Mason–Weaver equation
The Mason–Weaver equation (named after Max Mason and Warren Weaver) describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field.
[1] Assuming that the gravitational field is aligned in the z direction (Fig.
1), the Mason–Weaver equation may be written where t is the time, c is the solute concentration (moles per unit length in the z-direction), and the parameters D, s, and g represent the solute diffusion constant, sedimentation coefficient and the (presumed constant) acceleration of gravity, respectively.
The Mason–Weaver equation is complemented by the boundary conditions at the top and bottom of the cell, denoted as
These boundary conditions correspond to the physical requirement that no solute pass through the top and bottom of the cell, i.e., that the flux there be zero.
The cell is assumed to be rectangular and aligned with the Cartesian axes (Fig.
1), so that the net flux through the side walls is likewise zero.
Hence, the total amount of solute in the cell is conserved, i.e.,
A typical particle of mass m moving with vertical velocity v is acted upon by three forces (Fig.
, where g is the acceleration of gravity, V is the solute particle volume and
At equilibrium (typically reached in roughly 10 ns for molecular solutes), the particle attains a terminal velocity
Since V equals the particle mass m times its partial specific volume
The flux J at any point is given by The first term describes the flux due to diffusion down a concentration gradient, whereas the second term describes the convective flux due to the average velocity
A positive net flux out of a small volume produces a negative change in the local concentration within that volume Substituting the equation for the flux J produces the Mason–Weaver equation The parameters D, s and g determine a length scale
, the Mason–Weaver equation becomes subject to the boundary conditions at the top and bottom of the cell,
This partial differential equation may be solved by separation of variables.
, we obtain two ordinary differential equations coupled by a constant
is a constant, the Mason–Weaver equation is reduced to solving for the function
The ordinary differential equation for P and its boundary conditions satisfy the criteria for a Sturm–Liouville problem, from which several conclusions follow.
that satisfy the ordinary differential equation and boundary conditions.
(In our case, the lowest eigenvalue is zero, corresponding to the equilibrium solution.)
Third, the eigenfunctions form a complete set; any solution for
are constant coefficients determined from the initial distribution
function satisfies the ordinary differential equation and boundary conditions at all values of
(as may be verified by substitution), and the constant B may be determined from the total amount of solute To find the non-equilibrium values of the eigenvalues
The P equation has the form of a simple harmonic oscillator with solutions
as new variables, the second-order equation for P is factored into two simple first-order equations Remarkably, the transformed boundary conditions are independent of
Therefore, we obtain an equation giving an exact solution for the frequencies
, and comprise the set of harmonics of the fundamental frequency
Taken together, the non-equilibrium components of the solution correspond to a Fourier series decomposition of the initial concentration distribution
