Saffman–Delbrück model

The Saffman–Delbrück model describes a lipid membrane as a thin layer of viscous fluid, surrounded by a less viscous bulk liquid. This picture was originally proposed to determine the diffusion coefficient of membrane proteins, but has also been used to describe the dynamics of fluid domains within lipid membranes. The Saffman–Delbrück formula is often applied to determine the size of an object embedded in a membrane from its observed diffusion coefficient, and is characterized by the weak logarithmic dependence of diffusion constant on object radius.

In a three-dimensional highly viscous liquid, a spherical object of radius a has diffusion coefficient

by the well-known Stokes–Einstein relation. By contrast, the diffusion coefficient of a circular object embedded in a two-dimensional fluid diverges; this is Stokes' paradox. In a real lipid membrane, the diffusion coefficient may be limited by:

Philip Saffman and Max Delbrück calculated the diffusion coefficient for these three cases, and showed that Case 3 was the relevant effect.

The diffusion coefficient of a cylindrical inclusion of radius ${\displaystyle a}$ in a membrane with thickness ${\displaystyle h}$ and viscosity ${\displaystyle \mu _{m}}$, surrounded by bulk fluid with viscosity ${\displaystyle \mu _{f}}$ is:

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