Mason–Weaver equation

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 ${\displaystyle fv}$, the force of gravity ${\displaystyle mg}$ and the buoyant force ${\displaystyle \rho Vg}$, where g is the acceleration of gravity, V is the solute particle volume and ${\displaystyle \rho }$ is the solvent density. At equilibrium (typically reached in roughly 10 ns for molecular solutes), the particle attains a terminal velocity ${\displaystyle v_{\text{term}}}$ where the three forces are balanced. Since V equals the particle mass m times its partial specific volume ${\displaystyle {\bar {\nu }}}$, the equilibrium condition may be written as

${\displaystyle fv_{\text{term}}=m(1-{\bar {\nu }}\rho )g\ {\stackrel {\mathrm {def} }{=}}\ m_{b}g}$

where ${\displaystyle m_{b}}$ is the buoyant mass.

We define the Mason–Weaver sedimentation coefficient ${\displaystyle s\ {\stackrel {\mathrm {def} }{=}}\ m_{b}/f=v_{\text{term}}/g}$. Since the drag coefficient f is related to the diffusion constant D by the Einstein relation

${\displaystyle D={\frac {k_{B}T}{f}}}$,

the ratio of s and D equals

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

where ${\displaystyle k_{B}}$ is the Boltzmann constant and T is the temperature in kelvins.

The flux J at any point is given by

${\displaystyle J=-D{\frac {\partial c}{\partial z}}-v_{\text{term}}c=-D{\frac {\partial c}{\partial z}}-sgc.}$

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 ${\displaystyle v_{\text{term}}}$ of the particles. A positive net flux out of a small volume produces a negative change in the local concentration within that volume

${\displaystyle {\frac {\partial c}{\partial t}}=-{\frac {\partial J}{\partial z}}.}$

Substituting the equation for the flux J produces the Mason–Weaver equation

${\displaystyle {\frac {\partial c}{\partial t}}=D{\frac {\partial ^{2}c}{\partial z^{2}}}+sg{\frac {\partial c}{\partial z}}.}$

The parameters D, s and g determine a length scale ${\displaystyle z_{0}}$

${\displaystyle z_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {D}{sg}}}$

and a time scale ${\displaystyle t_{0}}$

${\displaystyle t_{0}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {D}{s^{2}g^{2}}}}$

Defining the dimensionless variables ${\displaystyle \zeta \ {\stackrel {\mathrm {def} }{=}}\ z/z_{0}}$ and ${\displaystyle \tau \ {\stackrel {\mathrm {def} }{=}}\ t/t_{0}}$, the Mason–Weaver equation becomes

${\displaystyle {\frac {\partial c}{\partial \tau }}={\frac {\partial ^{2}c}{\partial \zeta ^{2}}}+{\frac {\partial c}{\partial \zeta }}}$

subject to the boundary conditions

${\displaystyle {\frac {\partial c}{\partial \zeta }}+c=0}$

at the top and bottom of the cell, ${\displaystyle \zeta _{a}}$ and ${\displaystyle \zeta _{b}}$, respectively.

This partial differential equation may be solved by separation of variables. Defining ${\displaystyle c(\zeta ,\tau )\ {\stackrel {\mathrm {def} }{=}}\ e^{-\zeta /2}T(\tau )P(\zeta )}$, we obtain two ordinary differential equations coupled by a constant ${\displaystyle \beta }$

${\displaystyle {\frac {dT}{d\tau }}+\beta T=0}$

${\displaystyle {\frac {d^{2}P}{d\zeta ^{2}}}+\left[\beta -{\frac {1}{4}}\right]P=0}$

where acceptable values of ${\displaystyle \beta }$ are defined by the boundary conditions

${\displaystyle {\frac {dP}{d\zeta }}+{\frac {1}{2}}P=0}$

at the upper and lower boundaries, ${\displaystyle \zeta _{a}}$ and ${\displaystyle \zeta _{b}}$, respectively. Since the T equation has the solution ${\displaystyle T(\tau )=T_{0}e^{-\beta \tau }}$, where ${\displaystyle T_{0}}$ is a constant, the Mason–Weaver equation is reduced to solving for the function ${\displaystyle P(\zeta )}$.

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 ${\displaystyle P_{k}(\zeta )}$ that satisfy the ordinary differential equation and boundary conditions. Second, the corresponding eigenvalues ${\displaystyle \beta _{k}}$ are real, bounded below by a lowest eigenvalue ${\displaystyle \beta _{0}}$ and grow asymptotically like ${\displaystyle k^{2}}$ 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 ${\displaystyle c(\zeta ,\tau )}$ can be expressed as a weighted sum of the eigenfunctions

${\displaystyle c(\zeta ,\tau )=\sum _{k=0}^{\infty }c_{k}P_{k}(\zeta )e^{-\beta _{k}\tau }}$

where ${\displaystyle c_{k}}$ are constant coefficients determined from the initial distribution ${\displaystyle c(\zeta ,\tau =0)}$

${\displaystyle c_{k}=\int _{\zeta _{a}}^{\zeta _{b}}d\zeta \ c(\zeta ,\tau =0)e^{\zeta /2}P_{k}(\zeta )}$

At equilibrium, ${\displaystyle \beta =0}$ (by definition) and the equilibrium concentration distribution is

${\displaystyle e^{-\zeta /2}P_{0}(\zeta )=Be^{-\zeta }=Be^{-m_{b}gz/k_{B}T}}$

which agrees with the Boltzmann distribution. The ${\displaystyle P_{0}(\zeta )}$ function satisfies the ordinary differential equation and boundary conditions at all values of ${\displaystyle \zeta }$ (as may be verified by substitution), and the constant B may be determined from the total amount of solute

${\displaystyle B=N_{\text{tot}}\left({\frac {sg}{D}}\right)\left({\frac {1}{e^{-\zeta _{b}}-e^{-\zeta _{a}}}}\right)}$

To find the non-equilibrium values of the eigenvalues ${\displaystyle \beta _{k}}$, we proceed as follows. The P equation has the form of a simple harmonic oscillator with solutions ${\displaystyle P(\zeta )=e^{i\omega _{k}\zeta }}$ where

${\displaystyle \omega _{k}=\pm {\sqrt {\beta _{k}-{\frac {1}{4}}}}}$

Depending on the value of ${\displaystyle \beta _{k}}$, ${\displaystyle \omega _{k}}$ is either purely real (${\displaystyle \beta _{k}\geq {\frac {1}{4}}}$) or purely imaginary (${\displaystyle \beta _{k}<{\frac {1}{4}}}$). Only one purely imaginary solution can satisfy the boundary conditions, namely, the equilibrium solution. Hence, the non-equilibrium eigenfunctions can be written as

${\displaystyle P(\zeta )=A\cos {\omega _{k}\zeta }+B\sin {\omega _{k}\zeta }}$

where A and B are constants and ${\displaystyle \omega }$ is real and strictly positive.

By introducing the oscillator amplitude ${\displaystyle \rho }$ and phase ${\displaystyle \varphi }$ as new variables,

${\displaystyle u\ {\stackrel {\mathrm {def} }{=}}\ \rho \sin(\varphi )\ {\stackrel {\mathrm {def} }{=}}\ P}$

${\displaystyle v\ {\stackrel {\mathrm {def} }{=}}\ \rho \cos(\varphi )\ {\stackrel {\mathrm {def} }{=}}\ -{\frac {1}{\omega }}\left({\frac {dP}{d\zeta }}\right)}$

${\displaystyle \rho \ {\stackrel {\mathrm {def} }{=}}\ u^{2}+v^{2}}$

${\displaystyle \tan(\varphi )\ {\stackrel {\mathrm {def} }{=}}\ v/u}$

the second-order equation for P is factored into two simple first-order equations

${\displaystyle {\frac {d\rho }{d\zeta }}=0}$

${\displaystyle {\frac {d\varphi }{d\zeta }}=\omega }$

Remarkably, the transformed boundary conditions are independent of ${\displaystyle \rho }$ and the endpoints ${\displaystyle \zeta _{a}}$ and ${\displaystyle \zeta _{b}}$

${\displaystyle \tan(\varphi _{a})=\tan(\varphi _{b})={\frac {1}{2\omega _{k}}}}$

Therefore, we obtain an equation

${\displaystyle \varphi _{a}-\varphi _{b}+k\pi =k\pi =\int _{\zeta _{b}}^{\zeta _{a}}d\zeta \ {\frac {d\varphi }{d\zeta }}=\omega _{k}(\zeta _{a}-\zeta _{b})}$

giving an exact solution for the frequencies ${\displaystyle \omega _{k}}$

${\displaystyle \omega _{k}={\frac {k\pi }{\zeta _{a}-\zeta _{b}}}}$

The eigenfrequencies ${\displaystyle \omega _{k}}$ are positive as required, since ${\displaystyle \zeta _{a}>\zeta _{b}}$, and comprise the set of harmonics of the fundamental frequency ${\displaystyle \omega _{1}\ {\stackrel {\mathrm {def} }{=}}\ \pi /(\zeta _{a}-\zeta _{b})}$. Finally, the eigenvalues ${\displaystyle \beta _{k}}$ can be derived from ${\displaystyle \omega _{k}}$

${\displaystyle \beta _{k}=\omega _{k}^{2}+{\frac {1}{4}}}$

Taken together, the non-equilibrium components of the solution correspond to a Fourier series decomposition of the initial concentration distribution ${\displaystyle c(\zeta ,\tau =0)}$ multiplied by the weighting function ${\displaystyle e^{\zeta /2}}$. Each Fourier component decays independently as ${\displaystyle e^{-\beta _{k}\tau }}$, where ${\displaystyle \beta _{k}}$ is given above in terms of the Fourier series frequencies ${\displaystyle \omega _{k}}$.

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