An Interface-Particle Interaction Approach for Evaluation of the Co-Encapsulation Efficiency of Cells in a Flow-Focusing Droplet Generator

Abstract

Droplet-based microfluidics offers significant advantages, such as high throughput and scalability, making platforms based on this technology ideal candidates for point-of-care (POC) testing and clinical diagnosis. However, the efficiency of co-encapsulation in droplets is suboptimal, limiting the applicability of such platforms for the biosensing applications. The homogeneity of the bioanalytes in the droplets is an unsolved problem. While there is extensive literature on the experimental setups and active methods used to increase the efficiency of such platforms, passive techniques have received less attention, and their fundamentals have not been fully explored. Here, we develop a novel passive technique for investigating cell encapsulation using the finite element method (FEM). The level set method was used to track the interfaces of forming droplets. The effects of walls and the droplet interfaces on relatively large cells were calculated to track them more accurately during encapsulation. The static surface tension force was used to account for the effects of the interfaces on cells. The results revealed that the pairing efficiency is highly sensitive to the standard deviation (SD) of the distance between the cells in the entrance channel. The pairing efficiency prediction error of our model differed by less than 5% from previous experiments. The proposed model can be used to evaluate the performance of droplet-based microfluidic devices to ensure higher precision for co-encapsulation of cells.

Keywords: droplet generator; encapsulation efficiency; finite element method; flow focusing; microfluidics; particle interaction.

Figure 1

(a) Model and dimensions. Inertial focusing devices are used to control the injection of particles into the computational domain. (b,c) The lateral equilibrium locations of the particles in straight and spiral flow-focusing devices, respectively. In spiral microchannels with specific aspect ratios and particle densities, the equilibrium location of the particles changes to near the outer side of the spiral. The rectangular red frames show the cross-sections in which particles lie to better emphasize their local positions. (d) Microchannels with spiral and (e) straight flow-focusing devices. The magnified part (f) shows the droplet formation junction, which is separated with a red rectangular frame in (d,e).

Figure 2

The modified algorithm for calculating the lift force exerted on particles in the entrance channels. The initial values of the U_p and ω_x, ω_y are set from the velocity and angular velocity by solving the Navier–Stokes equations (NS) for laminar incompressible flow in a channel without particles. With these initial conditions, the NS will be solved in a channel with particles in the x_p and y_p positions. In the inner loop, after each iteration, the values of (V,p) are used as the initial conditions for the next iteration.

Figure 3

The 12 cross-sections along the length of the channel are used to calculate the lift forces on particles considering trilinear interpolation between these sections. Particles are dominated by the drag force in the right portion of the design after the tapered section; therefore, the effect of lift forces on particle encapsulation is negligible in this zone. Here, $x_o$ stands for the local x-coordinate laid in the direction of the oriented channel, and subscript t shows the local x-coordinate of the tapered section.

Figure 4

The interaction between a particle and an interface. (a) Green cells are those with volume fractions between 0 and 1 that are touched by the particle, or those for which the distance between their center and the center of the particle is less than the particle radius. The total volume of these cells equals NV_σ. (b) The dimensions and parameters used to calculate the surface tension force exerted on the particle by the interface. The dashed red lines show the faces of touched cells.

Figure 5

Grid study on (a) the droplet diameter and frequency and (b) the velocity of the center of the incoming channel. The mesh number effect on the droplet diameter is not significant (less than 1%), and thus the lowest number of mesh cells needed for calculations is determined by other variables, such as the velocity field. (c) The W/O interface in the midplane of the domain in t = 0.0005 s with different mesh numbers, which shows the independence of the mesh.

Figure 6

The lift forces on three different cross-sections: (a) on x = 20 μm, (b) on x = 0, and (c) on x₀ = 88.32 μm (see Figure 3). The x-y coordinate is a local coordinate whose dimensions are shown on each rectangle. The circle only exhibits the scale of the particle dimensions in comparison with those of the channel. All dimensions are in μm.

Figure 7

Here, θ_wall = 5°, σ = 0.06 N/m, and EF = 2.63. The pictures numbered 1 to 5 show the W/O common interface at different time instants in the injection tip (Video S1).

Figure 8

(a) Here, σ = 0.06 and (b) σ = 0.05207 N/m for EF = 2.63. The average droplet diameter for the first 10 droplets. The superhydrophobic wall contact angles produce a less stable regime. The red color shows the medium phase.

Figure 9

(a) The effect of the geometrical EF on the droplet formation stability and size with respect to different contact angles (σ = 0.05207 N/m]) and (b) droplet diameters with respect to σ. The greater the surface tension coefficient, the larger the droplets.

Figure 10

(a,b) Encapsulation efficiency of spiral and straight inertial focusing devices for two SD values of 5 and 10 μm. The scattering of the droplet contents shows similar productivity for both devices; other than small variations in the cell pair scattering, changing the focusing channel type does not affect the efficacy of the encapsulation. (c) The pairing efficiency increases by lowering the frequency. The frequency is reduced by increasing σ to 0.065 N/m. (d) The efficiency values and scattering cell pairs are from [2], with an approximate SD of 7 μm.