Colloidal%20Sedimentation

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Colloidal Sedimentation
The interplay of Brownian Hydrodynamic Forces
Ard Louis Dept. of Chemistry
Movie from Paddy Royall (Utrecht) Polystyrene
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Sedimentation of colloids
The bigger the particles the faster they sediment
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Sedimentation
Brownian Dynamics 240 discs in a closed container
Pe10
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Hydrodynamic forcesare long-ranged

• G.K. Batchelor, J. Fluid Mech. 52, 245 (1972)

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Sedimentation
Hydrodynamics? 240 discs in a closed container
200,000 small fluid particles to generate
Brownian and hydrodynamic forces Hydrodynamics
induces correlated velocity fluctuations
Pe10
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Hydrodynamic fluctuations?
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Hydrodynamic Screening?

• Still a mystery!
• Walls on sides?
• Wall at bottom?
• Stratification?
• Noise induced phase transitions?

Segre et al. PRL 79,2574 (1997)
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What about Mr Brown?
Brownian and Hydrodynamic fluctuations? Thermal
v.s. non-thermal noise?
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The problem
Colloid
Solvent molecule

• Colloids gtgt solvent molecules
• Stupendous amount of solvent molecules E.g 1011
  water molecules per R1 micron colloid.
• Coarse-graining is necessary

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Computational methods

• Stokesian Dynamics (SD)
• Dissipative Particle Dynamics (DPD)
• Lattice Boltzmann (LB)
• Stochastic Rotation Dynamics (SRD)

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Stokesian Dynamics

• Approximate solution of Stokesequation 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)

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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)

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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

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Stochastic Rotation Dynamics

• a.k.a.Multi-Particle Collision Dynamics
• a.k.a.
• Malevanets-Kapral Method

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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)

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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)

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How does it work?

• Streaming step

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Collision step Coarse-grain the system into
cells Let all SRD particles in a cell collide
with each other
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Streaming step
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Collision step Coarse-grain the system into
cells Let all SRD particles in a cell collide
with each other
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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
conserved This generates Navier Stokes
hydrodynamics
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Many parameters!
A different random rotation axis for each cell

• SRD mass m
• Rotation angle a
• Cell size a
• Average density g
• Temperature kT
• Collision interval dt

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Adding Colloids
System can be viewed as a 2-component MD scheme
WCA (hard sphere like)
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Physically important parameters?
SRD particles are not individual molecules they
are a Navier Stokes solver with thermal noise
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Physically important timescales?
Solvent relaxation time tf10-14
Brownian relaxation time tBm/g10-9
Diffusion time tDR2/Dgtgt tB
Whats important is that they are separated If
tfSRD time-step then tB20 tf
tD2000 tf
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Simulation of 3D sedimentation

• a/R was varied and tested with full velocity
  field around single colloid a/R2 gives 2
  error
• Hydrodynamic radius the same as from friction
• N8 to 800 colloids
• 500,000 SRD particles
• 3-D Box p.b.c. LxLx14 R, Lz42 R
• From 200 to 30,000 Stokes times tS

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Average Sedimentation velocityinfluence of
Brownian forces
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Average Sedimentation velocityinfluence of
Brownian forces
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Average Sedimentation velocityinfluence of
Brownian forces
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Average Sedimentation velocityinfluence of
Brownian forces
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Average Sedimentation velocityinfluence of
Brownian forces
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Spatial correlations
Swirls?
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Spatial correlations
Scaled with (vsed)2 Swirls are dominated by
hydrodynamics
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Temporal CorrelationsBrownian timescales
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Temporal CorrelationsHydrodynamic timescales
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3-D sedimentation
See J.T. Padding and A.A.L, cond-mat/0409133 or
PRL (to appear)

• 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
  Pe30000)
• For Pegt5 long time non-equilibrium fluctuations
  behave just like infinite Pe limit
• Neither Brownian nor hydrodynamic interactions
  can be ignored

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Other fun things to try?

• We now have a flexible method to do simulations

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Example 1 (Nc 2)
Pe 8
Pe 40
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Example 2

• Sedimentation of 1024 (2D) spheres at high
  concentration in a system with periodic
  boundaries
• Reminiscent of Rayleigh-Taylor instability

Pe 40
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Example 3 Lane formation
Pe 50 Brownian Dynamics
Pe50 SRD
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Credits

• Person who did the work
• Dr. Johan Padding
• Details of sedimentation cond-mat/0409133 to
  appear PRL (2004)
• www-louis.ch.cam.ac.uk
• For more stuff if youd like to join us
• Thank you for listening

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