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

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## Abstract

**:**

## 1. Introduction

## 2. Model Description and Methods

_{t}= 44 μm and the height of the output channel was h

_{o}. The ratio of h

_{o}to h

_{t}is called the geometry expansion factor (EF). Based on these dimensions and the definition of We and Ca numbers, $W{e}_{m}=\frac{{\rho}_{m}{V}_{m}^{2}{D}_{h}}{\sigma}$ and $C{a}_{o}=\frac{{\mu}_{o}{V}_{o}}{\sigma}$, in which $\rho $, $\mu $, and $\sigma $ are the density, viscosity, and surface tension, respectively. Subscript m refers to the medium and o denotes the oil phase. Here, $V$ is the mean velocity in the throat and ${D}_{h}$ is the hydraulic diameter.

#### 2.1. Interface Tracking

_{st}) plus the drag force (F

_{d}) reaction of particles divided by the total volume of computational cells containing the particle ($N{V}_{p}$), and the reaction force of the surface tension exerted on the particles (F

_{σ}) divided by the total volume of interface cells that were touched by the particle (NV

_{σ}), which will be discussed later in Section 2.2.2.

^{−8}s.

#### 2.2. Particle Tracking

_{p}(steady particle velocity). The rotational velocities of the particle (ω

_{x}, ω

_{y}) are set by the angular velocity of the fluid in that location. The inlet and outlet velocity profiles are set as U

_{p}–U

_{xy}

_{,}and the walls are assumed to have a velocity of −U

_{p}The fluid flow is then solved for this configuration in order to calculate the net reaction forces and torques on the particles from the fluid. If the values meet the following criteria (Equation (12)), the internal loop is considered to be converged, and the values of forces and torques are stored:

_{p}are estimated based on Equation (13):

_{px}and I

_{py}refer to the mass inertia of the particles around the x- and y-axes, respectively (the z-axis is along the direction of flow). The inner loop continues until Equation (12) holds true and then the particle is situated in another location. These steps are repeated for the entire network of discrete locations in the cross-section.

_{y}) in the local coordinates of the cross-sections does not vary between these halves, but (F

_{x}) is in the opposite direction with equal magnitude. After the cross-section at x = 190 μm (locally x

_{t}= 24 μm), because two streams of the flow intersect, the maximum velocity increases. Based on Equation (14), as the difference between the particle and fluid velocity increases, the magnitude of F

_{d}increases. We assume that after this cross-section, only the drag and interface reaction control the particles’ trajectory.

#### 2.2.1. Gaussian Random Number Generator (GRNG)

#### 2.2.2. Interaction of Particles with the Interface of Water in Oil (W/O) Droplets

_{σ}in Equation (15) by ignoring the gravity term [44]. Additionally, the drag force exerted on particles is added by Stokes assumption (Equation (11)). The direction of this force is perpendicular to the interface of the fluid, which is in the opposite direction of the normal vector calculated by Equation (6). The θ in this equation is shown in Figure 4b, R

_{p}is the radius of the particle, and σ is the coefficient of the surface tension:

## 3. Results and Discussion

#### 3.1. Grid Study and Time Independency

^{5}mesh number. The number of meshes was chosen near 4 × 10

^{5}as a conservative approach.

#### 3.2. Lift Forces

#### 3.3. Droplet Formation

_{wall}was considered in the range of 5° (superhydrophobic) to 75°, and σ in the range of 0.05207–0.06 N/m. Two expansion factors of 1.71 and 2.63 were also simulated to calculate the first 10 droplet diameters to investigate the stability.

_{wall}decreases. However, at higher contact angles, the variations in the droplet diameters are negligible. Moreover, reducing σ results in a less stable regime, as can be understood from regions of stability. The stable region for σ = 0.06 N/m is 35° to 75°, while it is 45° to 75° for σ = 0.05207 N/m. More simulations of lower surface tension coefficients led to unstable regimes at all contact angles.

_{wall}= 65°) does not change with EF. Moreover, the droplets with lower EF are uniform at different contact angles, which indicates that if the constant wall contact angle cannot be applied as the boundary condition, this geometry will generate droplets with high uniformity.

_{wall}= 15°, which is illustrated in Figure 9b. By increasing the value of σ, the droplet diameter will increase. Meanwhile, the stability is not subjected to change in this chart.

#### 3.4. Encapsulation

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

^{6}CPU-hours of calculation time with a high-performance computer (HPC).

## Conflicts of Interest

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**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 x

_{0}= 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.

Item/Properties | Density (kg/m^{3}) | Viscosity (Pa.s) | Surface Tension (N/m) |
---|---|---|---|

Entry 1 Medium | 998.2 | 1.003 × 10^{−3} | 0.043 |

Entry 2 Oil | 866.0 | 5.076 × 10^{−4} |

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**MDPI and ACS Style**

Yaghoobi, M.; Saidi, M.S.; Ghadami, S.; Kashaninejad, N.
An Interface–Particle Interaction Approach for Evaluation of the Co-Encapsulation Efficiency of Cells in a Flow-Focusing Droplet Generator. *Sensors* **2020**, *20*, 3774.
https://doi.org/10.3390/s20133774

**AMA Style**

Yaghoobi M, Saidi MS, Ghadami S, Kashaninejad N.
An Interface–Particle Interaction Approach for Evaluation of the Co-Encapsulation Efficiency of Cells in a Flow-Focusing Droplet Generator. *Sensors*. 2020; 20(13):3774.
https://doi.org/10.3390/s20133774

**Chicago/Turabian Style**

Yaghoobi, Mohammad, Mohammad Said Saidi, Sepehr Ghadami, and Navid Kashaninejad.
2020. "An Interface–Particle Interaction Approach for Evaluation of the Co-Encapsulation Efficiency of Cells in a Flow-Focusing Droplet Generator" *Sensors* 20, no. 13: 3774.
https://doi.org/10.3390/s20133774