Static interface profiles for contact lines on an elastic membrane with the Willmore energy

Area of Science:

• Soft Matter Physics
• Materials Science
• Fluid Dynamics

Background:

• Understanding the interplay between fluid interfaces and elastic materials is crucial in various scientific and engineering fields.
• The behavior of such systems is governed by complex energy contributions, including surface energies and membrane bending (Willmore energy).
• Previous studies often simplified membrane properties or interface interactions, necessitating a more comprehensive analysis.

Purpose of the Study:

• To derive equilibrium conditions for a fluid interface in contact with an elastic membrane by minimizing total system energy.
• To analyze the static profiles of both the fluid interface and the elastic membrane.
• To investigate the influence of membrane bending rigidity on the contact line behavior and interface morphology.

Main Methods:

• Energy minimization principle applied to the coupled system of fluid interface and elastic membrane, incorporating volume constraints.
• Derivation of equilibrium equations governing the static profiles.
• Asymptotic analysis of solutions in two-dimensional systems for limiting cases of reduced bending modulus (ν → +∞ and ν → 0).

Main Results:

• The membrane is locally flat at the contact line, with the contact angle obeying the Young-Dupré equation.
• A discontinuity (jump) in the gradient of the membrane's mean curvature is observed across the contact line, balancing fluid surface tension.
• In the stiff membrane limit (ν → +∞), behavior resembles a droplet on a rigid substrate. In the soft limit (ν → 0), a transition layer forms, and apparent contact angles follow Neumann's law.

Conclusions:

• The study provides a detailed theoretical framework for fluid-membrane interactions, highlighting the critical role of membrane bending.
• The derived asymptotic solutions offer valuable insights into system behavior across a range of elastic properties.
• Numerical validation confirms the accuracy of the theoretical predictions for static interface and membrane profiles.