Stokes Radius

Einstein Relation

The '''Einstein relation''' is a connection revealed by Albert Einstein in his 1905 paper on Brownian motion:

(1)
\begin{align} D = {\mu k_B T} \end{align}

linking ''D'', the Diffusion constant (see Fick's Law), and ''μ'', the mobility of the particles; where ''$k_{B}$'' is Boltzmann's constant, and ''T'' is the absolute temperature.

The mobility ''μ'' is the ratio of the particle's terminal drift velocity to an applied force, ''$\mu = \frac{v_{d}}{F}$''.

Diffusion of particles

In the limit of low Reynolds number, the mobility ''μ'' is the inverse of the drag coefficient ''γ''. For spherical particles of radius ''r'', the Stokes law gives

(2)
\begin{align} \gamma = 6 \pi \eta r \end{align}

where ''η'' is the viscosity of the medium. Thus the Einstein relation becomes:

(3)
\begin{align} D=\frac{k_B T}{6 \pi \eta r} \end{align}

Stokes Radius

The “Stokes radius”, “Stokes-Einstein radius”, or hydrodynamic radius $R_{H}$, named after George Gabriel Stokes, is not the effective radius of a hydrated molecule in solution. It is the radius of a hard sphere that diffuses at the same rate as the molecule. The behavior of this sphere includes hydration and shape effects. Since most molecules are not perfectly spherical, the Stokes radius is smaller than the effective radius (or the rotational radius). A more extended molecule will have a larger Stoke's radius compared to a more compact molecule of the same molecular weight.

In liquids where there are considerable interactions between solute and solvent molecule, Stokes radius is proportional to frictional coefficient f and inverse proportional to viscosity $\eta$ ($R_{H}\ \alpha \ f / \eta$). The frictional coefficient is determined by the size and shape of the molecule under consideration and the diffusion constant can be evaluated by:

(4)
\begin{align} D=\frac{k_B T}{6 \pi \eta R_{H}} \end{align}

For example, a 100 kDalton protein, we obtain $D = 10^{-10} \ m^{2}s{-1}$, assuming a "standard" protein
density of about $1.2 \ 10^{3} kg m^{-3}$.

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