Droplet Breakup in Expansion-contraction Microchannels

We investigate the influences of expansion-contraction microchannels on droplet breakup in capillary microfluidic devices. With variations in channel dimension, local shear stresses at the injection nozzle and focusing orifice vary, significantly impacting flow behavior including droplet breakup locations and breakup modes. We observe transition of droplet breakup location from focusing orifice to injection nozzle, and three distinct types of recently-reported tip-multi-breaking modes. By balancing local shear stresses and interfacial tension effects, we determine the critical condition for breakup location transition, and characterize the tip-multi-breaking mode quantitatively. In addition, we identify the mechanism responsible for the periodic oscillation of inner fluid tip in tip-multi-breaking mode. Our results offer fundamental understanding of two-phase flow behaviors in expansion-contraction microstructures, and would benefit droplet generation, manipulation and design of microfluidic devices.


Legends of Supplementary Movies S1 to S2
1.1 Supplementary Movie S1:

Tip Morphology Transformation and Breakup at Injection Nozzle
This video demonstrates the transformation of inner liquid tip with enlarging orifice distance L and the occurrence of breakup at injection orifice in a capillary microfluidic device.
The inner phase is a mixture of 70 wt.% glycerol and 30 wt.% distilled water, and outer phase is silicone oil. The inner and outer phase flow rates are 0.5 mL h -1 and 3.5 mL h -1 , respectively. The video is recorded with 500 frames per second (fps) and played with 10 fps.

Supplementary Movie S2: Droplet Size Distribution Affected by Orifice Distance L
This video shows three different size distributions in tip-multi-breaking mode influenced by orifice distance L. Droplets are generated sequence by sequence periodically in this situation. The inner phase is a mixture of 70 wt.% glycerol and 30 wt.% distilled water, and outer phase is silicone oil. The inner and outer phase flow rates are 0.02 mL h -1 and 3.5 mL h -1 , respectively. The video is recorded with 500 fps and played with 10 fps.

Confirming Constant Inner Flow Rate
To confirm the constant inner flow rate in our experiments, the inner and outer flow rates are set by syringe pumps with Q in = 4.5 μL h -1 and Q out = 3 mL h -1 , respectively. The channel dimension are D f = 197 μm and D i = 147 μm. Various droplet generation processes are obtained with the increase of orifice distance L (Fig. S5).
To estimate the inner flow rate by analyzing the captured video, we build a general model to calculate the volume of droplets in one sequence (Fig. S6). The three types of droplet size distribution (descending, constant-decreasing, and increasing-constant-decreasing) are distinguished by the values of n 1 , n 2 and b ( Fig. S6 and Table S2). Considering the symmetry, we argue that θ 1 = θ 2 for droplet sequence of increasing-constant-decreasing mode.
For the left and right sequences, we assume that droplet size obeys a geometrical progression, with the first droplet size R 1 = R. So we have for both of the left and right sequences. Common factor is calculated by = (1 − )/(1 + ) 1 . Therefore, the volumes of droplets for left and right sequences are Subscript "k" can be either "1" or "2", indicating the left and right sequence, respectively. For the middle part with constant droplet size distribution, the volume is calculated as follows: Thus, the total volume for one droplet sequence gives as follows, Accordingly, the inner flow rate Q in is estimated by the following equation: with T being the period of the droplet sequence generation. All the measurements and related calculations can be found in Table S2.
We plot the results of Q in versus L in Fig. S7. It suggests that within the tested range of L, inner flow rate is fairly constant, bounded in the range of 4 μL h -1 to 5 μL h -1 , consistent with the value monitored by syringe pump of 4.5 μL h -1 .

Determining the Most Unstable Mode of a Viscous Jet
Here we consider that an initial infinitesimal perturbation grows on a long cylindrical viscous jet with viscosity in  immersed in another viscous liquid with viscosity out  , and the growing perturbation is proportional to () i t kz e   . According to Tomotika 2 , the dispersion relation gives, where  is frequency of the perturbation,  the interfacial tension, 0 R the radius of the unperturbed jet.
with   In Eqs. (8) and (9), () Ix and () Kx are modified Bessel Functions, with subscripts "0" and "1" representing the order. In addition, 1  , 2  , 3  and 4  are functions of x and  , given in the following form: The most unstable mode corresponds to i with the biggest value, and i is proportional to so for a situation with viscosity ratio  fixed, the most unstable jet matches the value 0 x that yields the maximum 0 ( , ) Fx  . Since ( , ) Fx has only one maximum value in the whole domain of x , as shown in Fig. S8, we find the maximum of ( , ) Solving Eq. (15) in MATLAB, we obtain the value of 0( ) x  . We thus have, by using 0 Letting orifice distance L   , and injection orifice diameter as the condition for the occurrence of breakup at injection nozzle. Eq. (17)    S10 Figure S5. Snapshots of droplet sequence generation with enlarging orifice distance L.
The value of L is provided below each image. Inner and outer flow rates are held constant by the syringe pumps with Q in = 4.5 μL h -1 and Q out = 3 mL h -1 (the pump we used can achieve controlling flow rate as low as 1 μL h -1 by using 1cc Terumo plastic syringe). Droplet size distribution changes from descending, to constant-decreasing, and to increasing-constant-decreasing mode with the enlarging L. Scale bar, 300 μm. Figure S6. Model for calculating the inner flow rate Q in . The droplet sequence is divided into three parts: the left sequence with droplet numbers n 1 and apex angle 2θ 1 , the middle part with constant droplet size of number b, and the right sequence with droplet numbers n 2 and apex angle 2θ 2 . For droplet sequence with "descending" size distribution, n 2 = b = 0; for the sequence with "constant-decreasing" size distribution, n 2 = 0; for that with "increasing-constant-decreasing" size distribution, none of n 1 , n 2 and b is zero. R is the radius of the droplet with constant size, and is also the first droplet size "R 1 " for the left and right sequences.      Table S2. Data for calculating the inner flow rate Q in measured from the video. In the row of "Type", "De", "Con-de" and "In-con-de" represent droplet sequence with size distribution in the form of "Descending", "Constant-decreasing" and "Increasing-constant-decreasing", respectively. For L = 439.4 μm, b = -1 means that the left sequence with apex angle 2θ 1 and the right sequence with 2θ 2 share the same biggest droplet.

Supplementary Tables
In this case, the droplet firstly increases in size, and then directly decreases without keeping constant.