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MasonWeaver equationThe MasonWeaver equation 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 MasonWeaver equation may be written where t is the time, c is the solute concentration (moles per unit length in the zdirection), and the parameters D, s, and g represent the solute diffusion constant, sedimentation coefficient and the (presumed constant) acceleration of gravity, respectively. The MasonWeaver equation is complemented by the boundary conditions 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 is conserved, i.e., dN_{tot} / dt = 0.
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Derivation of the MasonWeaver equationA typical particle of mass m moving with vertical velocity v is acted upon by three forces (Fig. 1): the drag force fv, the force of gravity mg and the buoyant force ρVg, where g is the acceleration of gravity, V is the solute particle volume and ρ 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 , the equilibrium condition may be written as where m_{b} is the buoyant mass. We define the MasonWeaver 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 k_{B} is the Boltzmann constant and T is the temperature in kelvin. 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 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 Substituting the equation for the flux J produces the MasonWeaver equation The dimensionless MasonWeaver equationThe parameters D, s and g determine a length scale z_{0} and a time scale t_{0} Defining the dimensionless variables and , the MasonWeaver equation becomes subject to the boundary conditions at the top and bottom of the cell, ζ_{a} and ζ_{b}, respectively. Solution of the MasonWeaver equationThis equation may be solved by separation of variables. Defining , we obtain the two equations coupled by a constant β where acceptable values of β are defined by the boundary conditions at the upper and lower boundaries, ζ_{a} and ζ_{b}, respectively. Since the T equation has the solution T(τ) = T_{0}e^{ − βτ}, where T_{0} is a constant, the MasonWeaver equation is reduced to solving for the function P(ζ). The ordinary differential equation for P and its boundary conditions satisfy the criteria for a SturmLiouville problem, from which several conclusions follow. First, there is a discrete set of orthonormal eigenfunctions P_{k}(ζ) that satisfy the ordinary differential equation and boundary conditions. Second, the corresponding eigenvalues β_{k} are real, bounded below by a lowest eigenvalue β_{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(ζ,τ) can be expressed as a weighted sum of the eigenfunctions where c_{k} are constant coefficients determined from the initial distribution c(ζ,τ = 0) At equilibrium, β = 0 (by definition) and the equilibrium concentration distribution is which agrees with the Boltzmann distribution. The P_{0}(ζ) function satisfies the ordinary differential equation and boundary conditions at all values of ζ (as may be verified by substitution), and the constant B may be determined from the total amount of solute To find the nonequilibrium values of the eigenvalues β_{k}, we proceed as follows. The P equation has the form of a simple harmonic oscillator with solutions where Depending on the value of β_{k}, ω_{k} is either purely real () or purely imaginary (). Only one purely imaginary solution can satisfy the boundary conditions, namely, the equilibrium solution. Hence, the nonequilibrium eigenfunctions can be written as
where A and B are constants and ω is real and strictly positive. By introducing the oscillator amplitude ρ and phase φ as new variables, the secondorder equation for P is factored into two simple firstorder equations Remarkably, the transformed boundary conditions are independent of ρ and the endpoints ζ_{a} and ζ_{b} Therefore, we obtain an equation giving an exact solution for the frequencies ω_{k} The eigenfrequencies ω_{k} are positive as required, since ζ_{a} > ζ_{b}, and comprise the set of harmonics of the fundamental frequency . Finally, the eigenvalues β_{k} can be derived from ω_{k} Taken together, the nonequilibrium components of the solution correspond to a Fourier series decomposition of the initial concentration distribution c(ζ,τ = 0) multiplied by the weighting function e^{ζ / 2}. Each Fourier component decays independently as , where β_{k} is given above in terms of the Fourier series frequencies ω_{k}. See alsoReferences


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "MasonWeaver_equation". A list of authors is available in Wikipedia. 