Coarse-graining dynamics by telescoping down time-scales: comment for Faraday FD144

Consider a buoyant colloid of mass Mc and a radius a = 1μm in H2O. As described in more detail in [1], its behaviour is governed by a series of different timescales shown in table I. If you are only interested in the behaviour of the colloids, then the two fastest time-scales, the solvent collision time τcol and the solvent relaxation time τf , can be ignored as long as they are shorter than any other colloidal time-scales. The first physically relevant time-scale is the Fokker Planck time-scale τFP ≈ 10 −13 over which the colloid loses memory of the short-time forces acting on it [2]. For the example colloid, the next time-scale up is the sonic time tcs ≈ 6.7×10 s. Then comes the Langevin time τB ≈ 2.2 × 10 s that measures the exponential decay time of the velocity autocorrelation function within the Langevin approximation. Interestingly, for colloids this time-scale is artificial and does not have direct physical meaning (see appendix of [1]). Next up is the kinematic time τν ≈ 10 s over which vorticity diffuses away from the colloid. If your colloid moves a significant fraction of its radius within the time τν , then the colloid will feel the effects of its own motion from a time τν back, and finite Reynolds number (Re) effects start to kick in. For that reason, it needs to be kept small compared to time-scales of colloidal diffusion or advection. The largest time-scale we consider here is the diffusion time τD ≈ 5s. However, if the colloid also moves under an external force with a velocity vs, then there is an additional time-scale ts = a/vs that measures how long it takes to advect over its radius, and you can then also define a related Peclet number Pe = ts/τD that measures the relative importance of convection over diffusion.