Helfrich Membrane Mechanics

Helfrich free energy (1973): F = ∫[½κ(2H−c₀)² + κ_G K + σ] dA, where H = mean curvature, K = Gaussian curvature, c₀ = spontaneous curvature, σ = surface tension. The persistence length ξ_p = a·exp(4πκ/3k_BT) sets the scale over which the membrane remains flat — for typical lipid bilayers (κ ≈ 20 k_BT), ξ_p ~ microns to millimeters. Thermal fluctuations (undulations) renormalize κ downward via κ_eff = κ − 3k_BT/4π·ln(L/a). Nonzero c₀ drives spontaneous curvature — this controls vesicle shapes (stomatocyte → oblate → prolate). The shape equation (Ou-Yang & Helfrich 1989) governs equilibrium vesicle shapes.