Lamm equation

TheLamm equationdescribes the sedimentation and diffusion of asoluteunderultracentrifugationin traditionalsector-shaped cells. (Cells of other shapes require much more complex equations.) It was named afterOle Lamm,later professor of physical chemistry at theRoyal Institute of Technology,who derived it during his Ph.D. studies underSvedbergatUppsala University.

The Lamm equation can be written:

${\frac {\partial c}{\partial t}}=D\left[\left({\frac {\partial ^{2}c}{\partial r^{2}}}\right)+{\frac {1}{r}}\left({\frac {\partial c}{\partial r}}\right)\right]-s\omega ^{2}\left[r\left({\frac {\partial c}{\partial r}}\right)+2c\right]$

wherecis the solute concentration,tandrare the time and radius, and the parametersD,s,andωrepresent the solute diffusion constant, sedimentation coefficient and the rotorangular velocity,respectively. The first and second terms on the right-hand side of the Lamm equation are proportional toDandsω²,respectively, and describe the competing processes ofdiffusionandsedimentation.Whereassedimentationseeks to concentrate the solute near the outer radius of the cell,diffusionseeks to equalize the solute concentration throughout the cell. The diffusion constantDcan be estimated from thehydrodynamic radiusand shape of the solute, whereas the buoyant massm_bcan be determined from the ratio ofsandD

${\frac {s}{D}}={\frac {m_{b}}{k_{B}T}}$

wherek_BTis the thermal energy, i.e., Boltzmann's constantk_Bmultiplied by thetemperatureTinkelvins.

Solutemoleculescannot pass through the inner and outer walls of the cell, resulting in theboundary conditionson the Lamm equation

$D\left({\frac {\partial c}{\partial r}}\right)-s\omega ^{2}rc=0$

at the inner and outer radii,r_aandr_b,respectively. By spinning samples at constantangular velocityωand observing the variation in the concentrationc(r,t), one may estimate the parameterssandDand, thence, the (effective or equivalent) buoyant mass of the solute.