= Sedimentation coefficient
{title2=$s=v/(\omega^2r)$}
The <sedimentation coefficient> $s$ is radial drift <velocity> divided by local <centrifugal acceleration>. It has units of time and combines particle properties with those of the surrounding liquid. In a dilute, deterministic <mathematical model> with negligible particle <inertia>, an effective displaced volume $V$, particle <mass> $m$, local liquid <density> $\rho_f$ and <linear friction coefficient> $f>0$ give
$$
f v=(m-\rho_f V)\omega^2r,\qquad s=\frac{v}{\omega^2r}=\frac{m-\rho_f V}{f}.
$$
The subtraction represents <buoyancy>. Equal <mass> does not imply equal $s$: changing shape changes <drag force>, and changing the medium changes both <buoyancy> and <viscosity>.https://www.beckman.com/resources/technologies/centrifugation/ultracentrifugation/[An Introduction to Ultracentrifugation, The Physics Behind the Technology: mass, size, shape, medium density and viscosity]{ref}
For an isolated <sphere> of radius $b$ and uniform <density> $\rho_p$, <Stokes's law> supplies $f=6\pi\eta b$, where $\eta$ is dynamic <viscosity>, and $m=(4\pi/3)b^3\rho_p$. Therefore
$$
s=\frac{2b^2(\rho_p-\rho_f)}{9\eta}.
$$
This original force-balance check predicts fourfold $s$ on doubling $b$ with everything else fixed, and zero drift when $\rho_p=\rho_f$. It applies to dilute isolated <spheres> in <Stokes flow>, not arbitrary folded <proteins> or crowded suspensions. If $s$ and $\omega$ remain constant in a uniform medium, the <ordinary differential equation> $dr/dt=s\omega^2r$ has the <exponential function> solution $r(t)=r(0)e^{s\omega^2t}$. Arrival at an outer boundary therefore depends on starting position as well as $s$; a separation has no universal size cutoff.
Back to article page
