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other shapes require much more complex equations.)

The Lamm equation can be written

:
rac{\partial c}{\partial t} =
D \left[ \left( rac{\partial^{2} c}{\partial r^2} ight) +
rac{1}{r} \left( rac{\partial c}{\partial r} ight) ight] -
s \omega^{2} \left r \left( rac{\partial c}{\partial r} ight) + 2c ight


where ''c'' is the solute concentration, ''t'' and ''r'' are the time and radius, and
the parameters ''D'', ''s'', and \omega
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 s\omega^{2}, 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''
:
rac{s}{D} = rac{m_{b}}{k_{B}T}

where k_{B}T 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
:
D \left( rac{\partial c}{\partial r} ight) - 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 \omega 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