The Mason-Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. Assuming that the gravitational field is aligned in the z direction (Fig. 1), the Mason-Weaver equation may be written

$$

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

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

$$

D \frac{\partial c}{\partial z} + s g c = 0 at the top and bottom of the cell, denoted as $z\_\{a\}$ and $z\_\{b\}$, 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

$$

N_{tot} = \int_{z_{b}}^{z_{a}} dz \ c(z, t) is conserved, i.e., $dN\_\{tot\}/dt\; =\; 0$.

The Mason-Weaver equation was first described in the paper, "The Settling of Small Particles in a Fluid", by Max Mason and Warren Weaver Rev., 23, 412-426 (1924).

Derivation of the Mason-Weaver equation

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

$$

f v_{term} = m (1 - \bar{\nu} \rho) g \equiv m_{b} g where $m\_\{b\}$ is the buoyant mass.

We define the Mason-Weaver sedimentation coefficient $s\; \backslash equiv\; m\_\{b\}\; /\; f\; =\; v\_\{term\}/g$. Since the drag coefficient f is related to the diffusion constant D by the Einstein relation $D\; =\; \backslash frac\{k\_\{B\}\; T\}\{f\}$, the ratio of s and D equals

$$

\frac{s}{D} = \frac{m_{b}}{k_{B} T} where $k\_\{B\}$ is the Boltzmann constant and T is the temperature in kelvin.

The flux J at any point is given by

$$

J = -D \frac{\partial c}{\partial z} - v_{term} c = -D \frac{\partial c}{\partial z} - s g c 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 $v\_\{term\}$ of the particles. A positive net flux out of a small volume produces a negative change in the local concentration within that volume

$$

\frac{\partial c}{\partial t} = -\frac{\partial J}{\partial z}

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

$$

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

The dimensionless Mason-Weaver equation

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

$$

z_{0} \equiv \frac{D}{sg}

and a time scale $t\_\{0\}$

$$

t_{0} \equiv \frac{D}{s^{2}g^{2}}

Defining the dimensionless variables $\backslash zeta\; \backslash equiv\; z/z\_\{0\}$ and $\backslash tau\; \backslash equiv\; t/t\_\{0\}$, the Mason-Weaver equation becomes

$$

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

subject to the boundary conditions

$$

\frac{\partial c}{\partial \zeta} + c = 0 at the top and bottom of the cell, $\backslash zeta\_\{a\}$ and $\backslash zeta\_\{b\}$, respectively.

Solution of the Mason-Weaver equation

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This equation may be solved by separation of variables. Defining $c(\backslash zeta,\backslash tau)\; \backslash equiv\; e^\{-\backslash zeta/2\}\; T(\backslash tau)\; P(\backslash zeta)$, we obtain the two equations coupled by a constant $\backslash beta$

$$

\frac{\partial T}{\partial \tau} + \beta T = 0

$$

\frac{\partial^{2} P}{\partial \zeta^{2}} + \left\beta - \frac{1}{4} \right P = 0

where acceptable values of $\backslash beta$ are defined by the boundary conditions

$$

\frac{dP}{d\zeta} + \frac{1}{2} P = 0 at the upper and lower boundaries, $\backslash zeta\_\{a\}$ and $\backslash zeta\_\{b\}$, respectively. Since the T equation has the solution $T(\backslash tau)\; =\; T\_\{0\}\; e^\{-\backslash beta\; \backslash tau\}$ where $T\_\{0\}$ is a constant, the Mason-Weaver equation is reduced to solving for the function $P(\backslash zeta)$.

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

$$

c(\zeta, \tau) = \sum_{k=0}^{\infty} c_{k} P_{k}(\zeta) e^{-\beta_{k}\tau}

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

$$

c_{k} = \int_{\zeta_{a}}^{\zeta_{b}} d\zeta \ c(\zeta, \tau=0) e^{\zeta/2} P_{k}(\zeta)

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

$$

e^{-\zeta/2} P_{0}(\zeta) = B e^{-\zeta} = B e^{-m_{b}gz/k_{B}T} which agrees with the Boltzmann distribution. The $P\_\{0\}(\backslash zeta)$ function satisfies the ordinary differential equation and boundary conditions at all values of $\backslash zeta$ (as may be verified by substitution), and the constant B may be determined from the total amount of solute

$$

B = N_{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 $\backslash beta\_\{k\}$, we proceed as follows. The P equation has the form of a simple harmonic oscillator with solutions $P(\backslash zeta)\; =\; e^\{i\backslash omega\_\{k\}\backslash zeta\}$ where

$$

\omega_{k} = \pm \sqrt{\beta_{k} - \frac{1}{4}} Depending on the value of $\backslash beta\_\{k\}$, $\backslash omega\_\{k\}$ is either purely real ($\backslash beta\_\{k\}\backslash geq\backslash frac\{1\}\{4\}$) 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

$$

P(\zeta) = A \cos{\omega_{k} \zeta} + B \sin{\omega_{k} \zeta}

where A and B are constants and $\backslash omega$ is real and strictly positive.

By introducing the oscillator amplitude $\backslash rho$ and phase $\backslash phi$ as new variables,

$$

u \equiv \rho \sin(\phi) \equiv P

$$

v \equiv \rho \cos(\phi) \equiv - \frac{1}{\omega} \left( \frac{dP}{d\zeta} \right)

$$

\rho \equiv u^{2} + v^{2}

$$

\tan(\phi) \equiv v / u the second-order equation for P is factored into two simple first-order equations

$$

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

$$

\frac{d\phi}{d\zeta} = \omega

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

$$

\tan(\phi_{a}) = \tan(\phi_{b}) = \frac{1}{2\omega_{k}}

Therefore, we obtain an equation

$$

\phi_{a} - \phi_{b} + k\pi = k\pi = \int_{\zeta_{b}}^{\zeta_{a}} d\zeta \ \frac{d\phi}{d\zeta} = \omega_{k} (\zeta_{a} - \zeta_{b})

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

$$

\omega_{k} = \frac{k\pi}{\zeta_{a} - \zeta_{b}}

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

$$

\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 $c(\backslash zeta,\; \backslash tau=0)$ multiplied by the weighting function $e^\{\backslash zeta/2\}$. Each Fourier component decays independently as $e^\{-\backslash beta\_\{k\}\backslash tau\}$, where $\backslash beta\_\{k\}$ is given above in terms of the Fourier series frequencies $\backslash omega\_\{k\}$.

See also

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• Sedimentation
• the Lamm equation

External links

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Laboratory techniques | Partial differential equations