An asymmetric flow-focusing droplet generator promotes rapid mixing of reagents

Nowadays droplet microfluidics is widely used to perform high throughput assays and for the synthesis of micro- and nanoparticles. These applications usually require packaging several reagents into droplets and their mixing to start a biochemical reaction. For rapid mixing microfluidic devices usually require additional functional elements that make their designs more complex. Here we perform a series of 2D numerical simulations, followed by experimental studies, and introduce a novel asymmetric flow-focusing droplet generator, which enhances mixing during droplet formation due to a 2D or 3D asymmetric vortex, located in the droplet formation area of the microfluidic device. Our results suggest that 2D numerical simulations can be used for qualitative analysis of two-phase flows and droplet generation process in quasi-two-dimensional devices, while the relative simplicity of such simulations allows them to be easily applied to fairly complicated microfluidic geometries. Mixing inside droplets formed in the asymmetric generator occurs up to six times faster than in a conventional symmetric one. The best mixing efficiency is achieved in a specific range of droplet volumes, which can be changed by scaling the geometry of the device. Thus, the droplet generator suggested here can significantly simplify designs of microfluidic devices because it enables both the droplet formation and fast mixing of the reagents within droplets. Moreover, it can be used to precisely estimate reaction kinetics.

where • ρ is the density (kg/m 3 ); • u is the velocity vector (m/s); • p is the pressure (Pa); • I is the identity matrix; • μ is the dynamic viscosity (Pa·s); • F st is the surface tension force (N/m 3 ).
The spatial distribution of the two phases was described by a phase field φ, which takes values from -1 to 1. To determine the phase-field evolution minimization of the system's free energy was performed by the Cahn-Hilliard equation, which is a 4th-order PDE. The Phase Field interface decomposed the Cahn-Hilliard equation into two second-order PDEs using help variable ψ: The quantity λ (N) is the mixing energy density and ε (m) is a capillary width that scales with the thickness of the interface. ε was set as a half of the maximum mesh size. These two parameters are related to the surface tension coefficient, σ (N/m), through the equation and mobility γ (m 3 ·s/kg) is related to ε through 2 γ χε = where χ (m·s/kg) is the mobility tuning parameter.
The surface tension force was added to the Navier-Stokes equations as a body force by multiplying the chemical potential of the system G by the gradient of the phase field variable: Transport of Diluted Species interface was added to the consideration by solving convection-diffusion equation: where c is the concentration of a reagent (mol/m 3 ), D is the diffusion coefficient (m 2 /s), u is the velocity vector (m/s) calculated from (1) and (2).
To improve the quality of the obtained results quadratic shape functions were used to interpolate u, c, φ and ψ between the mesh nodes, and a linear shape function was used for pressure interpolation p. We used triangular mesh as the design of the droplet generator had acute and obtuse angles. The study of convergence with the decrease of the mesh size was conducted for asymmetric design with Q d = 0.2 and Q d = 0.7 μl/min. In both cases Q c = 1 μl/min and D = 3.5·10 -10 m 2 /s (fig. S1). Mesh size equal 1 μm in a pinch region and 1.5 μm in others (154949 degrees of freedom) provided reasonably accurate results without significant increase in computing requirements. So we have chosen it for the simulations. Figure S1. Mixing index obtained during simulation for convergence study. Q d = 0.5, Q c = 1 ul/min and D = 3.5·10 -10 m 2 /s The advantage of the phase field method is that it provides the opportunity to calculate contact line displacement with no slip boundary condition for fluid velocity. It reduces pressure jumps at the corners and prevents artificial vortex in the area of channel crossing. Also the phase field method, as an interface capturing method, provides the opportunity to resolve droplet breakup, but Cahn-Hilliard diffusion may shift the interface contour and effectively change the size of a drop. This leads to escape of some reagent quantity from one phase to another. To minimize such effect the reagent was injected along the longer side of the channel decreasing its contact with liquid-liquid interface ( Figure S6. Distribution of the dimensionless value K = div(V 2D )*h/|V 2D | that characterize three dimensional flow during the filling stage of droplet formation, where V 2D -2D velocity of the flow, measured by PIV; h -channels depth. a) Symmetric geometry, channels depth is 40 μm, b) symmetric geometry, channels depth is 60 μm, c) asymmetric geometry, channels depth is 40 μm, d) asymmetric geometry, channels depth is 60 μm. The scale bar is 30 μm.

Video captions
Video V1. 2D simulation of a droplet formation process: disperse phase velocity profile (top) and during concentration distribution (bottom) during droplet formation in an asymmetric droplet generator. Continuous phase flow rate is 1 μl/min, dispersed phase flow rate is 0.2 μl/min. The scalar bar is 30 μm.
Video V2. 2D simulation of a droplet formation process: disperse phase velocity profile (top) and concentration distribution (bottom) during droplet formation in a symmetric droplet generator. Continuous phase flow rate is 1 μl/min, dispersed phase flow rate is 0.2 μl/min. The scalar bar is 30 μm.
Video V3. Droplet formation process in an asymmetric droplet generator with channels depth h = 40 μm. Disperse phase contains 1 um tracer particles for PIV measurements. Continuous phase flow rate is 1 μl/min, dispersed phase flow rate is 0.2 μl/min. The scalar bar is 30 μm.
Video V4. Droplet formation process in an asymmetric droplet generator with channels depth h = 60 μm. Disperse phase contains 1 um tracer particles for PIV measurements. Continuous phase flow rate is 1 ul/min, dispersed phase flow rate is 0.2 μl/min. The scalar bar is 30 μm.
Video V5. Droplet formation process in an symmetric droplet generator with channels depth h = 40 μm. Disperse phase contains 1 μm tracer particles for PIV measurements. Continuous phase flow rate is 1 μl/min, dispersed phase flow rate is 0.2 μl/min. The scalar bar is 30 μm.
Video V6. Droplet formation process in an symmetric droplet generator with channels depth h = 60 μm. Disperse phase contains 1 um tracer particles for PIV measurements. Continuous phase flow rate is 1 μl/min, dispersed phase flow rate is 0.2 μl/min. The scalar bar is 30 μm.