StokesEinstein

The Stokes-Einstein relation is a fundamental equation in physical chemistry and soft matter physics that connects the translational diffusion of a spherical particle to the temperature, solvent viscosity, and particle size. In a simple, dilute suspension, the translational diffusion coefficient D of a rigid particle with hydrodynamic radius r in a Newtonian fluid of viscosity η at temperature T is given by D = k_B T / (6 π η r). This form applies in the limit of low Reynolds number, where inertial effects are negligible.

The relation arises from combining Einstein’s theory of Brownian motion with Stokes’ law for viscous drag. Einstein showed that random thermal motion can be characterized by a diffusion coefficient, while Stokes’ law provides the drag force on a sphere moving through a viscous fluid, yielding a friction coefficient f = 6 π η r. By equating thermal energy to work against friction in a stochastic description, the diffusion coefficient becomes inversely related to viscosity and particle size, giving the Stokes-Einstein expression.

Applications of the Stokes-Einstein relation include estimating particle sizes from measured diffusion coefficients and inferring solvent viscosity from diffusion data. The equation assumes dilute suspensions of roughly spherical, rigid particles in a Newtonian fluid and neglects hydrodynamic interactions, non-sphericity, surface slip, and non-Newtonian effects. Deviations occur at the nanoscale, in crowded or viscoelastic environments, or when particles interact strongly with boundaries.

Extensions include the rotational diffusion form D_r = k_B T / (8 π η r^3) for spheres, and generalized Stokes-Einstein relations that incorporate frequency-dependent viscosity or complex geometries.