Lipid vesicle

A lipid vesicle is a closed lipid bilayer enclosing fluid. Its shape and thermal membrane undulations depend on membrane tension, membrane bending modulus, and constraints such as fixed enclosed volume.

For a tension-dominated cylindrical lipid vesicle of fixed length $L$ and volume, a real axisymmetric sine amplitude $u_q$ has stiffness $K_q=\pi \sigma L[(q{R}_0{)}^2-1]/R_0$. Fixed volume requires a mean-radius shift $-u_q^2/(4R_0)$. Stable modes have $⟨u_q^2⟩=k_{B}T/K_q$ by the equipartition theorem. Modes with $q{R}_0<1$ have the Rayleigh–Plateau instability, rather than a negative equilibrium variance. Bending modifies very short wavelengths and near-marginal modes.

An axisymmetric cylindrical membrane deformation is invariant under rotations about the cylinder axis, so its radius depends only on the axial coordinate. Its surface area is $2\pi \int R\sqrt{1+R_z^2}dz$, and its enclosed volume is $\pi \int R^{2}dz$. These scalar shape constraints are the basis of the fixed-volume capillary spectrum of a cylindrical membrane.