and interfacial tension, and subscript i and o stand for inner and outer phase, respectively. In
Stokes regime, capillary effect is counteracted by inner and outer viscous dissipations, while fluid inertia is
negligible, which gives a linear scaling of Rmin with ? = 116?18. The neck profile is featured by the asymmetric
double-cone shape. The cone slopes only depend on the viscosity ratio ? (? = ?i/?o) of the two-fluid system16?19,
independent of other parameters, such as nozzle diameter, interfacial tension, and density difference18. If the
thinning neck radius can decrease down to the molecular scales, a transition to the thermal-fluctuation
dominated pinch-off occurs when Rmin < LT20,21. Here LT = kbT /? is the thermal length scale, with kb and T being
Boltzmann?s constant and temperature, respectively. The neck profile is symmetric and the liquid-liquid interface
is rough due to the effects of thermal fluctuations. Both experimental20,21 and theoretical22 results suggest the
power ? ? 0.42 for thermal-fluctuation regime. However, LT is usually nanometric for simple liquids, where the
effects of thermal fluctuations are hardly to be observed at laboratory-scale. As such, Stokes flow is the final
asymptotic regime in viscous two-fluid pinch-off, provided the molecular scales are not reached.
Except for the pinch-off of complex liquids in air23,24 and zero-viscosity fluids (for example bubbles) inside
another highly viscous liquids25,26, the neck thinning dynamics of Newtonian fluids usually shows a universal
behavior, which means that the neck profile and scaling law depends on neither initial nor boundary conditions.
For instance, in liquid bridge experiments27, varying the stretching velocity can change the breakup locations28
and alter satellite droplet formation29, but the local dynamics of pinch-off remains unchanged and is identical to
any pinch-off event involving that fluid28?30. However, in microfluidics, the channel confinement and dynamic
process (such as flow rates) complicate the pinch-off scenario and would alter the neck scaling31?34 due to the
alteration in local force balance. Utilization of external forcing introduces more variables to be considered.
Previous experiments show that perturbing the inner fluid pressure displays an enhancement in droplet
breakup6,8. Assuming the minimum neck radius Rmin still obeys the scaling law Rmin = C (tc ? t)? (C is the
proportionality factor), the neck thinning velocity gives V r = dRmin/dt = C (tc ? t)??1. Influencing either the
proportionality factor C or the exponent ? would induce variations in the dynamics of droplet pinch-off, and thus the
neck thinning velocity Vr. Nevertheless, knowledge of pinch-off in microfluidics subjected to external forcing is
limited, leaving several questions unaddressed like whether the external forcing would affect C or ? or both and
to what extent, and how it varies satellite droplet formation.
Here, we explore the dynamics of viscous two-fluid pinch-off in a co-flow microfluidic capillary device where
the inner fluid velocity is oscillatory, actuated by mechanical perturbation (Fig.?1a). Our study lies into the Stokes
flow regime, where the actuated neck radius still scales linearly with time prior to pinch-off (?= 1). However,
the neck thinning velocity Vr is accelerated by the oscillation amplitude compared to that without oscillation
(Fig.?1b?d). By estimating the pulsatile velocity of inner fluid at the ejection nozzle, we derive a dimensionless
pre-factor to quantify the neck thinning velocity Vr, and rescale the perturbed pinch-off. The pulsatile velocity
reinforces the transient inner fluid inertia, which, in turn, counteracts inner viscous stress, thus enhancing the
pinch-off. Moreover, we find that shifts in pinch-off location, raised from the oscillation amplitude, impact the
formation of satellite droplets. Our work has significance in microscale hydrodynamic systems with external
actuation, such as surface-acoustic-wave5 and mechanically-perturbed6?8 microfluidics.
Materials and Methods
Experimental procedure. We conducted experiments in a co-flow capillary microfluidic device that
was fabricated by aligning one round glass capillary with a taped nozzle (outer diameter of the nozzle
Dn = 151.443 ?m, Fig.?1a,b) inside another intact capillary (inner diameter 580 ?m). Both inner and outer fluids
were injected by syringe pumps (Longer Pump) into the microcapillary device, Fig.?1a. The outer fluid flow rate
was kept as Qo = 1 mL h?1, while inner flow rate Qi was varied (0.02 mL h?1 ? Qi ? 0.45 mL h?1). In the operation
window of Qo and Qi, only dripping was observed, no jetting occurred. A mechanical vibrator (Pasco Scientific,
Model SF-9324) perturbed the inner fluid microtubing (inner-wall radius Riw = 0.43 mm) in the gravitational
direction with a displacement of y = ?0 sin(2?ft). The perturbed microtubing was put nearly straight and
meanwhile enabled to be displaced freely without deformation by the vibrator. The two ends of the microtubing were
held still during experiments, one connected to the syringe needle and the other to the device (Fig.?1a). The
distance L (Fig.?1a) between the perturbing location and nozzle was constant to be L = 40 cm. We studied the
oscillation amplitude ?0 between 0 and 5 mm, with the upper bond limited by the vibrator, and examined frequency
f ? 100 Hz. When frequency is higher than 100 Hz, enhancement of pinch-off decays significantly35,36, probably
due to two reasons: first, amplitude ?0 dissipates when frequency is too high; second, according to linear stability
analysis14, when frequency is larger than a critical value, the cut-off wavenumber kR0 > 1 (k is the wavenumber
and R0 is the unperturbed jet radius) and thus prevents the external perturbation from growing (see the
Supplementary Information for validation). The flow was visualized, monitored and recorded (images and
videos) by an inverted microscope (Nikon Eclipse TS100, Inverted Microscope) equipped with a high-speed
camera (MotionPro? X4, IDT, Taiwan). Captured images and videos were analyzed by ImageJ.
Water-in-oil two-phase flow was examined in experiments, where outer fluid was silicone oil (viscosity:
?o = 492.875 mPa s), and inner fluids were various glycerol-water mixtures (viscosity: 0.992 mPa s
? ?i ? 831.07 mPa s). This choice of two-fluid systems enables a wide range of viscosity ratios (10?3 < ? < 10),
but narrow variations in interfacial tension (24.2 mN m?1 ? ? ? 32 mN m?1) and fluid density (0.998 g cm?3
? ?i ? 1.261 g cm?3), so as to isolate the effect of viscosity ratio on pinch-off dynamics. Viscosity was measured by
a viscometer (microVISCTM, RheoSense, Inc.), interfacial tension by a ring tensiometer (Surface Tensiometer 20,
Cole-Parmer), and density by quantifying the volume of a known-mass fluid.
Estimate of the inner fluid oscillatory velocityVn. With perturbation, the inner phase pressure is disturbed
periodically, so that the velocity Vn of the inner fluid ejected from the nozzle pulsates with time. We have made a
simple analysis to determine the pulsatile velocity Vn. Before accounting for the effect of mechanical perturbation,
we first consider the unperturbed case, in which the inner phase flow rate Qi is estimated by Poiseuille?s law37,
where Riw is inner-wall radius of the microtubing (Riw = 0.43 mm), ?i is the viscosity of inner fluid, Pp and Pn are
pressures respectively at the perturbing location and the nozzle, and L is the length between the two locations.
Based on Qi, we evaluate the steady ejection velocity of the inner fluid as,
For perturbed case, we simply assume that the gravitational displacement of y = ?0 sin(2?ft) modifies the
steady pressure Pp into the time-dependent Pp(t),
with inner fluid density being ?i and the gravitational acceleration being g. Since the linear pinching time-scale tl
(tl ~ 1 ms, which is shown in the next section, Fig.?2) is much smaller than the perturbation period tp (tp ? 10 ms
for f ? 100 Hz in experiments), we assume that the Poiseuille?s law is still valid to estimate the fluctuating inner
flow rate Qi(t) in the linear pinch-off regime. When Pp is replaced by Pp(t) in Eq.?(1), we obtain Qi(t) as a function
of time t,
Qi (t) = Qi ??1 +
?Ri4w ?ig ?0 sin(2?ft) ??
V n = V 0(1 + ? sin(2?ft)),
Now the oscillatory velocity Vn discharged from the nozzle is approximated to beV n = 4Qi (t)/?Dn2, and hence
in the form of
where ? = ?Ri4w ?ig ?0/8Qi?iL is the dimensionless amplitude of the fluctuation term.
Although we ignore several effects that might influence the transient velocity Vn, such as shaking and
deformation of the microtubing, and fluid acceleration in the microtubing, the simplified form of Qi(t) in Eq.?(4)
(as well as Vn in Eq.?(5)) does characterize essentially how oscillation amplitude ?0, in association with material
properties (?i and ?i) and geometrical parameters (Riw and L), modulates the ejection velocity. We verify Eq.?(4)
by comparing the experimental measurements to theoretical predictions of the inner fluid discharged volume
(see supplementary Figs S1 and S2). Moreover, we will show in the next section that pinch-off is enhanced by the
vibration in the gravitational direction, while no enhancement is identified when the vibration is parallel to the
horizontal plane (see supplementary Fig. S3). This difference suggests the dominant effect of gravity in impacting
neck pinch-off. As a result, the above analysis paves a way to evaluate the average momentum transferred into the
fluid system, to quantify the acceleration of pinch-off with perturbation, to rescale the perturbed neck pinch-off,
and to account for the non-negligible inner fluid inertia in perturbed case, which are shown in the next section.
Results and Discussion
Scaling of classical viscous two-fluid pinch-off. We start from the case without oscillation but with the
influence of viscosity ratio ?. Keeping silicone oil as the outer fluid, and choosing various glycerol-water mixtures
as the inner fluid, we can tune viscosity ratio ? over three magnitudes (10?3 < ? < 10). Fluid system with smaller
? means less viscous inner fluid in the context, and vice versa. Figure?1e compares unperturbed minimum neck
diameter Dmin (Fig.?1b, Dmin = 2Rmin) versus time ? (? = tc ? t, time remaining to neck pinch-off) for different
viscosity ratios ?. Fluid system with smaller viscosity ratio pinches off faster (Fig.?1e) due to the less inner viscous
resistance. In the vicinity of pinch-off, a linear scaling of the neck diameter is presented (inset of Fig.?1f). In this
linear pinch-off scenario, two-fluid Stokes flow dominates, where inner axial viscous stress ?iu/z (with the
kinematic estimate being u ~ z/?, z the length scale of the pinching neck and u the axial velocity of fluid flowing out
of the neck), outer radial viscous stress ?ou/Rmin, and capillary pressure ?/Rmin compete with each other15. This
viscous-capillary force balance yields the scaling of neck diameter as Dmin = F(?)V ?? (see the Supplementary
Information for derivations), where V ? = ?/?o is viscous-capillary velocity, and F(?) is a dimensionless
proportionality factor that can be evaluated from linear stability analysis16?18,38. Normalizing Dmin by the nozzle diameter
Dn, we obtain a linear scaling of the minimum neck diameter in the form of
with viscous time scale t? = Dn/V ?. Based on the analysis by Tomotika38, we evaluate F(?) for different viscosity
ratios (see the Supplementary Information for the estimate of F(?)), and replot the data in Fig.?1f, which is in
good agreement with previous study of viscous two-fluid drop pinch-off15?18.
Perturbation accelerated pinch-off. Now we consider the fluid system with fixed viscosity ratio
(? = 0.0387) under periodic perturbation. When undergoing pinch-off subjected to an oscillatory flow, the
droplet initially grows in its size until the neck appears. Afterwards, a sudden suction of the inner fluid, caused by the
pulsatile velocity Vn (Eq.?(5)), triggers and subsequently accelerates the neck pinch-off (exampled in Fig.?1d, and
see supplementary Movies S1?S5). The above process is identical to any individual pinch-off with oscillation (see
supplementary Movie S6)). As a result, actuated neck thinning velocity Vr is faster than the case without
actuation, while the linear neck thinning regime features both cases in the vicinity of pinch-off (Fig.?2a,b). The neck
thinning velocity Vr increases apparently with the oscillation amplitude ?0 (Fig.?2a). However, the pinching
velocity weakly depends on the oscillation frequency for f ? 100 Hz (Fig.?2b), probably due to the mismatch between
the two time scales, the linear pinching time scale tl (tl ~ 1 ms; see dashed line in Fig.?2a,b) and the perturbation
period tp (tp? 10 ms for f ? 100 Hz).
We now quantify the acceleration of perturbed pinch-off by estimating the extra momentum contribution
from mechanical vibration. After simply calculating the root mean square of the fluctuation term in Eq. (5),
? V ~ f ?01/f V02?2 sin2(2?ft)dt ~ V02?2/2 ~ 2 ?V 0/2, we extrapolate that the average momentum is
increased by a ratio of 2 ?/2 due to mechanical vibration, and hypothesize that the neck thinning velocity is
accelerated by a ratio of1 + 2 ?/2 as an estimate. To verify this hypothesis, we replot Fig.?2a,b by rescaling Dmin
versus the product of time ? and the dimensionless factor1 + 2 ?/2. After rescaling, all data essentially collapse
onto a single master curve in the vicinity of pinch-off (Fig.?2c,d). The enhancement of pinch-off comes thus from
the additional momentum transportation induced by mechanical vibration, and is proportional to the
dimensionless amplitude ?.
Rescaling perturbed neck thinning with various viscosity ratios. In subsequent tests, we focus on
the influence of viscosity ratio on fluid systems with oscillation. In comparison to unactuated case, the
oscillation-enhanced pinch-off has a faster neck thinning rate for all fluid systems examined, as shown in Fig.?3a.
Note that Fig.?2b only indicates a dependence of neck thinning velocity on oscillation amplitude ?0, but Fig.?3a
further clarifies the dependence on the dimensionless amplitude ?, in that larger enhancement of pinch-off is
observed for fluid system with the same ?0 but smaller viscosity ratio ? (smaller ?i in experiments). Therefore,
Eq.?(6) fails in scaling the perturbed neck thinning with arbitrary oscillation amplitudes and viscosity ratios. From
previous discussions, neck thinning velocity is accelerated by a ratio of 1 + 2 ?/2. We then rescale the neck
shrinkage behavior via multiplying the right hand side of Eq.?(6) by the dimensionless pre-factor1 + 2 ?/2:
Dmin/Dn = (1 +
as shown in Fig.?3b (data rescaled from Fig.?3a). To highlight this rescaling, we replot representative data close to
pinch-off in the inset of Fig.?3b, where all data falls onto a single master straight line with unit-slope, in excellent
agreement with Eq.?(7).
Non-negligible inner fluid inertia. Having quantified the accelerated pinch-off, we now explore the origin
of the acceleration in pinch-off with oscillatory flow. For conventional pinch-off scenario, two-fluid Stokes flow
dominates near the singularity. However, oscillation modifies the force balance via disturbing the flow
velocity, therefore affecting the whole pinch-off dynamics. A recent study proposed that, when pinching in air, fluid
thread passes through a plethora of transient regimes before transferring into the final inertial-viscous regime39.
arbitrary ? is developed (Fig.?3e) based on the formula: F(?l) = (1 + 2 ?/2)F(?). According to above analysis, we
can even modulate a glycerol-water mixture undergoing ?less viscous? pinch-off dynamics than distilled water by
exerting appropriate oscillation amplitude (Fig.?4). The equivalence of pinch-off dynamics indicates potential in
breaking highly viscous liquid in an easier manner by applying an oscillatory flow. Our findings show that
dynamics of neck thinning in the linear regime is controlled by both fluid properties and oscillation amplitude. By
characterizing the combined effects of viscosity ratio and dimensionless amplitude (Eq.?(7) and Fig.?3e), we can
predict the dynamics of linear neck pinch-off.
Shifts of pinch-off location with perturbation. Besides neck thinning dynamics, pinch-off location
also changes with applied oscillation. When breaking up, the liquid thread that connects the main drop and the
residual fluid can pinch off either at its front or rear side40,41 (Fig.?5a, and see arrows in Fig.?5b?f). Compared with
unactuated case (Fig.?5b), applied oscillation shifts pinch-off location upstream towards the nozzle (Fig.?5c?f,
and see supplementary Movies S1?S5) due to the pulsatile velocity Vn. More remarkably, while front pinch-off
occurs before rear pinch-off for unperturbed case (Fig.?5b), large driving amplitude ?0 can introduce the reversed
case (Fig.?5d?f ), between which a symmetric neck forms with front and rear pinch-offs occurring
simultaneously (Fig.?5c). Meanwhile, oscillation can also induce a transition of pinch-off location from outside the nozzle
(Fig.?5b?d) to inside the nozzle (Fig.?5f).
We map the pinch-off locations in Fig.?5g by tracing the location of minimum neck Z0 (Fig.?1b) versus time
(t ? tc) before pinch-off. Two boundaries are emphasized in Fig.?5g: symmetric neck formation (Z0 bifurcates,
pentagon) and pinch-off at the nozzle (Z0 = 0 when t = tc, circle). The transition of pinch-off location is the result
of momentum transport. Because axial velocity u of inner fluid scales as u ~ V r ~ (1 + 2 ?/2), it is reasonable
to assume that the two boundaries correspond to certain critical values of ?. Experimentally, we find that, in the
tested range of viscosity ratios, ?c = 0.110 and ?c = 0.225 (Fig.?5h) are the two critical values for the symmetric
neck formation and pinch-off at the nozzle, respectively. These results provide a method to precisely manipulate
pinch-off location by using oscillatory velocity.
Being susceptible to the details of breakup40,41, satellite droplet formation varies with the shift of pinch-off
location. The multi-breakup of liquid thread generates multiple satellite and subsatellite droplets (circles in
Fig.?5b?f ) owing to the self-repeated neck formation42,43. With increased oscillation amplitude, the number of
thread fragmentation rises, inducing more satellites and subsatellites (Fig.?5b?f ); for instance, as many as ten
satellites are generated with ?0 = 5 mm (Fig.?5f ). In addition, the size distribution of satellite drops relies on the
symmetry of neck formation. Symmetric pinch-off induces symmetric size distribution (Fig.?5c); asymmetrical
neck formation otherwise renders the size distribution asymmetric (Fig.?5b,d,e). Astonishingly, rear pinch-off
inside the nozzle produces satellite drops with descending size distribution (Fig.?5f), which are similar to those
generated by the tip-multi-breaking, a recently reported droplet breakup mode44,45. Despite their non-uniformity,
the satellites depend on oscillation, which may provide an additional handle to harness the formation of
droplets with various volumes, a required feature in some applications. For example, in multi-volume droplet digital
polymerase chain reaction (MV-dPCR)46, droplets with various volumes enable simultaneous measurements of a
sample at different copies per droplet. Compared to single-volume digital PCR, MV-dPCR achieves higher
detection reproducibility, wider dynamic range and better resolution while reducing the total number of droplets/wells
required for the measurements46?48.
In conclusion, we have experimentally examined the dynamics of pinch-off in viscous two-fluid systems with
oscillatory velocity actuated by mechanical perturbation. Attributed to the oscillatory flow, an enhanced suction
of inner fluid towards the nozzle occurs in the last stage of pinch-off. In this scenario, the inner fluid thread thins
radially faster prior to breakup compared to the unactuated case, still in a linear way though. The enhancement of
pinch-off by external actuation depends on the oscillation amplitude. We rescale the actuated neck radius by a
dimensionless pre-factor 1 + 2 ?/2, where ? is the dimensionless amplitude. Actuating the fluid system
modulates the local force balance via increasing the effects of inner fluid inertia. Therefore, the enhanced fluid inertia is
non-negligible and responsible for the accelerated neck thinning. Meanwhile, the actuated pinch-off displays a
?less viscous? behavior due to the counterbalance between the enhanced inertia and the inner viscous resistance.
Such a ?less viscous? performance would enable robust control over breaking viscous liquid filament more easily.
Moreover, the enhanced inner fluid inertia shifts the pinch-off location upstream towards the nozzle, which
afterwards affects satellite droplet formation. By quantifying the relationship between the oscillation amplitude and
the pinch-off location, novel control over satellites formation by oscillation could probably be developed.
Beyond technical benefit in modulating droplet formation resulted from our study, this work raises several
questions of hydrodynamic interest but remaining to be addressed. Among them, two issues are of the most
importance. One is that if the oscillation effect could dominate capillary effects, a new regime might occur, which
may be similar to the thermal fluctuation dominant regime20,21. Another is that the existence of transient regimes
during two-fluid pinch-off from the initial to final regime remains unexplored. Deeper understanding towards
these issues calls for experimental studies in association with numerical simulations and theoretical explanations.
Supplementary information accompanies this paper at http://www.nature.com/srep
Competing financial interests: The authors declare no competing financial interests.
How to cite this article: Zhu, P. et al. Pinch-off of microfluidic droplets with oscillatory velocity of inner phase
flow. Sci. Rep. 6, 31436; doi: 10.1038/srep31436 (2016).
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The financial support from the Research Grants Council of Hong Kong (GRF 17237316 , 17211115 , 17207914, and GRF HKU717613E) and the University of Hong Kong (URC 201511159108, 201411159074 and 201311159187) is gratefully acknowledged. The work is also supported in part by the Zhejiang Provincial, Hangzhou Municipal and Lin'an County Governments . In particular, the authors wish to thank Dr. Ho Cheung Shum for the generous use of mechanical vibrator .