1. Mainz-02-11-04 1 Colloidal Sedimentation Ard Louis Dept. of Chemistry Movie from Paddy Royall (Utrecht); Polystyrene The interplay of Brownian & Hydrodynamic Forces

2. 2 Sedimentation of colloids The bigger the particles the faster they sediment buoyant mass

3. Mainz-02-11-04 3 George Gabriel Stokes (1851) Albert Einstein (1905) Peclet Robert Brown (1827) Brownian motion

4. Mainz-02-11-04 4 Pe=10 Brownian Dynamics 240 discs in a closed container Sedimentation

5. Mainz-02-11-04 5 G.K. Batchelor, J. Fluid Mech. 52, 245 (1972) Hydrodynamic forces are long-ranged

6. Mainz-02-11-04 6 Pe=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

7. Mainz-02-11-04 7 Hydrodynamic fluctuations? Segre et al. PRL 79,2574 (1997) Caflish & Luke, Phys. Fluids 28, 259 (1985) Uncorrelated concentration fluctuations induce velocity fluctuations

8. Mainz-02-11-04 8 Hydrodynamic Screening? Segre et al. PRL 79,2574 (1997) Still a mystery! Walls on sides? Wall at bottom? Stratification? Noise induced phase transitions?

9. Mainz-02-11-04 9 What about Mr Brown? Brownian and Hydrodynamic fluctuations? Thermal v.s. non-thermal noise?

10. Mainz-02-11-04 10 The problem Colloids >> solvent molecules Stupendous amount of solvent molecules; E.g 10 11 water molecules per R=1 micron colloid. Coarse-graining is necessary ColloidSolvent molecule

11. Mainz-02-11-04 11 Computational methods Stokesian Dynamics (SD) Dissipative Particle Dynamics (DPD) Lattice Boltzmann (LB) Stochastic Rotation Dynamics (SRD)

12. Mainz-02-11-04 12 Stokesian Dynamics Approximate solution of Stokes’ equation for many spheres in 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)

13. Mainz-02-11-04 13 Dissipative Particle Dynamics Each DPD particle represents a 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)

14. Mainz-02-11-04 14 Lattice Boltzmann Solvent hydrodynamics emerges from 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

15. Stochastic Rotation Dynamics a.k.a. Multi-Particle Collision Dynamics a.k.a. Malevanets-Kapral Method

16. Mainz-02-11-04 16 Stochastic Rotation Dynamics Solvent hydrodynamics emerges from collisions in coarse-grained cells 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)

17. Mainz-02-11-04 17 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)

18. Mainz-02-11-04 18 How does it work? Streaming step:

19. Mainz-02-11-04 19 Collision step Coarse-grain the system into cells Let all SRD particles in a cell collide with each other

20. Mainz-02-11-04 20 Streaming step:

21. Mainz-02-11-04 21 Collision step Coarse-grain the system into cells Let all SRD particles in a cell collide with each other

22. Mainz-02-11-04 22 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

23. Mainz-02-11-04 23 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!

24. Mainz-02-11-04 24 Adding Colloids System can be viewed as a 2-component MD scheme WCA (hard sphere like)

25. Mainz-02-11-04 25 Physically important parameters? SRD particles are not individual molecules: they are a Navier Stokes solver with thermal noise Schmidt number = Sound speed (compressibility effects)

26. Mainz-02-11-04 26 Physically important timescales? Solvent relaxation time: t f ~10 -14 Brownian relaxation time: t B =m/ g ~10 - 9 Diffusion time t D =R 2 /D>> t B What’s important is that they are separated If t f ~SRD time-step then: t B ~20 t f t D ~2000 t f

27. Mainz-02-11-04 27 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 t S

28. 28 Average Sedimentation velocity influence of Brownian forces

29. 29 Average Sedimentation velocity influence of Brownian forces

30. 30 Average Sedimentation velocity influence of Brownian forces

31. 31 Average Sedimentation velocity influence of Brownian forces

32. 32 Average Sedimentation velocity influence of Brownian forces

33. Mainz-02-11-04 33 Spatial correlations Swirls?

34. Mainz-02-11-04 34 Spatial correlations Scaled with (v sed ) 2 Swirls are dominated by hydrodynamics

35. Mainz-02-11-04 35 Temporal Correlations: Brownian timescales Long time tail: Hydrodynamic fluctuations

36. Mainz-02-11-04 36 Temporal Correlations: Hydrodynamic timescales

37. Mainz-02-11-04 37 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)

38. Mainz-02-11-04 38 Other fun things to try? We now have a flexible method to do simulations

39. Mainz-02-11-04 39 Example 1 (N c = 2) Pe = 8Pe = 40

40. Mainz-02-11-04 40 Example 2 Sedimentation of 1024 (2D) spheres at high concentration in a system with periodic boundaries Reminiscent of Rayleigh-Taylor instability Pe = 40

41. Mainz-02-11-04 41 Example 3: Lane formation Pe ~ 50: Brownian DynamicsPe~50: SRD

42. Mainz-02-11-04 42 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 you’d like to join us Thank you for listening

43. Mainz-02-11-04 43