Droplet shape relaxation in a four-channel microfluidic hydrodynamic trap

Two-phase liquid-liquid systems are prevalent in a range of commercial and environmental applications. Understanding the behavior of liquid-liquid systems under various processing conditions requires the study of droplet dynamics under precisely controlled flow fields. Here we trap and control the position of droplets in a microfluidic trap to study their dynamics using hydrodynamic forces alone without an external field. The hydrodynamic trap is adapted from a previously implemented “Stokes trap” by incorporating a drop-on-demand system to generate droplets at a T-junction geometry on the same microfluidic chip. Using the hydrodynamic trap, confined droplet dynamics in response to perturbation are studied by applying a millisecond-pressure pulse to deform trapped droplets. Droplet shape relaxation after cessation of the pressure pulse follows an exponential decay. The characteristic droplet shape relaxation time is obtained from the shape decay curves, for aqueous glycerol droplets of varying viscosities in the dispersed phase with light and heavy mineral oils in the continuous phase. Systems were chosen to provide similar equilibrium interfacial tensions (5–10 mN/m) with wide variations of viscosity ratios. It is found that the droplet shape relaxation in the moderately confined regime shows a strong dependence on droplet radius, and a weaker dependence on the ratio of dispersed to continuous phase viscosity. An empirical scaling relationship is developed, and relaxation times from the experiments are compared to theoretical relaxation times for the limiting regime of unconfined droplets. The droplet response in the moderately confined regime differs from both limiting regimes of unconfined and highly confined droplets with regards to the radius scaling. Droplet shape relaxation time can be used inform the response of droplets in an emulsion when subjected to transient flows in various processing conditions. Finally, an application of this platform for directly visualizing droplet coalescence in planar extensional flow is presented. The microfluidic four-channel hydrodynamic trap can thus be applied for studying the fundamental physics of droplet deformation and droplet-droplet interactions on the microscale.

Figure 1

(a) Experimental setup of the hydrodynamic trap for droplets, showing the microfluidic device mounted on an inverted microscope with the rigid tubing (red) used to increase flow resistance. (b) Schematic of the trapping setup, showing (1) microfluidic chip, (2) microscope stage, (3) centrifuge tubes used as reservoirs for the fluids, (4) continuous phase fluid (oil), (5) dispersed phase fluid (aqueous), (6) tubing used to deliver oil to the device, (7) tubing used to deliver aqueous phase to the device, (8) analog-controlled pressure regulators, and (9) tubing used to deliver pressurized air to the reservoirs.

Figure 2

(a) Microfluidic device design for the modified hydrodynamic trap, showing the location of the T-junction for droplet formation on the microfluidic chip and the four-channel cross-slot for droplet trapping on the same chip. (b) Dimensions of the T-junction for droplet formation (c) Dimensions of the cross-slot for droplet trapping, including the designations ${\mathrm{P}}_1\text{–}{\mathrm{P}}_4$ for the four channels forming the cross-slot. (d) Image of a droplet trapped in the modified hydrodynamic trap. The red square is an overlay used to indicate the desired position or “set point” for trapping, and the red circle is used to indicate the current or actual position of the droplet.

Figure 3

(a) Interfacial tension measurements using pendant drop tensiometry for light mineral oil with 0.05% v/v SPAN 80 in the continuous phase with 0–70% aqueous glycerol in the dispersed phase. (b) Interfacial tension measurements using pendant drop tensiometry for heavy mineral oil with 0.05% v/v SPAN 80 in the continuous phase with 0–70% aqueous glycerol in the dispersed phase.

Figure 4

(a)–(d) Droplet shape perturbation after application followed by cessation, of a pressure pulse at various instants of time, showing deformation and relaxation of droplet shape. The original undeformed projected droplet diameter $d_p$ and the lengths of the major and minor axes of the deformed droplet ($d_{\mathrm{major}}$ and $d_{\mathrm{minor}}$) are indicated in panels (a) and (c), respectively. (e) Sketch of the droplet confined by the walls of the microfluidic device, showing the direction of the applied pressure pulse (blue arrows) and the channel height $h$ compared with the projected drop diameter $d_p$.

Figure 5

(a) Droplet deformation $D$ as a function of time for the system containing 50% glycerol-water solution in the dispersed phase with light mineral oil in the continuous phase ($\lambda =0.257$). Different symbols indicate droplets with different initial projected radii ($r_p=d_p/2$), and the channel height ($h$) is taken to be 90 µm. Droplets attain highly deformed shapes, followed by relaxation back to equilibrium. (b) Maximum deformation as a function of projected diameter for aqueous glycerol systems (0%, 20%, 50%, 70%) in light mineral oil. (c) Maximum deformation as a function of projected diameter for aqueous glycerol systems (0%, 20%, 50%, 70%) in heavy mineral oil.

Figure 6

(a), (b) Dimensionless droplet shape relaxation time as a function of confinement ratio for (a) light mineral oil and (b) heavy mineral oil in the continuous phase with 0%, 20%, 50%, and 70% aqueous glycerol in the dispersed phase. The error bars indicate propagated uncertainty arising from uncertainty in the relaxation time fits and interfacial tension and channel height measurements. The solid lines are examples of power-law fits to the relaxation time data with corresponding colors.

Figure 7

(a) Droplet deformation versus time for the light mineral oil–50% aqueous glycerol system ($\lambda =0.257$) at various droplet sizes. The channel height $h$ is taken to be 90 µm. (b) Normalized deformation $\frac{D}{D_{\max }}$ (left axis) as a function of rescaled time $t_1=t{\left(\frac{h}{r_p}\right)}^{0.47}$, for the same system. The black line is an exponential decay fit to the collapsed droplet shape relaxation curve. The residuals for the fit are indicated on the right axis, with a goodness-of-fit parameter $R^2=0.98$.

Figure 8

Dimensionless scaled relaxation time ${\tau }^{\prime }=\tau {\left(\frac{h}{r_p}\right)}^{0.47}$, as a function of viscosity ratio. The solid line indicates a power-law fit to the data, with a goodness-of-fit parameter $R^2=0.8$. The error bars on the data points indicate propagated uncertainty arising from uncertainty in the relaxation time fits and interfacial tension and channel height measurements.

Figure 9

(a) Theoretical relaxation time for unconfined droplets as a function of droplet diameter over the range of droplet diameters and viscosity ratios in this study. (b) Theoretical relaxation time for unconfined droplets as a function of viscosity ratio for a droplet of equivalent sphere diameter 157 µm with an interfacial tension of 6 mN/m.

Figure 10

Coalescence of an incoming droplet with a droplet trapped at the center of the cross-slot showing the various stages of coalescence. The system shown here consists of confined water droplets in light mineral with 0.01% v/v SPAN 80, studied in detail in Narayan et al. [59].