Helfrich energy

The Helfrich energy of a fluid membrane combines surface tension with curvature elasticity. Without spontaneous curvature or a topological term,

A thermal membrane undulation is a shape fluctuation excited by thermal energy. In a planar membrane, a Fourier mode has stiffness $k_c{q}^4+\sigma q^2$, so tension suppresses long-wavelength excess area while bending suppresses short wavelengths.

The membrane bending modulus sets the energy cost of curvature. A larger $k_c$ suppresses thermal undulations and resists wrapping around small objects.

Membrane tension is the free-energy cost per added projected area. Together with bending elasticity it selects a length scale and controls the radius of a pulled membrane tube.

The membrane elastic length is the distance over which bending-dominated deformation relaxes under tension. In a small-slope one-dimensional membrane it is $\xi =\sqrt{k_c/\sigma }$.

For a nearly flat membrane represented by height $h(x)$, the quadratic bending-and-tension energy per invariant transverse length isIts free equilibrium equation is $k_c{h}_{xxxx}-\sigma h_{xx}=0$.

For a membrane of bending modulus $k_c$ adhering with energy per area $\mathcal{U}$ to a cylinder of radius $R$, $U=R^2\mathcal{U}/k_c$ compares adhesion with bending cost.

A membrane-mediated interaction potential is the separation-dependent energy produced when deformation fields around two membrane-bound objects overlap. Its sign and range depend on membrane tension, bending stiffness, inclusion geometry, and which side of the membrane each object occupies.

Minimizing the Helfrich energy per length of an unconstrained cylindrical membrane gives $r_0=\sqrt{k_c/(2\sigma )}$, balancing the area cost against bending curvature.

When a tether draws area from the thermal undulations of a finite vesicle, extension raises membrane tension rather than drawing area from a fixed-tension reservoir. The tether therefore narrows and its required pulling force increases with length.

Thermal membrane undulations generate Helfrich repulsion from a nearby particle. For a sphere of radius $R$ tightly enclosed by a high-tension membrane,

For a sphere of radius $R$ moving through a membrane tube with a uniform narrow gap $\delta \ll R$, lubrication theory givesIt exceeds the unbounded-fluid Stokes drag law coefficient by $2R^2/{\delta }^2$.