Mathematics of band centrifugation: Concentration‐independent sedimentation and diffusion in shallow density gradients

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Abstract
Integral expressions for concentration as a function of time and distance are derived from the continuity equation for centrifugation in a sector‐shaped cell for a macro‐molecular solute initially contained in a finite upper layer and a solute of low molecular weight in the supporting liquid. Computer patterns based on the sedimentation and diffusion coefficients of sucrose and of spherical and randomly coiled model solutes illustrate: (1) the time course of redistribution of both banded and supporting solutes from initial uniform concentrations; (2) the influence of the initial concentration, width, and solute concentration of the upper band; and (3) the effect of restricted diffusion at the meniscus on subsequent band shape. A Gaussian, approximation to band shape is derived and graphically tested. Rapid methods, not requiring computers, are out lined for the estimation of sedimentation and diffusion coefficients, where their concentration dependence is negligible, by band centrifugtion. The theoretical resolution of mixtures attainable by this technique is compared with moving‐boundary centrifugation, with the use of both integral (interferotmetric or absorption) and derivative (schlieren) optics.