How this calculator works
Apply the Einstein relation between thermal energy and viscous drag in a continuum Newtonian liquid at low Reynolds number. The disk is an ideal zero-thickness no-slip disk. Ellipsoid inputs are semiaxes, not full diameters, and use prolate spheroidal drag. The tumbling choice averages translational mobility over orientations; it is not a rotational diffusion coefficient. Rotation about a disk normal is reported separately.
Formula
D = k_B T/zeta. Sphere: zeta = 6 pi eta r; disk face: 16 eta r; disk edge: (32/3) eta r. Disk rotational zeta = (32/3) eta r^3.
Worked example
For a 1 nm sphere at 300 K in viscosity 0.001 Pa s, D is about 2.197371 x 10^-10 m2/s. Doubling the radius halves D.
Advertisement
Frequently asked questions
Does this work for molecular-scale particles?
Continuum hydrodynamics can become inaccurate at molecular scales; measured hydrodynamic radius and solvent conditions matter.
What is the ellipsoid orientation average?
It is the mean of one parallel and two perpendicular mobilities, suitable for freely sampled orientations.
Why does rotation have different units?
Rotational diffusion describes angular spreading rather than displacement, so its coefficient has rad2/s units.