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 and , respectively (Fig. 1). 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., .

Derivation of the Mason–Weaver equationEdit

Figure 1: Diagram of Mason–Weaver cell and Forces on Solute

A typical particle of mass m moving with vertical velocity v is acted upon by three forces (Fig. 1): the drag force  , the force of gravity   and the buoyant force  , where g is the acceleration of gravity, V is the solute particle volume and   is the solvent density. At equilibrium (typically reached in roughly 10 ns for molecular solutes), the particle attains a terminal velocity   where the three forces are balanced. Since V equals the particle mass m times its partial specific volume  , the equilibrium condition may be written as


where   is the buoyant mass.

We define the Mason–Weaver sedimentation coefficient  . Since the drag coefficient f is related to the diffusion constant D by the Einstein relation


the ratio of s and D equals


where   is the Boltzmann constant and T is the temperature in kelvins.

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   of the particles. 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 dimensionless Mason–Weaver equationEdit

The parameters D, s and g determine a length scale  


and a time scale  


Defining the dimensionless variables   and  , the Mason–Weaver equation becomes


subject to the boundary conditions


at the top and bottom of the cell,   and  , respectively.

Solution of the Mason–Weaver equationEdit

This partial differential equation may be solved by separation of variables. Defining  , we obtain two ordinary differential equations coupled by a constant  


where acceptable values of   are defined by the boundary conditions


at the upper and lower boundaries,   and  , respectively. Since the T equation has the solution  , where   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. First, there is a discrete set of orthonormal eigenfunctions   that satisfy the ordinary differential equation and boundary conditions. Second, the corresponding eigenvalues   are real, bounded below by a lowest eigenvalue   and grow asymptotically like   where the nonnegative integer k is the rank of the eigenvalue. (In our case, the lowest eigenvalue is zero, corresponding to the equilibrium solution.) Third, the eigenfunctions form a complete set; any solution for   can be expressed as a weighted sum of the eigenfunctions


where   are constant coefficients determined from the initial distribution  


At equilibrium,   (by definition) and the equilibrium concentration distribution is


which agrees with the Boltzmann distribution. The   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  , we proceed as follows. The P equation has the form of a simple harmonic oscillator with solutions   where


Depending on the value of  ,   is either purely real ( ) or purely imaginary ( ). Only one purely imaginary solution can satisfy the boundary conditions, namely, the equilibrium solution. Hence, the non-equilibrium eigenfunctions can be written as


where A and B are constants and   is real and strictly positive.

By introducing the oscillator amplitude   and phase   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   and the endpoints   and  


Therefore, we obtain an equation


giving an exact solution for the frequencies  


The eigenfrequencies   are positive as required, since  , and comprise the set of harmonics of the fundamental frequency  . Finally, the eigenvalues   can be derived from  


Taken together, the non-equilibrium components of the solution correspond to a Fourier series decomposition of the initial concentration distribution   multiplied by the weighting function  . Each Fourier component decays independently as  , where   is given above in terms of the Fourier series frequencies  .

See alsoEdit

  • Lamm equation
  • The Archibald approach, and a simpler presentation of the basic physics of the Mason–Weaver equation than the original.[2]


  1. ^ Mason, M; Weaver W (1924). "The Settling of Small Particles in a Fluid". Physical Review. 23: 412–426. Bibcode:1924PhRv...23..412M. doi:10.1103/PhysRev.23.412.
  2. ^ Archibald, William J. (1938-05-01). "The Process of Diffusion in a Centrifugal Field of Force". Physical Review. American Physical Society (APS). 53 (9): 746–752. doi:10.1103/physrev.53.746. ISSN 0031-899X.