= Finite-time rate-zonal separation
Consider two dilute populations of <spheres> with the same <buoyant density> and radii $b$ and $2b$. Adopt the fixed linear <density gradient> and constant local <centrifugal acceleration> of <linear-gradient isopycnic focusing>, with constant dynamic <viscosity> and negligible <diffusion>. <Stokes's law> makes $V/f$ proportional to $b^2$, so doubling radius multiplies the migration rate parameter by four, although the <mass> increases eightfold.
Use $u=z/L$, $\tau=t/T$ and $q=(\rho_b-\rho_0)/(GL)=2$. The liquid occupies $0\leq u\leq1$, so the matching <density> position lies beyond its outer boundary. Both populations begin at $u(0)=0.1$; choose $\lambda_A=0.15$ and $\lambda_B=0.6$. Their band-center trajectories are
$$
u_A(\tau)=2-1.9e^{-0.15\tau},\qquad u_B(\tau)=2-1.9e^{-0.6\tau}.
$$
At $\tau=0.5$, they lie at approximately $0.2373$ and $0.5924$; at $\tau=1$, at $0.3647$ and $0.9573$. Equal <buoyant density> has not prevented a useful finite-time separation. Solving $u_B(\tau)=1$ using the <natural logarithm> gives first boundary arrival at $\tau=\ln(1.9)/0.6\simeq1.0698$. The corresponding arrival for A is $\ln(1.9)/0.15\simeq4.2790$. Trajectories describe migration only until contact with the boundary; subsequent collection into a <centrifugation pellet> is not represented by extending the curve beyond $u=1$.
\Image[/rate-zonal-bands-before-boundary-arrival.png]
{title=<Rate-zonal centrifugation>: equal <buoyant density>, different finite-time migration}
{description=Original schematic tubes and trajectories for the stipulated <finite-time rate-zonal separation> model. B represents <spheres> with twice A's radius. Both matching <density> positions lie outside the column. Tube shading shows an imposed increasing <density gradient>; band thickness is schematic. The marked boundary-arrival time is calculated from this model, not an experimental run time or an operating instruction.}
This is the useful contrast with <isopycnic centrifugation>: stopping time helps determine the recovered positions in <rate-zonal centrifugation>, whereas a stationary matching <density> position sets the limiting band center in <isopycnic centrifugation>. Real resolution also depends on initial zone width, <diffusion> and variations in particle properties; distinct calculated centers alone do not prove that fractions will be pure.
Back to article page
