Mason-Weaver equation

The Mason-Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field.cite journal | last = Mason | first = M | coauthors = Weaver W | year = 1924 | title = The Settling of Small Particles in a Fluid | journal = Physical Review | volume = 23 | pages = 412–426 | doi = 10.1103/PhysRev.23.412] 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 = 0at 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.

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 ar{ u}, the equilibrium condition may be written as

:f v_{term} = m (1 - ar{ u} ho) g stackrel{mathrm{def{=} m_{b} g

where m_{b} is the buoyant mass.

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

:D = 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

The parameters "D", "s" and "g" determine a length scale z_{0}

:z_{0} stackrel{mathrm{def{=} frac{D}{sg}

and a time scale t_{0}

:t_{0} stackrel{mathrm{def{=} frac{D}{s^{2}g^{2

Defining the dimensionless variables zeta stackrel{mathrm{def{=} z/z_{0} and au stackrel{mathrm{def{=} t/t_{0}, the Mason-Weaver equation becomes

:frac{partial c}{partial au} =frac{partial^{2} c}{partial zeta^{2 + frac{partial c}{partial zeta}

subject to the boundary conditions

:frac{partial c}{partial zeta} + c = 0at the top and bottom of the cell, zeta_{a} and zeta_{b}, respectively.

olution of the Mason-Weaver equation

This equation may be solved by separation of variables. Defining c(zeta, au) stackrel{mathrm{def{=} e^{-zeta/2} T( au) P(zeta), we obtain the two equations coupled by a constant eta

:frac{partial T}{partial au} + eta T = 0

:frac{partial^{2} P}{partial zeta^{2 + left [ eta - frac{1}{4} ight] P = 0

where acceptable values of eta are defined by the boundary conditions

:frac{dP}{dzeta} + frac{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^{-eta 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 criteriafor 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 eta_{k} are real, bounded below by a lowest
eigenvalue eta_{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^{-eta_{k} au}

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

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

At equilibrium, eta=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( frac{sg}{D} ight) left( frac{1}{e^{-zeta_{b - e^{-zeta_{a} ight)

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

:omega_{k} = pm sqrt{eta_{k} - frac{1}{4

Depending on the value of eta_{k}, omega_{k} is either purely real (eta_{k}geqfrac{1}{4}) or purely imaginary (eta_{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

: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 stackrel{mathrm{def{=} ho sin(phi) stackrel{mathrm{def{=} P

:v stackrel{mathrm{def{=} ho cos(phi) stackrel{mathrm{def{=} - frac{1}{omega} left( frac{dP}{dzeta} ight)

: ho stackrel{mathrm{def{=} u^{2} + v^{2}

: an(phi) stackrel{mathrm{def{=} v / u

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

:frac{d ho}{dzeta} = 0

:frac{dphi}{dzeta} = omega

Remarkably, the transformed boundary conditions are independent of ho and the endpoints zeta_{a} and zeta_{b}

: an(phi_{a}) = an(phi_{b}) = frac{1}{2omega_{k

Therefore, we obtain an equation

:phi_{a} - phi_{b} + kpi = kpi = int_{zeta_{b^{zeta_{a dzeta frac{dphi}{dzeta} = omega_{k} (zeta_{a} - zeta_{b})

giving an exact solution for the frequencies omega_{k}

:omega_{k} = frac{kpi}{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} stackrel{mathrm{def{=} pi/(zeta_{a} - zeta_{b}). Finally, the eigenvalues eta_{k} can be derived from omega_{k}

:eta_{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(zeta, au=0)multiplied by the weighting function e^{zeta/2}. Each Fourier component decays independently as e^{-eta_{k} au}, where eta_{k} is given above in terms of the Fourier series frequencies omega_{k}.

ee also

* Sedimentation
* Lamm equation

References


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