Colloidal Hydrodynamics and Interfacial Effects

Size: px

Start display at page:

Download "Colloidal Hydrodynamics and Interfacial Effects"

Dortha Harrison
5 years ago
Views:

1 Colloidal Hydrodynamics and Interfacial Effects Gerhard Nägele (1) and Maciej Lisicki (2) (1) Institute of Complex Systems (ICS-3), Forschungszentrum Jülich, Germany (2) Faculty of Physics, University of Warsaw, Poland SOMATAI Summer School in Berlin, Germany, August 21,

2 Content 1. Inertia-free dynamics in microworld 2. Hydrodynamic interactions 3. Particle near an interface 4. Active swimmers near a surface 2

3 1. Inertia-free dynamics in microworld 3

4 Length scales of colloids and micro-organisms Colloidal dispersions (including proteins & viruses) Atoms Molecules Bacteria, protozoa ( µm) human cell: 10 µm 1 nm 10 nm 100 nm 1 µm radius non - Brownian (L > 5 µm) Colloids, proteins, bacteria, single-celled eukaryotes share friction-dominated hydrodynamics 4

5 Low-Reynolds-number fluid flow Inertial forces tiny as compared to friction forces: Reynolds - # << 1 ur () = 0 V incompressibility in water: Re = inertial forces viscous forces = ρ flv η 0 wale: 10 8 human swimmer: colloid particle: bacteria /cells: p() r +η ur () + fr () = 0 linear Stokes equation inertia-free force balance 5

Likewise: inertia free microparticle dynamics M d V (t) dt ζ 0 V (t) M 5 t >> τ B = 10 s ζ0 x >> V τ L Re 1 0 B stopping distance V0 30 50 µ m

6 Likewise: inertia free microparticle dynamics M d V (t) dt ζ 0 V (t) M 5 t >> τ B = 10 s ζ0 x >> V τ L Re 1 0 B stopping distance V µ m / s Rhodospirillum bacteria (length L 5 µm) Inertia free motion on coarse-grained colloidal time- and length scales Bacteria need no breakes 6

7 1. Implication: linear particle force velocity relation Forced sphere with no-slip BC V t = µ 0 F h µ = t 0 1 6πη a 0 V F = F h translational mobility r No force no motion (Aristotles) ur () 1 r long-range decay (momentum conservation) u dr = 0 streamlines parallel to local fluid velocity 7

8 2. Implication: Linearity V = µ F V = µ F F 1 µ µ 2 F * = m g V F (transl. mobilities for L >> D) V = V + V Experiment: needle in syrup, no rotation Linear superposition of particle velocities and flow fields allowed 8

3. Implication: Kinematic reversibility G.I. Taylor, Cambridge https://web.mit.edu/hml/ncfmf.

9 3. Implication: Kinematic reversibility G.I. Taylor, Cambridge Flow determined by instantaneous boundary configuration and velocities Force reversal and application history retracing : Fluid elements retrace motion along unchanged streamlines 9

10 ink t 1 t 2 t 3 Ink (colloid) particle motion across circular streamlines induced by: - Brownian motion - Inertia effects (Re 1) - Many - particle HI in concentrated systems (chaotic trajectories) - Reversibility - breaking direct interactions D. Pine et al., Nature 438 (2005) J. Gollub and D. Pine, Physics Today, August

11 Kinematic reversibility + symmetry = useful tool { u, V, Ω } { u, V, Ω } V =0 Lift force arises here only if non- zero inertial contributions: du/dt 0 V 0 also for: Non-spherical rigid or flexible particle (polymer, droplet), flexible wall 11

12 Two homework problems w/o math gravity V Is rod re-orienting itself while settling with a sidewise velocity component? Is settling sphere moving towards / away from rigid wall? Is it rotating? 12

13 Particle forces balance in microworld d M V = 0= F + F + F + F dt i i h ext int B i i i i nr () σr () nr () h Fi = ds σ( r;x) n( r) S i fluid force / area on sphere surface element ds S i σ αβ() r = p() r δ αβ +η0 αu() β r + βu α() r u, p from Stokes Eqs. & BC Hydrodynamic drag force determined by fluid boundary configuration and velocities plus BCs Direct Interactions and Brownian forces break reversibility 13

14 Brownian motion Movie: E.R. Weeks, Austin Mean-squared displacement 14

15 Big particles E.R. Weeks, Austin Small particles a = 1.5 µm 1.0 µm 0.5 µm Stokes - Einstein - Sutherland: 2 a τ a = a D t 0 3 Brownian motion negligible (too slow) if a > 5 µm 15

16 Surface- and interfacial boundary conditions z = 0 η 1 η 2 l s l s= Liquid interface Partial slip wall Perfect slip wall = free surface Liquid liquid interface: Rigid partial-slip surface: du η =η dz (1) (2) x,y x,y 1 2 (1) (2) z z du dz u = u = 0 u = u (1) (2) x,y x,y u x,y = l s uz = 0 du dz x,y 16

17 Sedimentation of a sphere: no-slip and perfect slip (gas bubble) No-slip sphere: outside flow 2 1 a 1 ur () = ( 1+ rr ˆˆ) ( 3rr ˆˆ 1) r r F πη 0 r Forced point particle = Stokeslet flow: 0 { ( )} + rr 1 1 ust() r = 1 ˆˆ F = T() r F 8πη r Gas bubble: outside flow: Oseen tensor gas { 1 ( )} F ˆˆ r 4πη0 u ( r) = 1+ rr, (r > a) V V gas F t = = µ 0( 6 stick ) F πη a 0 F = = µ 4πη a 0 t 0 ( slip) F Flow around no-slip sphere is superposition of Stokeslet and Source doublet 17

1 a F 1 8 0 ( ( ˆˆ ) F ) ( 3ˆ ˆ πη u r = 1+ rr e rr 1) e r 3 3 r Stokeslet 2 Source doublet 1 O(r ) 3 O(r ) Arbitrary outside

18 1 a F ( ( ˆˆ ) F ) ( 3ˆ ˆ πη u r = 1+ rr e rr 1) e r 3 3 r Stokeslet 2 Source doublet 1 O(r ) 3 O(r ) Arbitrary outside particle flows can be superposed by Stokeslet and higher order singular elemental flow solutions (certain derivatives of Stokeslet) 18

19 Application: Boundary layer method r Rigid particle of arbitrary shape with no - slip BC: S [ ] ur ( ) = ds' Tr ( r') σ n ( r') S V Ω R r' Stokeslet superposition Inserting BC gives surface integral equation for stress: ( ) ') [ ] V + Ω r R = ds' Tr ( r σ n ( r') ( r S) S { } [ ] { } { h h } V, Ω σ n F, T, u (Mobility problem) 19

Particle with complex surface: Triangularization in N triangles N ( ) = Tr r [ ] V + Ω r R ( ) σ n ( r ) i i j j= 1 3N 3N inversion j tt tr V μ μ F = rt rr h Ω μ μ T h r i r j Mobility matrix:

20 Particle with complex surface: Triangularization in N triangles N ( ) = Tr r [ ] V + Ω r R ( ) σ n ( r ) i i j j= 1 3N 3N inversion j tt tr V μ μ F = rt rr h Ω μ μ T h r i r j Mobility matrix: Translation and rotation are coupled Mobility matrix depends on particle shape, orientation and size only Key quantity in theory and simulations What about many particles and their hydrodynamic motion coupling? 20

21 2. Hydrodynamic interactions 21

22 Too complicated? Make it spherical! 22

23 Bead modeling of complex shaped particles Stokesian Dynamics simulation J.W. Swan et al. Phys. Fluids 23 (2011) 23

24 N freely rotating spheres N tt V = μ (X) F 3 3 mobility tensors 3 i j = 1 i j j V V 2 F 2 V = µ tt ( X) F mobility problem V 1 R 1 Mobility matrix depends on configuration X Long-ranged 1/r Many body character Positive definiteness is crucial Approximations { } X = R,..., R 1 N (,..., ) V= V V 1 N tt tt μ11 μ 1N tt µ = tt tt μn1 μ NN 3N x 3N matrix 24

Mobility approximations Point particles approximation: t (i) t i = µ i i + inc( ) = µ i i + ( i j) j j i V F u r F TR R F Reflects far distance behavior only Can violate positve definiteness

25 Mobility approximations Point particles approximation: t (i) t i = µ i i + inc( ) = µ i i + ( i j) j j i V F u r F TR R F Reflects far distance behavior only Can violate positve definiteness for close particles Rotne Prager approximation: Far distance only, but positive definiteness preserved (see notes) Full hydrodynamic interactions (e.g. HYDROMULTIPOLE code) numerically expensive 25

26 Sedimentation of spherical cloud of non - Brownian beads leakage toroidal circulation in cloud rest frame (cf. liquid drop) V = µ 0 t sed 0 F cloud 0 N 1 Vsed Vsed + F 5 πη R 0 M.J. Ekiel-Jezewska, Phys. Fluids 18 (2006), B. Metzger et al., J. Fluid Mech. 580, 238 (2007) Cloud sediments faster than single bead (point model overestimates velocity) Instability for large N and large settling times 26

27 Evolution: spherical cloud torus breakup in two clouds t = t/ τ * sed R * t = 0 * t = 700 * t = 400 * t = 800 E. Guazzelli and J.F. Morris, A Physical Introduction to Suspension Dynamics, Cambridge (2012) 27

28 Reason: Chaotic fluctuations due to many - body HI (-1.1,0,1.16) (-1.1, 0, 1.20) z/σ d X(t) tt =µ X(t) F dt ( ) } no DI no BM non - linear in X gravity μ tt ij η 1 R - R 0 i j x/σ x/σ Sensitive dependence on initial configuration for N > 2 deterministic chaotic trajectories Courtesy: M. Ekiel-Jezewska & E. Wajnryb, Phys. Rev. E 83, (2011) 28

29 To relax: just two non - Brownian microspheres sedimenting V t 0 = µ 0 F gravity V 0 F 29

30 The sedimentation race: bet which pair settles fastest, and how? 30

31 r gravity constant sep. vector r V pair 0 t sed Vsed 0 > =µ F 31

32 Sedimentation in homogeneous suspension gravity V V sed 0 sed HS V / V 1 1.8φ sed 0 sed 1/3 salt CS Suspension sedimentation velocity smaller than that of isolated particle Slower sedimentation of charged clay particles (river - delta) φ Banchio & Nägele, JCP 127,

33 Plausibility argument g(r) 1 neutral. charged p = nf sa Pressure gradient drives mean fluid backflow Bottom wall important Increased friction with backflowing fluid rm n 1/3 r 33

34 3. Particle near an interface 34

35 Influence of free planar surface: method of images ( σ n)ds h - h z X du (z = 0) x,y dz = 0 u z (z = 0) = 0 ( σ n) im ds Surface superposition of image Stokeslets S [ σ n] r r r ) [ σ n] ur ( ) = ds' Tr ( r') ( ') ds' T( ' ( r ') S im im 35

36 Point particle (Stokeslet) near a free surface 36

37 Stokeslet near a no-slip wall (I) To enforce zero tangential slip: more complex image system required Faster than O(1/r) decay for z : momentum taken out from system (screening) 37

38 Stokeslet near a no-slip wall (II) r 2 ur ( ) O(r ), z + 3 ur ( ) O(r ) No-slip particle near a no-slip wall: S [ σ n] r [ ] u( r) = ds' T( r r') ( ') ds' T ( r r') σ n ( r') S im im im 38

39 Hydrodynamic screening Long-range forced flow decay can be screened by fixed obstacles in the flow Examples: flow due to a point force parallel to walls / perpendicular to axis: Obstacles 1. cylindrical pipe 2. two parallel walls 3. electrophoresis Flow asymptotics e αr 2 1/r 3 1/r A.J. Banchio, G. Nägele et al., Phys. Rev. Lett. 96 (2006) H. Diamant, Israel Journal of Chemistry 47 (2007) 39

40 Anisotropic mobilities of sphere close to wall h Perkins & Jones, Physica A 189 (1992) Cichocki & Jones, Physica A 258 (1998) a / h Experimental verification: Holmqvist, Dhont, Lang, JCP 128 (2007) Only rotational mobility for vertical axis non-zero at contact Lubrication 40

41 Lubrication: squeezing motion of no-slip sphere towards no-slip wall l rough < h < 0.01a F dh dt µ =µ (h)f rel t 0 F h a t µ 0 F h(t) h0 exp t a Strong pressure gradient inside gap determines dynamics (viscous stress negligible) Finite contact time due to: surface roughness, van der Waals attraction, 41

380 t 0 Experimental realization: Particles at water air interface of

42 In-touch motion along a planar free interface z x y F F r Ω r 0 r =µ T µ / µ µ t V = µ F t / µ t 0 Experimental realization: Particles at water air interface of hanging drop Extension to Q2D system of interacting microspheres: Cichocki, Ekiel-Jezwska, G. N., Wajnryb, JCP 121 (2004) 42

43 Apparent like - charge attraction of particles near boundary figure taken from: Squires & Brenner, PRL (2000) charged glass wall or liquid - air interface F - F? Attractive wall: apparent repulsion 43

44 4. Active swimmers near a surface sea urchin sperm bacterium rotating rigid flagellum wave-like flagellum motion 44

45 Purcell s scallop theorem for autonomous microswimmers Internal forces and torques only: F h = 0 = F T h = 0 = T time Purcell s microscallop theorem: For net displacement after one shape cycle: - Non - reciprocal sequence of body deformations: - At least 2 - parametric deformations - Skew symmetric flagellar rotation, e.g. Additionally required: Friction asymmetry φmin φmax E.M. Purcell, Life at low Reynold s number, Am. J. Phys. 45, 3 (1977) 45

46 Re << 1: microrganism in water or macroscopic swimmer in glycerin G.I. Taylor, Cambridge 46

47 MPC simulation of swimming E. coli in bulk (full HI and Brownian motion included) Courtesy: R. G. Winkler, J. Hu and G. Gompper, FZ Juelich, work submitted (2014) 47

48 Far distance flow field around a no - slip particle Expand around inside point: S [ ] [ ] ur ( ) = ds' Tr ( ) r' Tr ( )... σ n ( r') r' r r ' Split in symmetric and anti-symmetric parts: S r ( 1 rrr) h h h 3 ˆ ˆ ˆˆˆ 2 2 ur () Tr () F + r T r 3 : S + O(r ) 8πη r 8πη r Freely mobile particle (force- and torque-free): h h F = 0 = T autonomous microswimmer Creates O(r -2 ) flow disturbance by its symmetric force dipole S = ds' σ nr ( ') r' + σ nr ( ') 1Tr rσ ' nr ( ') h S ( ) surface - moment tensor 48

49 Symmetric force dipole (pusher: p > 0) [ ] u( r) = T ( r d e ) T ( r+ d e ) Fe 0 z 0 z z r h 1 S = p e zez 1 3 p = 2Fd η V L dipole moment 2 θ Far field flow : cosθ= e z rˆ p u D( r ) 3cos θ 1 ˆ 2 8πη r r 0 2 Swimmer - created flow has form of symmetric force dipole for distances > L 49

50 Pusher: p > 0 Puller: p < 0 Production stroke E. Coli, salmonella, sperm, Algae Chlamydomonas, Propelling part at rear Propelling part on head side Tend to attract each other. Tend to repel each other ( asocial ). 50

51 Hydrodynamic interaction of dipole swimmers u R u R R e p R e R ˆ ( ˆ ) 2 adv 1 ( 1) = D( 1 2; 2) = πη(r 12) adv 2 2 = D u ( R ) u ( R R ; e ) z e 1 e 2 X R 12 1 Ω R u R e E R e 2 ( ) ( ) adv 1 ( 1) inc ( 1) + 1 inc( 1) 1 Lauga & Powers Rep. Prog. Phys. 72 (2009) Two pushers on not too diverging course tend to attract and reorient each other into parallel side-by-side motion Two pullers reorient antiparallel, swimming away from each other 51

52 Dipole swimmer near a surface e im D 0 D 0 im u() r = u ( r R ;) e + u ( r R ; e ) h free surface z X adv im im z R0 = D,z R0 R0 e u ( ) u ( ; ) p 2 = 1 3cos θ 2 32πηh im e ( 2 ) adv 3p sin θ 2 Ω y ( R0) = 2 cos θ 3 128πηh cosθ= e e x Interface always attracts pusher and orients it parallel to surface Interface always repels puller, orienting it perpendicular to interface 52

53 These qualitative features are valid also for a partial-slip wall and liquid interface Berke et al., Phys. Rev. Lett. 101 (2008) Lauga & Powers, Rep. Prog. Phys. 72 (2009) Lopez & Lauga, Phys. Fluids 26 (2014) 53

54 No qualitative surface sensitivity of swimmer? 54

55 Bacterium moves in circles, with orientation dependent on surface BC 55

56 MPC simulations of swimming E. coli (HI and Brownian motion included) Courtesy: R. G. Winkler, J. Hu and G. Gompper, FZ Juelich, work submitted (2014) 56

57 Qualitative explanation Di Leonardo et al., Phys. Rev. Lett. 106 (2011) 57

58 Approximative singularity solutions superposition calculation Rotlet dipole added to force dipole: Applies for wall distances h > L u RD 3 ( r) O(r ) Lopez & Lauga, Phys. Fluids 26 (2014) 58

59 Critical values for partial slip wall Lopez & Lauga Phys. Fluids 26 (2014) l slip h slip effect negligible aspect ratio of swimmer Applications: Biosensor for surface slip; guiding bacteria by surface stripes; Sorting bacteria according to circle radius 59

60 Just one concluding remark Studying Low-Reynolds-number dynamic phenomena is fun, and often leads to surprising findings. 60

61 Thank you for your attention, and to Peter Lang (ICS-3, FZ Jülich) for his kind invitation 61

62 62

64 64

65 65

66 66

67 5. Phoretic particle motion in external field 67

Non - conducting charged sphere in electrolyte exposed to uniform electric field + E _ V E E el =µ el + O ( 2 ) Hückel limit λ D a V el Q< 0 0 0 0 el h 0 el F + F = QE 6πη av = 0 a 0 el µ = Q 6πη a 0

74 Non - conducting charged sphere in electrolyte exposed to uniform electric field + E _ V E E el =µ el + O ( 2 ) Hückel limit λ D a V el Q< el h 0 el F + F = QE 6πη av = 0 a 0 el µ = Q 6πη a 0 λ D oppositely charged diffuse layer electrokinetic effects for non-dilute diffuse microion layer only Retarding electro-osmotic drag by cross-streaming of oppositely charged fluid near sphere surface Retarding polarization effect force due to distortion of EDL away from spherical symmetry. Restructuring by microion diffusion and conduction, and by solvent convection is non-instantaneous 74

75 Smoluchowski limit and flat plane electro-osmosis Helmholtz Smoluchowski limit (1879 & 1903) E n = ˆrˆ λ << a D Ε out () r field within ultrathin EDL tangentially curved around non conducting sphere ( ) ( ) Eout r EDL 1 nn E Electro-osmosis: flow of charged electrolyte fluid past stationary (particle) surface 75

76 Electro - osmotic plug flow in an open capillary tube glass wall (negatively charged) Anode plug flow Cathode 2h U E Electroosmosis used in microfluid devices to drive aqueous media through narrow micro - channels where Low-Reynolds number fluid dynamics applies to 76

77 Open (a) versus closed (b) electro-osmotic cells 3λ D 3λ D ds u= 0 A 0 h y h / 3 if h λ Absolute electrophoretic velocity measured in zero flow plane (Malvern Zeta-sizer) D 77

78 Sphere in quiescent, infinite fluid, no-slip BC Sphere stationary (rest frame) u = 0 u = V 78

Hydrodynamic interactions and wall drag effect in colloidal suspensions and soft matter systems

Hydrodynamic interactions and wall drag effect in colloidal suspensions and soft matter systems Maciej Lisicki" Institute of Theoretical Physics" Faculty of Physics, University of Warsaw" Poland SOMATAI