The Lamm equation[O Lamm: (1929) "Die Differentialgleichung der Ultrazentrifugierung" Arkiv för matematik, astronomi och fysik 21B No. 2, 1–4] describes the sedimentation and diffusion of a solute under ultracentrifuge in traditional Circular sector-shaped cells. (Cells of
other shapes require much more complex equations.) It was named after Ole Lamm, later professor of physical chemistry at the Royal Institute of Technology, who derived it during his PhD studies under Svedberg at Uppsala 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
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
\frac{s}{D} = \frac{m_b}{k_\text{B} T}
where
kB T is the thermal energy, i.e., the Boltzmann constant
kB multiplied by
the absolute temperature
T.
Solute molecules cannot pass through the inner and outer walls of the
cell, resulting in the boundary conditions on the Lamm equation
D \left( \frac{\partial c}{\partial r} \right) - s \omega^2 r c = 0
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 (effective or equivalent) buoyant mass of the solute.
References and notes
