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

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 $\Gamma$, which induces a competitive mechanism between shear enhancement and hydraulic resistance penalty, and consequently leads to an optimal hydrodynamic balance at $\Gamma \approx 0.32$. 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.