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Mainz-02-11-041
Colloidal Sedimentation
Ard LouisDept. of Chemistry
Movie from Paddy Royall (Utrecht); Polystyrene
The interplay of Brownian & Hydrodynamic Forces
Mainz-02-11-042
Sedimentation of colloids
gFkT
F
F
2
6sed
mgV R
R
The bigger the particles the faster they sediment
sedF V mg
6 ( )R Stokes
buoyant mass
Mainz-02-11-043
George Gabriel Stokes (1851)
sed
mgv
Albert Einstein (1905)
kTD
advection
diffusionsed
B
RvPe
D
mgRk T
Peclet
Robert Brown (1827)
Brownian motion
Mainz-02-11-045
G.K. Batchelor, J. Fluid Mech. 52, 245 (1972)
! 0 0.15sedBig effect v for
Hydrodynamic forcesare long-ranged
1( )
8ˆˆv r F
rI rr
3
0 (1 6.55 )sed sedv v
1
r
Mainz-02-11-046Pe=10
Hydrodynamics?
240 discs in a closed container
+ 200,000 small fluid particles to generate Brownian and hydrodynamic forces
Hydrodynamics induces correlated velocity fluctuations
Sedimentation
Mainz-02-11-047
Hydrodynamic fluctuations?
Segre et al. PRL 79,2574 (1997)
Caflish & Luke, Phys. Fluids 28, 259 (1985)
22
2 2
1
ii
N
ii
fv
L
v v L
Uncorrelated concentration fluctuations induce velocity fluctuations
Mainz-02-11-048
Hydrodynamic Screening?
Segre et al. PRL 79,2574 (1997)
Still a mystery!
•Walls on sides?
•Wall at bottom?
•Stratification?
•Noise induced phase transitions?
Mainz-02-11-049
What about Mr Brown?
Brownian and Hydrodynamic fluctuations?
Thermal v.s. non-thermal noise?
Mainz-02-11-0410
The problem
Colloids >> solvent molecules Stupendous amount of solvent molecules;
E.g 1011 water molecules per R=1 micron colloid.
Coarse-graining is necessary
Colloid Solvent molecule
Mainz-02-11-0411
Computational methods Stokesian Dynamics (SD) Dissipative Particle Dynamics (DPD) Lattice Boltzmann (LB) Stochastic Rotation Dynamics (SRD)
Mainz-02-11-0412
Stokesian Dynamics Approximate solution of Stokes’
equation for many spheresin a solvent (“Oseen tensor”)
No explicit solvent Only correct at low densities of spheres Only correct in the bulk Non-spherical particles extremely difficult Relatively expensive
J.F. Brady and G. Bossis, Ann. Rev. Fluid Mech. 20, 111 (1988)
Mainz-02-11-0413
Dissipative Particle Dynamics
Each DPD particle representsa group of solvent molecules
Pairwise conservative forces Pairwise friction & random forces
Conservation of momentum(unlike traditional Brownian Dynamics)
R.D. Groot and P.B. Warren, J. Chem. Phys. 107, 4423 (1997)See also Sodderman, Dünweg and Kremer, PRE 69, 046702 (2003)
Mainz-02-11-0414
Lattice Boltzmann
Solvent hydrodynamics emergesfrom collisions on a lattice
Computationally cheap (order N) Discretisation problems with boundaries (walls
and colloid-solvent interactions) Brownian motion does not emerge naturally, but
must be added “by hand”A.J.C. Ladd and R. Verberg, J. Stat. Phys. 104, 1191 (2001)
See also Lobaskin & Dünweg NJP, 6, 54 (2004) and Cates et al. JPCM (2004) for ways to include Brownian forces
Mainz-02-11-0416
Stochastic Rotation Dynamics
Solvent hydrodynamics emergesfrom collisions in coarse-grainedcells
Computationally cheap (order N) Particles move in continuous space,
so no discretisation problems Brownian motion emerges naturally
A. Malevanets and R. Kapral, J. Chem. Phys. 110, 8605 (1999)T. Ihle and D.M. Kroll, Phys. Rev. E 67, 066705 (2003); ibid. 066706 (2003)
Mainz-02-11-0417
How does it work? Represent the solvent by N point-like
particles (SRD particles) In between collisions, the SRD particles do
not interact with each other (ideal gas)
Mainz-02-11-0419
Collision stepCoarse-grain the system into cellsLet all SRD particles in a cell collide with each other
Mainz-02-11-0421
Collision stepCoarse-grain the system into cellsLet all SRD particles in a cell collide with each other
Mainz-02-11-0422
0(N) Coarse-grained collision step
The velocities of SRD particles, relative to the centre-of-mass velocity of each cell, are rotated around an angle.momentum and energy are locally conservedThis generates Navier Stokes hydrodynamics
/i i ii cell i cell
i i
u mv m
v u v u
Mainz-02-11-0423
A different random rotation axis for each cell;
• SRD mass m
• Rotation angle
• Cell size a
• Average density
• Temperature kT
• Collision interval t
Many parameters!
Mainz-02-11-0424
Adding Colloids
21
( ) 44
( ); 24
( ); 6
( );
n n
ci
cc
cs
ss
V rr r
V r n
V r n
V r SRD coarse graining
System can be viewed as a 2-component MD scheme
WCA (hard sphere like)
Mainz-02-11-0425
Physically important parameters?SRD particles are not individual molecules:
they are a Navier Stokes solver with thermal noise
Schmidt number = 5f
momentum diffusion
mass diffusion D
cosf
kinematic vis ity
1 0.0016 Re 0.24sv RReynolds number
Sound speed (compressibility effects)
0.0067 0.1s
f
v
c
Mainz-02-11-0426
Physically important timescales?
Solvent relaxation time: f~10-14
Brownian relaxation time: B=m/~10-9
Diffusion time D=R2/D>> B
What’s important is that they are separated
If f~SRD time-step then:
B~20 f
D~2000 f
Mainz-02-11-0427
Simulation of 3D sedimentation
•a/R was varied and tested with full velocity field around single colloid – a/R~2 gives 2% error;
•Hydrodynamic radius the same as from friction
•N=8 to 800 colloids
•500,000 SRD particles
•3-D Box; p.b.c. Lx=Lx~14 R, Lz~42 R
•From 200 to 30,000 Stokes times tS
Mainz-02-11-0428
Average Sedimentation velocityinfluence of Brownian forces
0: (1 6.55 )s sBatchelor v v
Mainz-02-11-0434
Spatial correlations
( ) (0) ( )z z zC r v v r
Scaled with (vsed)2
Swirls are dominated by hydrodynamics
Mainz-02-11-0435
Temporal Correlations:Brownian timescales
1 3/ 212 ( )f BB k T Long time tail:
( / )B Bk T t
em
Hydrodynamic fluctuations
Mainz-02-11-0437
3-D sedimentation:
Average sedimentation velocity is dominated by hydrodynamics even for very small Pe (is this surprising?)
Short time fluctuations dominated by Brownian forces, but long time fluctuations by hydrodynamics for a wide range of Pe (ex of Pe*=30000)
For Pe>5 long time non-equilibrium fluctuations behave just like infinite Pe limit
Neither Brownian nor hydrodynamic interactions can be ignored
See J.T. Padding and A.A.L, cond-mat/0409133 or PRL (to appear)
Mainz-02-11-0440
Example 2 Sedimentation of
1024 (2D) spheres at high concentration in a system with periodic boundaries
Reminiscent of Rayleigh-Taylor instability
Pe = 40
Mainz-02-11-0442
Credits:
Person who did the work: Dr. Johan Padding Details of sedimentation:
cond-mat/0409133 – to appear PRL (2004) www-louis.ch.cam.ac.ukFor more stuff + if you’d like to join us Thank you for listening