The metastability of lipid vesicle shapes in uniaxial extensional flow

TL;DR

The paper analyzes metastability and bifurcations of deflated lipid vesicle shapes in uniaxial extensional flow. By refining the Helfrich energy framework and coupling it to axisymmetric Stokes flow, the authors show that all stationary vesicle configurations are metastable and derive a finite critical length for transitions to unbounded elongation, identifying a saddle-node bifurcation at a critical strain rate $\dot{\epsilon}_{c}$. They demonstrate that near the bifurcation, the stationary length obeys a square-root scaling and the growth rates of perturbations vanish as $\sqrt{1-\dot{\epsilon}/\dot{\epsilon}_{c}$, with the beginning of unbounded elongation exhibiting a logarithmic slowdown due to incompressible membrane effects. Direct numerical simulations corroborate the analytical predictions and reveal slow dynamics and metastability across a range of reduced volumes $\mathcal{V}$, clarifying discrepancies with earlier work and offering insights for experiments and flow-control techniques like Stokes traps.

Abstract

In this work, we investigate the elastic properties of deflated vesicles and their shape dynamics in uniaxial extensional flow. By analysing the Helfrich bending energy and viscous flow stresses in the limit of highly elongated shapes, we demonstrate that all stationary vesicle configurations are metastable. For vesicles with small reduced volume, we identify the type of bifurcation at which the stationary state is lost, leading to unbounded vesicle elongation in time. We show that the stationary vesicle length remains finite at the critical extension rate. The critical behaviour of the stationary vesicle length and of the growth rates of small perturbations is obtained analytically and confirmed by direct numerical computations. The beginning stage of the unbounded elongation dynamics is simulated numerically, in agreement with the analytical predictions.

Figures (12)

• Figure 1: A schematic representation of an axially symmetric vesicle (cross-section by the $x$--$z$ plane) in an extensional flow. $\mathbf{R}(\tau)$ parameterizes the curve — the vesicle boundary in the plane of cylindrical coordinates $(\rho,z)$.
• Figure 2: Qualitative picture of the bifurcation: For strain rates below the critical value, there exists a stability region ${L<L_u\sim L_0\dot \epsilon_c/\dot \epsilon}$ beyond which the vesicle undergoes unbounded elongation. (The characteristic effective barrier height at small strain rates is $\delta \mathcal{F}_{\mathrm{eff}}\sim \kappa L_0/R_0 (\dot \epsilon_c/\dot \epsilon)^2$.) For strain rates above the critical value, the stable region disappears via a saddle-node bifurcation.
• Figure 3: Stationary relative elongation as a function of strain rate $\dot \epsilon$, normalised by the critical value $\dot \epsilon_c$, for a vesicle with equilibrium elongation $L_0/R_0\approx 13 \,(\mathcal{V}\approx 0.31)$. The inset shows the same quantity in rescaled coordinates: for a saddle-node bifurcation, a square-root singularity is expected. Different colours correspond to different initial equilibrium elongations and reduced volumes $\mathcal{V}$. Half-lengths $L$ are measured in equilibrium radii $R_0$.
• Figure 4: Comparison of equilibrium configurations (half $z>0$ shown) of vesicles and nearly critical configurations in extensional flow for different equilibrium half-lengths $L_0$ (measured in central radii $R_0$).
• Figure 5: Marginal modes of the stationary vesicle with equilibrium elongation $L_0/R_0 \approx 53$ as functions of the strain rate (\ref{['fig:lambda_ov_eps_02']}, \ref{['fig:lambda_ov_sqrt_eps_02']}) and stationary vesicle length \ref{['fig:lambda_ov_L_02']}. Points show simulation data; lines indicate asymptotic behaviour obtained by fitting model laws. Growth rates $\lambda$ are measured in $t_\kappa^{-1}= \kappa/\eta R_0^{-3}$.