Lamm equation

The Lamm equation describes the sedimentation and diffusion of a solute under ultracentrifugation in traditional sector-shaped cells. (Cells of other shapes require much more complex equations.)

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The Lamm equation can be written

where c is the solute concentration, t and r are the time and radius, and the parameters D, s, and ω represent the solute diffusion constant, sedimentation coefficient and the rotor angular velocity, respectively. The first and second terms on the right-hand side of the Lamm equation are proportional to D and sω², respectively, and describe the competing processes of diffusion and sedimentation. Whereas sedimentation seeks to concentrate the solute near the outer radius of the cell, diffusion seeks to equalize the solute concentration throughout the cell. The diffusion constant D can be estimated from the hydrodynamic radius and shape of the solute, whereas the buoyant mass m_b can be determined from the ratio of s and D

where k_BT is the thermal energy, i.e., Boltzmann's constant k_B multiplied by the temperature T in kelvin.

Solute molecules cannot pass through the inner and outer walls of the cell, resulting in the boundary conditions on the Lamm equation

at the inner and outer radii, r_a and r_b, respectively. By spinning samples at constant angular velocity ω and observing the variation in the concentration c(r,t), one may estimate the parameters s and D and, thence, the buoyant mass and shape of the solute.

Derivation of the Lamm equation

Faxén solution (no boundaries, no diffusion)

See also

• Ultracentrifuge