Mason-weaver Equation

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Mason-weaver Equation

of solutes under a uniform Force , usually a Gravitation al field.
Assuming that the Gravitation al field is aligned in
the ''z'' direction (Fig. 1), the Mason-Weaver equation may be written

:

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

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

ho V g

, where ''g'' is the Acceleration of Gravity ,
''V'' is the Solute particle volume and

ho

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

\bar{
u}

,
the Equilibrium condition may be written as
:

f v_{term} = m (1 - \bar{
u} 
ho) g \equiv m_{b} g

where

m_{b}

is the Buoyant Mass .

We define the Mason-Weaver Sedimentation Coefficient

s \equiv m_{b} / f = v_{term}/g

.
Since the Drag Coefficient ''f'' is
related to the Diffusion Constant ''D'' by
the Einstein Relation

D = rac{k_{B} T}{f}

,
the ratio of ''s'' and ''D'' equals
:

rac{s}{D} = rac{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 rac{\partial c}{\partial z} - v_{term} c 
  =  -D rac{\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

:

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

Substituting the equation for the Flux ''J'' produces
the Mason-Weaver equation

:

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

THE DIMENSIONLESS MASON-WEAVER EQUATION

The parameters ''D'', ''s'' and ''g'' determine a length scale

z_{0}

:

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

and a time scale

t_{0}

:

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

Defining the Dimensionless variables

\zeta \equiv z/z_{0}

and

au \equiv t/t_{0}

,
the Mason-Weaver equation becomes

:

rac{\partial c}{\partial 	au} =
rac{\partial^{2} c}{\partial \zeta^{2}} + 
rac{\partial c}{\partial \zeta}

subject to the Boundary Conditions

:

rac{\partial c}{\partial \zeta} + c = 0

at the top and bottom of the cell,

\zeta_{a}

and

\zeta_{b}

, respectively.

SOLUTION OF THE MASON-WEAVER EQUATION

This equation may be solved by Separation Of Variables . Defining

c(\zeta,	au) \equiv e^{-\zeta/2} T(	au) P(\zeta)

, we obtain the two equations coupled by a constant

\beta

:

rac{\partial T}{\partial 	au} + \beta T = 0

:

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

where acceptable values of

\beta

are defined
by the Boundary Conditions
:

rac{dP}{d\zeta} + rac{1}{2} P = 0

at the upper and lower boundaries,

\zeta_{a}

and

\zeta_{b}

, respectively. Since the ''T'' equation
has the solution

T(	au) = T_{0} e^{-\beta 	au}

where

T_{0}

is a constant, the Mason-Weaver equation
is reduced to solving for the function

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

P_{k}(\zeta)

that satisfy the
Ordinary Differential Equation
and Boundary Conditions . Second, the corresponding
Eigenvalues

\beta_{k}

are real,
bounded below by a lowest
Eigenvalue

\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(\zeta, 	au)

can be
expressed as a weighted sum of the Eigenfunctions

:

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

where

c_{k}

are constant coefficients determined
from the initial distribution

c(\zeta, 	au=0)

:

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

At equilibrium,

\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}(\zeta)

function satisfies
the Ordinary Differential Equation and Boundary Conditions
at all values of

\zeta

(as may be verified by substitution), and the constant ''B''
may be determined from the total amount of Solute
:

B = N_{tot} \left( rac{sg}{D} 
ight) 
\left( rac{1}{e^{-\zeta_{b}} - e^{-\zeta_{a}}} 
ight)

To find the non-equilibrium values of the Eigenvalues

\beta_{k}

, we proceed as follows. The P equation has the form of a simple Harmonic Oscillator with
solutions

P(\zeta) = e^{i\omega_{k}\zeta}

where
:

\omega_{k} = \pm \sqrt{\beta_{k} - rac{1}{4}}

Depending on the value of

\beta_{k}

,

\omega_{k}

is either purely real (

\beta_{k}\geqrac{1}{4}

) or
purely imaginary (

\beta_{k} < rac{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

:

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

where ''A'' and ''B'' are constants
and

\omega

is real and strictly positive.

By introducing the oscillator Amplitude

ho

and
Phase

\phi

as new variables,

:

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

:

v \equiv 
ho \cos(\phi) \equiv - rac{1}{\omega} 
\left( rac{dP}{d\zeta} 
ight)

:

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

:

an(\phi) \equiv v / u

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

:

rac{d
ho}{d\zeta} = 0

:

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

Remarkably, the transformed Boundary Conditions are independent
of

ho

and the endpoints

\zeta_{a}

and

\zeta_{b}

:

an(\phi_{a}) = 
	an(\phi_{b}) = rac{1}{2\omega_{k}}

Therefore, we obtain an equation

:

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

giving an exact solution for the frequencies

\omega_{k}

:

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

The eigenfrequencies

\omega_{k}

are positive as required, since

\zeta_{a} > \zeta_{b}

, and comprise the set of Harmonics of the Fundamental Frequency

\omega_{1} \equiv \pi/(\zeta_{a} - \zeta_{b})

. Finally, the
Eigenvalues

\beta_{k}

can be derived
from

\omega_{k}

:

\beta_{k} = \omega_{k}^{2} + rac{1}{4}

Taken together, the non-equilibrium components of the solution
correspond to a Fourier Series decomposition of the initial
concentration distribution

c(\zeta, 	au=0)

multiplied by the
Weighting Function

e^{\zeta/2}

.
Each Fourier component decays independently as

e^{-\beta_{k}	au}

,
where

\beta_{k}

is given above in terms of the
Fourier Series frequencies

\omega_{k}

.

SEE ALSO

Sedimentation

the Lamm Equation

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