Flow of deformable droplets: self-pinned glasses and string-like flow

Technical Summary: Flow of Deformable Droplets: Self-Pinned Glasses and String-Like Flow

Problem Statement
The rheology of complex fluids, such as emulsions, foams, and biological cell suspensions, is governed by the collective organization of mesoscopic constituents rather than simple molecular interactions. A central challenge in soft matter physics is understanding the yield-stress transition and the subsequent flow dynamics of deformable particles. While hard-sphere colloids have been extensively studied, the role of particle deformability in controlling yielding and post-yielding morphology remains less explored. Specifically, the mechanisms by which deformable droplets transition from an arrested solid state to a flowing state, and the nature of the intermediate dynamical regimes, require further investigation.

Methodology
The authors investigate the rheology of a "dry" suspension of $N$ soft, deformable droplets under pressure-driven flow using numerical simulations.

• Model: The system is modeled using a multiphase-field approach. Each droplet $i$ is described by a phase field $\phi_i$. The system's free energy $F$ includes terms for defining the droplet shape, regulating the interface tension (favoring circularity), and imposing steric repulsion between droplets to penalize overlaps.
• Dynamics: The phase fields evolve according to Cahn-Hilliard equations coupled with a steady-state Poiseuille flow profile $v_x(y)$. Hydrodynamic interactions between droplets are neglected to isolate the effects of droplet-droplet interactions and geometry.
• Parameters: The study varies the Péclet number ($Pe$), representing the ratio of advective to diffusive timescales (forcing strength), and a dimensionless deformability parameter $d = \epsilon/\alpha$. The packing fraction is fixed at $\Phi = 0.67$.
• Observables: Key metrics include the mean center-of-mass velocity parallel to the flow ($\langle V_\parallel \rangle$), mean square displacement (MSD), excess kurtosis ($\gamma_2$) of the velocity distribution (to quantify non-Gaussianity), overlap profiles, and structural order parameters (laning $\Psi_L$, bond-orientational order $\Psi_6$, and velocity alignment $\Psi'$).

Key Contributions and Results
The simulations reveal that the suspension undergoes two distinct force-driven dynamical transitions as the forcing ($Pe$) increases:

1. Yielding Transition (Solid to Self-Pinned Glass):

   • At low forcing ($Pe < Pe^*$), the system behaves as a yield-stress material where droplets remain arrested in an amorphous solid-like state. The critical force $Pe^*$ scales inversely with deformability ($Pe^* \sim d^{-1}$).
   • Just above yielding, the system does not flow steadily. Instead, it enters an intermittent stick-slip regime.
   • Mechanism: This state is identified as a "self-pinned" glass. Persistent droplet overlaps create a slowly evolving, effective rugged energy landscape ($U_O$) that dynamically traps droplets in local cages. Motion occurs via activated events where droplets elastically load against these self-generated constraints before abruptly slipping.
   • Signatures: This regime is characterized by long-lived caging (slow neighbor exchange), non-Gaussian velocity statistics (high excess kurtosis $\gamma_2$), and bimodal velocity distributions. The dynamics resemble the depinning transition in the Prandtl-Tomlinson model, but the pinning potential is self-generated by the suspension rather than externally imposed.

2. Transition to String-Like Flow:

   • At higher forcing ($Pe \gg Pe^*$), a second transition occurs to a continuous, string-like flowing state.
   • Mechanism: Increased forcing allows droplets to deform sufficiently to continuously exchange neighbors. This "anneals" the overlap structure, erasing the rugged energy landscape and eliminating the long-lived caging.
   • Signatures: The system exhibits smooth, sustained motion with rapid decay of caging correlations. Topological defects (e.g., 5-7 disclination pairs) that are long-lived in the self-pinned phase are progressively annihilated. The flow profile transitions from a plug-like flow (in the self-pinned regime) to a more parabolic profile, and structural order parameters indicate the formation of aligned strings.

Phase Diagram and Scaling
The authors construct a phase diagram in the $(d, Pe)$ parameter space delineating three regimes: Stuck (solid), Self-Pinned Glass, and String-like Flow.

• The transitions are governed by the competition between the timescale of overlap-driven motion (relaxation) and the advection timescale.
• A single dimensionless control parameter, $Pe \cdot d$, effectively captures the system's state:
  • $Pe \cdot d \lesssim 0.39$: Stuck phase.
  • $0.39 \lesssim Pe \cdot d \lesssim 3.55$: Self-pinned glass phase.
  • $Pe \cdot d \gtrsim 3.55$: String-like flow phase.

Significance and Claims
The paper claims to identify the restructuring of overlap networks as a generic mechanism controlling flow in driven suspensions of deformable particles.

• It demonstrates that deformability introduces phenomena absent in hard-sphere systems, specifically the emergence of a "self-pinned" glassy state driven by dynamically generated constraints.
• The study clarifies that the transition from a glassy, stick-slip state to a flowing state is primarily dynamical in origin, characterized by the rapid decay of caging correlations and the annihilation of topological defects, rather than a sharp structural phase transition.
• The authors suggest that these mechanisms of self-generated pinning and long-lived caging may be relevant to other driven assemblies of deformable objects, such as active matter and dense cellular tissues, though they do not explicitly propose experimental validations for these specific systems within the text.