Fluid-Structure Interaction and Scaling Laws for Deterministic Encapsulation of Hyperelastic Cells in Microfluidic Droplets

Abstract

The precise encapsulation of deformable particles in multiphase flows involves complex transient Fluid-Structure Interactions (FSI) and topological interfacial changes. In the context of single-cell analysis, a numerical framework that couples the Cahn-Hilliard phase-field model with the Arbitrary Lagrangian-Eulerian (ALE) method is employed to investigate the dynamics of deformable cell encapsulation in flow-focusing microchannels. By resolving the coupling between the hyperelastic cell, carrier fluid, and evolving interface, we propose a unified dimensionless scaling law to predict the operational spatial window for the deterministic encapsulation quantitatively. Furthermore, the physical presence of cells modulates the droplet generation flow regime via a "geometric blockage effect", shifting the transition boundary from the squeezing to the dripping regime toward lower flow-rate ratios. The droplet generation period demonstrates a non-monotonic dependence on the cell blockage ratio $Γ$, which induces a competitive mechanism between shear enhancement and hydraulic resistance penalty, and consequently leads to an optimal hydrodynamic balance at $Γ\approx 0.32$. Finally, we find that while droplet periodicity is robust to variations in cell stiffness, the transient stress field within the cell is highly sensitive, particularly during the capillary pinch-off singularity. This work clarifies the fundamental interaction between hyperelastic cells and multiphase flows, and provides a quantitative framework for optimizing damage-free cell encapsulation systems.

Figures (11)

• Figure 1: Schematic diagram of the cross-linked microchannel containing cells. $W_d$ and $W_c$ represent the widths of the main channel and the side channel, respectively, and the length of the side channel is $2W_c$. $L_1$ and $L_2$ represent the lengths of the main channel on the dispersed phase inlet side and the main channel on the continuous phase outlet side. $Q_c$ and $Q_d$ are the inlet volumetric flow rates of the continuous phase and the dispersed phase. The distance from the center of the cell to the inlet in the $X$ direction is $D_c$.
• Figure 2: Validation of grid independence and comparison of simulation results. (a) Grid distribution of Mesh4 in the micro-droplet channel. The fluid region is divided into triangular elements. The enlarged plot shows local grid refinement near the cell. The red dot in the middle of the junction is the monitoring point $P_M$. (b) The $x$-direction velocity of the monitoring point $P_M$ during $t$ = 18.0 21.0 ms under different numbers of grid elements. (c) The diagram of flow regimes during droplet generation with the flow rate ratio $Q$ and the capillary number $Ca$. The pink dotted line and the purple dotted line represent the transitional boundary between the flow regimes DCJ and DC, DC and PF, respectively, as obtained by Liu et al.18. The pink solid line and the purple solid line are our corresponding results. The red triangles, green diamonds, and blue squares represent our simulation results corresponding to DCJ, DC, and PF regimes.
• Figure 3: Fluid interface (i), flow field (ii), and stress distribution (iii) during cell encapsulation at different times. (a) $t$ = 8.3 ms; (b) $t$ = 9.1 ms; (c) $t$ = 9.3 ms; (d) $t$ = 9.8 ms; (e) $t$ = 12.7 ms; (f) $t$ = 14.9 ms. The left column illustrates the time history of cell speed $V_{cell}$. The parameters for this simulation are $Ca$ = 0.004, $Re$ = 0.064, $Q$ = 1, $\Lambda$=1, $\lambda$=0.1, $L_{1}$ = 200 $\mu$m, $L_{2}$ = 1000 $\mu$m, and $D_{c}$ = 60 $\mu$m. The cell parameters are $C_{1}$ = 700 Pa, $C_{2}$ = 200 Pa, and $\Gamma$ = 0.30. The color bar scales for velocity and cell stress vary across different time instants. In particular, at $t = 12.7$ ms, the stress magnitude experienced by the cell is significantly higher than at other stages.
• Figure 4: (a) Schematic diagram of the initial cell location for successful encapsulation. Here, Zone I represents the advanced area before the encapsulation of the next droplet. Zone II represents the lagging area after the encapsulation of the current droplet. Zone III represents the normal encapsulation area of the current droplet. Zone IV represents the advanced area before the encapsulation of the current droplet. Zone V represents the lagging area after the encapsulation of the previous droplet. The distance from the left boundary of the successful encapsulation to the intersection of the microchannel is $D_{1}$$\mu$m, and the distance from the right boundary of the successful encapsulation to the intersection is $D_{2}$$\mu$m. (b) Von Mises stress vs. time curves of cells at different initial positions. The red dotted line corresponds to the cell initial position being in Zone II, the green double-dotted line in Zone III, and the blue solid line in Zone IV.
• Figure 5: Flow interfaces at different instants for the cell initially located in Zone II. (a) $t$ = 8.0 ms, (b) $t$ =8.2 ms, and (c) $t$ =9.8 ms. The simulation parameters are fixed at $Ca$ = 0.0018, $Re$ = 0.38, $Q$ = 0.5, $\Lambda$ = 1, $\lambda$ = 0.3, $L_{1}$ = 200 $\mu$m, and $L_{2}$ = 800 $\mu$m, $D_{c}$ = 65 $\mu$m, with cell mechanical properties set to $C_{1}$ = 700 Pa, $C_{2}$ = 200 Pa, and $\Gamma$ = 0.40.