Abstract

Lipid membranes are fundamental to cellular life, and their functional integrity is often linked to the lateral organization of their components, which can lead to phase separation into liquid-ordered ($L_o$) and liquid-disordered ($L_d$) domains. The structural and energetic properties of the interface, or domain boundary, between these phases are crucial determinants of membrane mechanics, domain growth, and protein partitioning. This study investigates the stable geometric configurations of the ordered domain boundary in a supported lipid membrane (SLB), a widely used model system for biophysical studies. Using a theoretical framework that minimizes the total free energy, which accounts for line tension, bending rigidity, and the substrate-induced pinning effect, we model the equilibrium shapes of the $L_o/L_d$ interface. Our results reveal that the solid support significantly influences the boundary configuration, leading to non-circular, often polygonal, shapes for ordered domains, particularly at smaller domain sizes. We identify specific stable configurations, such as elliptical or faceted boundaries, which depend on the balance between line tension and the anisotropic forces imposed by the substrate. These findings provide critical insights into the physical principles governing domain morphology in supported bilayers and have implications for understanding how cellular membranes maintain functional heterogeneity under mechanical constraints.