Gravitational Sedimentation Calculator
There is confusion between "dispersions" which are supposed to settle over time and "solutions" which don't. The distinction is false - the same sized nanoparticle and large molecule will settle with exactly the same rules, eventually reaching a steady, exponentially increasing concentration gradient. These rules aren't, at first, intuitive so the app illustrates what happens in a column of the right length, left undisturbed for the right length of time.
The calculations here are based on the famous Mason & Weaver1 paper from 1924. We have a column length l, a particle (or molecule) of radius r (and therefore volume V=4/3πr³) and density ρ (here assumed always to be larger than that of the solvent, with the difference being δ). For those with large polymers and biomaterials of known MWt, you can use the estimator2 to find an appropriate equivalent radius. The solvent viscosity is η and the initial (relative) concentration is 1 down the length of the column. Two important values are pre-calculated: α determines the ultimate equilibrium concentration at the bottom of the column - it depends on RT and Avogadro's number, N; β is simply the time that a particle would take to travel the length of the tube via the normal Stokes equation.
The equation for calculating the relative concentration (initially 1 throughout the column) is horrendous, with infinite sums of exponentials. But at its heart are two terms, where y is the relative distance from the top, going from 0 to 1, the first term is:
It's that term which gives us the concentration gradient (plotted) at infinite time - when the gravitational movement downwards is balanced by diffusion (controlled by the concentration gradient) upwards. This thermodynamic equilibrium is what is seen in ultracentrifuges for (e.g.) particle size or MWt measurements. It's highlighted here because few of us ever think of normal test tubes and ultracentrifuges in the same way. The app calculates that term for y=1 and provides it as Rel Conc. Max @ t=∞. It also outputs the relative concentrations at the top (min) and bottom (max)
The second term depends, obviously, on time, t, in the form of various exponentials involving t/β as well as y. The numerical implementation can blow up - if it does, then slide things around (e.g. increase α via a smaller l or increase t) till you get a reasonable graph - the conditions where the numerics blow up are of no practical interest anyway.
Those who handle nanoparticles will benefit from exploring the app, for two reasons. First, it gives some idea of timescales. Second, and more importantly, because there is a big difference from this thermodynamic concentration gradient and "crashing out" of particles via, say, flocculation. It is important for the nanoparticle world to understand that gravity is not a way to distinguish "soluble" molecules from "dispersed" particles. A large DNA molecule behaves identically to a similar-sized nanoparticle.
But there's another amazing reason. Via a trick using osmotic membranes, you can establish exactly the same gradient (i.e. the osmotic trick does not change the thermodynamics) in hours or days rather than months or years. So a 10cm column containing blue dextrin of MWt ~ 2million which would take years to settle normally, can produce a visible gradient in a day or so. This has been known for decades but has not been exploited recently - and deserves to be.
The inspiration for writing this app came from Dr Seishi Shimizu of U. York. Until he pointed it all out, I had been completely unaware of the theory or its significance in the debate about nanoparticles being "soluble" (which they are) rather than "dispersed" (an undefined and confusingly unhelpful term).
1M. Mason and W. Weaver, Phys. Rev., 23, 412 (1924), also W. Weaver, Phys. Rev., 27, 499 (1926).
2The volume, `V=(MWt)/(N.ρ)` so `r=(V/(1.333.π))^0.333`.