Table Of ContentCoarse-graining dynamics by telescoping down time-scales: comment for Faraday
FD144
Ard A. Louis1
1Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom
(Dated: January 7, 2010)
I briefly review some concepts related to coarse-graining methods for the dynamics of soft mat-
ter systems and argue that such schemes will almost always need to telescope down the physical
hierarchy of time-scales toa more compressed, but more computationally manageable, separation.
PACSnumbers:
Thequestionofhowtoproperlycoarse-grainadynam-
ical simulation is a very interesting one. I think there is
0 no single answer, but want to use an example here from 10−15s τ c o l − fluid collision
−14 τ
1 oursimulationsofcolloidalhydrodynamicstomakesome 10 s f − fluid de−correlation
−13 τ
0 points that I believe are of more general relevance. 10 s FP − Fokker−Planck
−12
2 10 s
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t] tthimeesoτlve,nctacnollbiseioingntoimreedτcaosl alonndgthaesstohlveeyntarreelasxhaotritoenr 10−6 s τνB −kinematic
f f
o than any other colloidal time-scales. The first physi-
s cally relevant time-scale is the Fokker Planck time-scale −1
at. τFP ≈10−13 over which the colloid loses memory of the 1100−0 ss τD −colloid diffusion
short-timeforcesactingonit[2]. Fortheexamplecolloid,
m
thenexttime-scaleupisthesonictimet ≈6.7×10−10s.
cs
d- Then comes the Langevin time τB ≈ 2.2×10−7s that
n measurestheexponentialdecaytimeofthevelocityauto-
FIG. 1: Telescoping down: The hierarchy of time-scales for
o correlationfunction within the Langevinapproximation.
a colloid (here the example taken is for a colloid of radius 1
c Interestingly, for colloids this time-scale is artificial and
[ µm in H20) is compressed in the coarse-grained simulations
does not have direct physical meaning (see appendix of to a more manageable separation. As long as the physically
1 [1]). Nextupisthekinematictimeτν ≈10−6soverwhich important times are clearly separated, the simulation should
v vorticity diffuses away from the colloid. If your colloid still generate the correct physical picture. Once the simu-
6 moves a significant fraction of its radius within the time lations are completed, they can be related in more detail to
6 τ , then the colloidwill feel the effects of its ownmotion particularexperimentsbytelescopingbackouttotherelevant
ν
1
from a time τ back, and finite Reynolds number (Re) experimental time-scales.
1 ν
effects start to kick in. For that reason, it needs to be
.
1 kept small compared to time-scales of colloidal diffusion
0
or advection. The largest time-scale we consider here is telescoped down to a more computationally manageable
0
1 the diffusion time τD ≈ 5s. However, if the colloid also separationinordermaximisesimulationefficiency,butin
: moves under an external force with a velocity vs, then such a way that the times are still sufficiently separated
v
thereis anadditionaltime-scalets =a/vs thatmeasures to correctly resolve the underlying physical behaviour.
i
X how long it takes to advect over its radius, and you can A good example of the rationale behind this thinking
r then also define a related Peclet number Pe = ts/τD can be illustrated with the sonic time tcs. Physically it
a thatmeasuresthe relativeimportance ofconvectionover needs to be muchsmaller thanthe diffusion time, or else
diffusion. locally you have supersonic behaviour. But if it is too
From the Fokker Planck time on up to the diffusion small,thesimulationwillspendmostofitstimeresolving
time covers 13 orders of magnitude. It is clearly not sound waves that may not be that interesting for the
possible to capture all of these in a simulation. Instead, colloidal behaviour you are trying to reproduce, making
what is needed is time-scale separation. As long as the the simulation very inefficient.
time-scales are properly separated, you should still be In order to correctlyinterpret the physical meaning of
simulating the correct underlying physics. This process you simulation you need to telescope the time-scale hi-
can visualized in Figure 1 (taken from ref. [1]) which erarchy back out to the physical one you want to study.
shows an example of how the hierarchy of time-scales is Forexample,ifyouareinterestedinphysicsthatisdom-
2
inated by diffusion, you would map onto your physical wearguein [1]thatmanyothercoarse-grainingmethods
diffusion time. A consequence of this strategy is that a for dynamics must make implicit use of the telescoping
single simulation can map onto many different physical downprocess. Youcanmakeyourmethodworkby care-
times. E.g. ifyourcolloidhasaradiusa=1µm,thenthe fully analyzing your time-scales, and then making sure
diffusion time is τ = 5s, and that fixes the time-scales you know how to telescope back out to the experimen-
D
for your simulation. On the other hand, if your colloid tal situation you want to emulate. It helps to do this
hasaradiusa=100nm,thenτ =5×10−3sandtheex- in terms of dimensionless variables. However sometimes
D
act same simulation would have a different fundamental you can’t compress the hierarchy to a computationally
time-scale. Thatmeans thatfor examplethe viscosityin achievable regime without bringing some time-scales too
your simulations would be different depending on what close to each other, or even switching the order of time-
colloid size you were mapping to, even though the fun- scales. It is then a matter of subtle judgement if the be-
damental physics is the same. This fact suggests that haviour that comes out of your simulation is physically
physical viscosity is often not such a good parameter to correct.
try and fit to in a coarse-grainedsimulation.
Youcanalsocorrectlymapthesamesimulationtodif- A similar analysis can be used to interpret systems
ferenttime-scaleseventhoughthecolloidalsystemisun- wherethedynamicsaredominatedbyenergybarriers. It
changed. Say that you are interested in the longer-time is going to be very hard to know what the real physical
behaviour of the velocity autocorrelation function that time-scales are here because processes depend exponen-
is dominated by the kinematic time τ . In that case if tially on barrier heights. In a coarse-grained dynamical
ν
you mapped to a physical system with a = 1µm, the simulation,itmaybeadvantageoustodramaticallylower
times would be quite different from what you would get (free)energybarriersinordertospeedupthesimulation.
when you mapped to the diffusion time of the same sys- Thehopeisthatyoukeeptherelativeorderthatcharac-
tem. So here the viscosities etc... would have different terizes the physical system you want to emulate, so that
values depending on what you processes were focusing at least the qualitative dynamic effects are correct. But
on. This is a good example of a no free lunch theorem. if, say, you try to interpret your time-scales from a mea-
If you coarse-graindynamics, you almost always need to surement of single particle diffusion, and then use that
do some kind of telescoping down, and that means that time to extract a physical time for a different process
it is hard to simultaneously match multiple time-scales in your simulation that is dominated by energy barri-
in your system. ers, then you will get those numbers completely wrong.
The particular example described here concerns a col- Clearly,there are many tricky subtleties that arise when
loid in suspension where it is relatively straightforward trying to coarse-grain the dynamics of soft-matter sys-
to work out what all the time-scales are. Nevertheless, tems.
[1] J. T. Paddingand A. A.Louis, Phys. Rev.E. 74, 031402 loids, Elsevier, Amsterdam, (1996).
(2006).
[2] J. K.G. Dhont,An Introduction to the Dynamics of Col-
3
TABLE I: Time-scales relevant for colloidal suspension; numerical values are for a buoyant colloid of radius a=1µm in H2O.
ForH2O,thespeedofsoundis1.48×109µm/s. Thekinematicviscosityν =1×106µm2/s,andmeasuresthediffusionconstant
with which vorticity diffuses away.
Solvent time-scales
Solvent collision time overwhich solvent molecules interact τcol ≈10−15 s
Solvent relaxation time overwhich solvent velocity correlations decay τ ≈10−14−10−13 s
f
Hydrodynamictime-scales
Sonic time overwhich sound propagates one colloidal radius tcs = a ≈6.7×10−10s
cs
Kinematic time overwhich momentum (vorticity) diffuses one colloidal radius τν = a2 ≈10−6s
ν
Brownian time-scales
Fokker-Planck timeover which force-force correlations decay τFP ≈10−13s
Brownian relaxation timeover which colloid velocity correlations decay in Langevin Eq. τB = Mc ≈2.2×10−7s
ξS
a2
Colloid diffusion timeover which a colloid diffuses overits radius τD = ≈5s
Dcol
Ordering of time-scales for colloidal particles
τcol < τf,τFP < tcs < τB < τν < τD
