Hydrodynamics of colloidal dispersions. Holger Stark Technische Universität Berlin

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1 Hydrodynamics of colloidal dispersions Holger Stark Technische Universität Berlin

2 Examples of Colloidal Dispersions applicadons: food (z.b. milk), paint, ink, fog, medicadon (z.b. nasal spray), cytoplasma of a cell, colloidal quantum dots, mountain lakes Parak (LMU)

3 mountain lake in Rocky Mountains Research Training Group (GRK1558): project area B colloidal pardcles, nano rods, and polymers in viscous environment (e.g. aqueous soludon) ê dynamics of colloidal pardcles in Newtonian fluids

4 World of small Reynolds numbers flow of a viscous fluid: è Navier- Stokes equadons flow field u( r,t ) shear viscosity η ρ du dt inertial forces = p + η 2 u + pressure forces viscous forces b volume forces micron scale: Reynolds number Re = inertial force viscous drag = ρua η 1 è Stokes equadons 0 = p + η 2 u + b

5 World of small Reynolds numbers 1. colloidal pardcle: Stokes friction force: 6πηav è no inerda! t = 0 v = 10 µm s 0.1 Å t = 1µs v 0 ( Aristotelian mechanics ) 2. Dme- reversal symmetry: 0 = p + η 2 u + b ( ) = u( r, t ) u r,t for p p b b

6 G.I. Taylor Coueae Zelle

7 Hydrodynamic interacdons = moving in a viscous environment u 0 wall passive: 1 r active: 1 r 2 passive colloids (bio- )polymers, beadng cilia/ flagella acdve pardcles microorganisms

8 Examples Semiflexible polymers in Poiseuille flow S. Reddig & H.S., JCP (2011)

9 Examples Semiflexible polymers in Poiseuille flow S. Reddig & H.S., JCP (2011) SynchronizaDon & metachronal waves: driven phase oscillators C. Wollin & H.S., EPJE (2011) F. Kendelbacher & H.S. (2012)

10 Swimmer in Poiseuille flow: HI with wall N dissipadve dynamics A. Zöal & H.S., PRL (2012)

11 Swimmer in Poiseuille flow: HI with wall N dissipadve dynamics CollecDve dynamics of swimmers A. Zöal & H.S., PRL (2012) swirling and swarming S. Thutupalli, R.Seemann, & S. Herminghaus, NJP (2011)

12 Solving the Stokes equadons 0 = p + η 2 v + ( ρb ) 0 = div v also: creeping flow equadon determines flow on the micron scale! 1. Implicit solvent: Hydrodynamic interacdons between colloidal pardcles: Method of mobilides (è this talk) - - advantage: simple, high degree of accuracy possible - - disadvantage: only spherical colloids, simple geometries (bulk, one/ two walls) 2. Explicit solvent: MulD- pardcle collision dynamics (MPCD) (è P.Kanehl) pardcle- based solver of Navier- Stokes equadons - - advantage: objects with complex shape, complex geometries - - inaccurate in near fields

13 HYDRODYNAMIC INTERACTIONS: I. METHOD OF MOBILITIES

14 1. Stokes equadon & Oseen tensor soludon for given bulk forces: 0 = p + η 2 v + ( ρb ) 0 = div v v ( x ) = d 3 x O( x x ) ( ) ρb x Oseen tensor: (infinite volume) O( x ) = 1 8πηr 1+ x x r 2 1 r velocity field of a point source: ρb( x ) = f 0 δ( x x ) 0 v ( x ) = O( x x 0 )f 0... Stokeslet

15 2. Hydrodynamic interacdons Fi ui Stokes eqn. are linear è F k colloidal suspension of N pardcles k u i = µ ik F k µ ik = µ ik x 1,,x i,,x k,,x N ( )... mobility tensors (i) µ ii self mobilities (isolated particle µ ii = 1 6πηa 1 ) (ii) µ ik cross mobilities (i k) (iii) all µ ik determine colloidal dynamics completely symmetry: µ T ik = µ ki ( Onsager- RelaDon )

16 2.1 Point pardcles limidng case: r ik = x ik = x k x i a self mobility: µ ii = 1 6πηa 1 cross mobilides? 2 particles with F 2 = 0 "point force" F 1 u 2 = v ( x ) 2 = O( x 2 x 1 )F 1 µ ik = O( x i x ) k µ ik 1 r ki HI are long-ranged! extended pardcles? rotadons?

17 problem: F p u p? x p 2.2 Faxén theorem v0 ( x )... soludon of the Stokes eqn. What is u p? transladon: u p = 1 F 6πηa p + ( a2 2 p )v ( 0 x ) p (i) v 0 ( x ) = 0 Stokes law (ii) F p = 0 u p = v 0 x p (iii) perturbation theory for HI ( ) a2 2 p v ( 0 x ) p!! rotadon: ( ) 1 Ω p = M 8πηa 3 p p v 0 x p vorticity Ω p? M p x p

18 Fi ui 2.3 Rotne- Prager approximadon F k mobility of isolated pardcle: 0 Stokes velocity field of pardcle k : v ( x ) = ( a2 2 k )O( x x k )F k µ = 1 6 a πη place pardcle i into this field: è Faxén theorem µ ( 2 2)( a 1 1 a ) ( ) u = F O x x F i 0 i 6 i 6 k i k k è mobilides: (transladon) µ ii = µ 0 1 ( ) a2 k 2 µ ik = a2 i 2 3 = µ 0 4 a r ik ( ) ( )O x i x k 1+ x ik x ik r ik (i) only 2- pardcle interacdons 3 (ii) expansion up to r 1 ik a r ik x ik x ik r ik 2

19 2.4 MulDpole expansion 1. systemadc approach for higher orders and many pardcles 2. imposed flow fields: e.g. Poiseuille flow 3. boundaries & lubricadon pardcle near contact: (i) lubrication theory asymptotic expansion in ε = d / a (ii) combine with multipole expansion a d

20 2- pardcles mobilides: x r = x µ ik = µ ik ( r ) x x + µ r 2 ik ( r ) 1 x x r 2 1/6 cross mobilities Rotne-Prager self mobilities Rotne-Prager r/a at r/a = 3 deviadons from Rotne- Prager < 3% r/a

21 2.5 Blake tensor J. Blake, J. Fluid. Mech. (1972) J.R. Blake & A.T. Chwang, J. Eng. Math. (1974)

22 HYDRODYNAMIC INTERACTIONS: II. MULTI- PARTICLE COLLISION DYNAMICS

23 efficient solver of Navier- Stokes equadons viscous fluid consists of coarse- grained fluid pardcles 1. streaming step 2. collision step è momentum conservadon v i v i Advantage: 1. well suited for complex geometries 2. easy implementadon of suspended, moving objects 3. includes thermal fluctuadons 4. analydc expressions for viscosity etc. needs computadonal resources: > 10 6 coarse- grained fluid pardcles parallel compudng

24 Examples of hydrodynamic interacdons in sov maaer and acdve pardcle systems using method of mobilides or MPCD I. Biopolymers and colloids in Poiseuille flow à non- linear dynamics II. InerDal microfluidics III. CollecDve dynamics of acdve pardcles in harmonic traps à Boltzmann distribudon in non- equilibrium

25 I. BIOPOLYMERS AND COLLOIDS IN POISEUILLE FLOW

26 1. Semiflexible polymer under Poiseuille flow F(ilamentous)- AcDn: under confinement and flow 8 nm experiments: L=8 μm persistence length: L p =13 μm Experiments: 2w = 10μm D.Steinhauser, S. Köster, T. Pfohl, ACS MacroLeaers (2012)

27 Lateral center- of- mass distribudon: y D cm 0? hydrodynamic repulsion from wall center wall à cross- streamline migradon in non- equilibrium

28 Numeric approach: hydrodynamic interacdons (HI) u P (r ) bead- spring chain with bending rigidity l forces F k on bead k è v i Brownian dynamics with HI of point pardcles: v i = µ ik ( F k + ) k mobility tensors µ ik = 1/ 6πηa for i = k two-wall Green tensor for i k

29 Numeric approach I: hydrodynamic interacdons (HI) u P (r ) bead- spring chain with bending rigidity l forces F k on bead k è v i Brownian dynamics with HI of point pardcles: v i = u P (r i ) + µ ik ( F k + k Poiseuille flow ξ k ) thermal force 1/ 6πηa for i = k mobility tensors µ ik = two-wall Green tensor for i k

30 Two- wall Green tensor R.B. Jones, J. Chem. Phys. (2004) F F

31 Results co- moving reference system! S. Reddig & H.S., J. Chem. Phys. (2012)

32 Lateral center- of- mass distribudon: analydc for needles strength of flow center wall S. Reddig & H.S., J. Chem. Phys. (2012)

33 DeterminisDc driv currents è centerline depledon wall wall S. Reddig & H.S., J. Chem. Phys. (2012) (determinisdc simuladon stardng from a typical configuradon)

34 2. Colloids in Poiseuille flow MoDvaDon: emulsion droplets in narrow channels courtesy R. Seemann simuladon: 1. full muldpole method up to l=5 for transladng and rotadng colloids 2. two- wall Green tensor 3. lubricadon correcdon

35 Two- pardcle trajectories: unbound states swapping trajectories cross- swapping trajectories co- moving reference frame

36 Two- pardcle trajectories: bound states paaern of eight oval shape co- moving reference frame

37 State diagram

38 stable: ParDcle trains unstable:

39 II. INERTIAL MICROFLUIDICS

40 Segré- Silberberg effect InerDal focussing & dilute pardcle suspensions! Re 1 wall centerline pardcle filtradon pardcle separadon è biological and medical applicadons D. Segré & A. Silberberg, Nature (1962) D. Di Carlo et al., PNAS (2007)

41 Results radial distribudon funcdon R Re=40 liv force è radial distribudon funcdon = Boltzmann distribudon C. Prohm, M. Gierlak & H.S. EPJE (2012)

42 III. COLLECTIVE DYNAMICS OF ACTIVE PARTICLES IN HARMONIC TRAPS

43 AcDve modon: Nature versus ardficial swimmers Dreyfus et al. Nature (2005) Chlamy- domonas Volvox algae Janus pardcle bacterial bath

44 Model 1. acdve pardcle i: acdve velocity v 0 p i force dipole moment: d radius: a 2. acdve Peclet number: Pe = v 0 a / D, 3. harmonic trap force: F i = - k 0 r i 4. pardcle horizon: r 0 = Pe k B T / (k 0 a) 5. Brownian dynamics simuladon: (i) v i è v i v 0 p i D = k B T 6πηa (ii) HI including dipole field of swimmers r 0 r i pardcle stops

45 MoDvaDon: study pump state R.W. Nash, R. Adhikari, J. Tailleur & M. E. Cates, PRL (2010). Pe = 100: hedgehog Pe = 500: pump nonequilibrium steady state no HI & Pe = 500 no pump! hedgehog

46 polar order mean polarizadon volume fracdon pump η N no pump constant horizon r 0 è mean volume fracdon η N

47 orientadonal distribudon funcdon Bolzmann distribudon: dipole in mean field è non- equilibrium steady state

48 Examples of hydrodynamic interacdons in sov maaer and acdve pardcle systems using method of mobilides or MPCD I. Biopolymers and colloids in Poiseuille flow à non- linear dynamics II. InerDal microfluidics III. CollecDve dynamics of acdve pardcles in harmonic traps à Boltzmann distribudon in non- equilibrium

Mesoscale fluid simulation of colloidal systems

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