Differential sedimentation — Stokes settling, size selectivity, Svedberg

Four centrifuges run side by side on the same medium so the d² law is visible as a result rather than asserted: 20, 50 and 100 nm protein particles at 200,000 × g, plus a 10.24 nm particle at 250,000 × g.

centrifuge is sized by Sigma theory, Σ = Q/(2 v_g), where v_g is the g-amplified Stokes settling velocity v = d²(ρ_p − ρ_m)g/(18η). So the reported sigma_m2 carries the settling velocity, recoverable as v = Q_liquid/(2Σ). Against the closed form the solver agrees to machine precision (relative difference 0 to 4e-16), which makes this a check of the tool and not just a demonstration of it.

What it shows: v(20 nm) = 1.53e-3 cm/s; the 100 nm particle settles exactly 4× faster than the 50 nm one, since velocity goes as the square of diameter and every other term cancels; and the 250,000 × g branch is sized so its velocity is 5e-4 cm/s, giving a sedimentation coefficient s = v/ω²r = 20.4 S — the Svedberg range real proteins occupy (catalase 11.3 S, ribosome ~70 S).

Bounded, and the bound matters. This is terminal Stokes velocity at constant field: no wall, no concentration gradient, no Boycott effect, and no hindered settling. Run the 50 and 100 nm particles for 30 minutes and the arithmetic says they separate by 51 cm, which no rotor can deliver — both pellet against the tube bottom first, the 100 nm one in about four minutes. The ratio is robust; the distance is what the formula says rather than what a centrifuge does.

It is also the reference wiring for a solids flowsheet: the particles arrive as their own phase: "solid" feed with a flat solids payload, wired directly to the centrifuge. Declaring them liquid or mixed, or routing them through a mixer first, leaves the unit with no solid-phase inlet and the solve fails.