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

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1 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 workshop, Jülich, March 2014

2 General Idea ML: 1st part, today :-) I. Introduction to hydrodynamics of suspensions" II. Near-wall dynamics of colloids" " " Gerhard Nägele: 2nd part, Summer School, Berlin" III. Charged colloidal systems" IV. Swimming of microorganisms" and many more interesting topics." " Joint lecture notes containing more details will be published afterwards.

3 Part I. Crash Course of Suspensions Hydrodynamics

4 Colloidal soft matter Emulsions, cosmetics, healthcare products, food, gels Active (Janus) particles, " motile microorganisms," bacteria Macromolecules, " micelles, lipid bilayers, proteins, engineered nanoparticles

5 Colloidal sizes granular media Colloidal dispersions (including proteins & viruses) molecules bacteria, protozoa atoms ( µm) human cell: 10 µm 1 nm 10 nm 100 nm 1 µm radius non - Brownian Brownian motion is important on these length scales!" " (L > 5 µm)

6 Mesoscopic description (N. van Kampen) Macro Thermodynamics Meso Brownian Motion Einstein s theory of fluctuations Separation of time scales! Micro Positions and momenta of particles etc.

7 Overview of time scales (particles with a = 100 nm in water) τ 2 mol τ sound τ B τ vort 10 t [ sec] τ = D a / D 0 molecular view - momentum relax. resolved - unsteady solvent flow diffusional relaxation time quasi - inertia free particles and fluid motion " Stokes flow Separation of time scales!" " Experimental resolution sets the minimal time scale, i.e. interval over which the observables are averaged.! " Theoretical description shall be compatible with the choice of time scale as well!! " Brownian motion - positional description, velocities irrelevant.

8 Microworld hydrodynamics Aim: description of the fluid flow field Volvox bacteria stirring the fluid (R. Goldstein, Cambridge) Conservation/balance laws: Continuity equation - incompressible flow Momentum balance equation (Navier - Stokes) +

9 Stokes Hydrodynamics Reynolds number Characteristic length & flow velocity Microworld: small U and l or: large viscosity G.G. Stokes O. Reynolds Stokes equations No inertia, no turbulence!

10 Low Reynolds Number Flows - the movie All movies were taken from: G I Taylor, Cambridge U." " National Committee for Fluid Mechanics Films (1960s)" Also on YouTube:

11 Properties of Stokes equations 1.Linearity - superposition principle! 2.Instantaneity (stationarity) on relevant time scales! 3.Kinematic reversibility World of Aristotle: no force, no velocity

12 Stokes Equations. Linearity: Two rods Example: two sedimenting rods Observation:

13 Stokes Equations. Linearity: Tilted rod How does it move? Settling angle Tilt angle " Superposition of motion in two perpendicular directions

14 Stokes Equations. Instantaneity. On the time scales of interest, the Stokes eqns. are instantaneous - infinite propagation of disturbances. The system immediately adjusts to new configuration. Flow field depends only on the current configuration of boundaries & particles

15 Stokes Equations. Kinematic reversibility. Reversal of forces reverses the trajectories!

16 Stokes Equations. Kinematic reversibility. (a) (b) (c) (d) Reversibility + symmetry = useful tools The particle sediments near a wall at a constant distance due to symmetry!

17 Sedimentation of a sphere - Stokes law A single sphere in an unbounded fluid settling under influence of a force (gravity) G.G. Stokes Long-ranged 1/r Flow disturbance caused by the sphere u 0 u = V =

18 Two particles - Hydrodynamic interactions Single-particle term Contribution to velocity of particle 2 from the force acting on particle 1 Higher-order terms

19 Two particles - Hydrodynamic interactions Linear relation between forces and velocities;! dependence on all other particles C. W. Oseen Oseen tensor T Single-particle term Contribution to velocity of particle 2 from the force acting on particle 1 Higher-order terms

20 The Oseen tensor - the point particles model Reflects far-field behaviour only can get very wrong for close particles (perpetuum mobile!)" Only two-body interactions" Unknown a " But:" Simple, intuitive picture!

21 Higher-order terms: Multiple reflections Linear relation between forces and velocities;! dependence on all other particles

22 Mobility matrix - Hydrodynamic interactions Velocity Mobility matrix Force Velocities and forces are related linearly, but the mobility matrix depends on the configuration of the whole system and describes hydrodynamic interactions (mediated via the solvent).! " Many-body! Dynamic! Long-ranged V 3 V 2 t F 2 V 1 HI s are complicated, yet crucial in dynamics of colloids

23 Three sedimenting non-brownian Spheres: asymmetric starting configuration (-1.1,0,1.16) (-1.1, 0, 1.20) z/σ d X tt X F e =µ dt ( ) non - linear in X gravity x/σ x/σ Sensitive dependence on initial configuration for N > 2 chaotic trajectories Courtesy: M. Ekiel-Jezewska & E. Wajnryb, Phys. Rev. E 83, (2011) 25

24 Complex system? Simplify!

25 Making a complex problem simple

26 In search for the mobility matrix Point force approximation (Oseen tensor) - may be unphysical (sometimes).! Higher-order approximation - works fine for large interparticle separations (Rotne-Prager tensor)! Evaluation of full hydrodynamic interactions - possible, but very expensive numerically (e.g. HYDROMULTIPOLE code)! For simple situations, analytical or approximate solutions available. leakage toroidal circulation in cloud rest frame (cf. liquid drop) V n

27 Particles of complex shape in flow

28 Particles of complex shape in flow Translational and rotational degrees of freedom P P pply force and/or torque and find velocities (mobility problem) Apply velocity and/or ang. velocity and find force and torque (friction problem)

29 Particles of complex shape in flow - mobility problem Generalized mobility matrix P " " Translations and rotations are coupled! Application of force may produce rotation, etc." The form of μ depends on the choice of coordinate system - be careful! Shape anisotropy = mobility anisotropy

model a H < a t 0 6 0aH ζ = πη microgel Porous hard-sphere" model λ slip

30 Different particle types and boundary conditions Solid particle Stick boundary conditions" on the surface core -shell a H a Spherical annulus" model a H < a t 0 6 0aH ζ = πη microgel Porous hard-sphere" model λ slip length Unifying description by effective hydrodynamic radius - Cichocki et al. (2014)

31 Part II. Walls, Walls Everywhere! Gaurav Goel et al J. Stat. Mech. (2009) 6 mm Adamczyk et al J. Colloid Interface Sci. Hulme et al, Lab Chip (2008) Cichocki & Jones, Physica A (1998)

32 The wall drag effect The 2003 Ig Nobel Prize

33 Wall hindrance effect Wall = additional boundary = more viscous dissipation = hindrance of the flow = drag force Measurement idea:" Apply force - measure velocity, e.g. U " Measure diffusion coefficients and use the fluctuation-dissipation theorem measure D:! LS, optical methods Smoluchowski Einstein

34 Point particle close to a wall Solid hard wall Free boundary (air-water, fluid-fluid) F F A B

35 Detour - method of images Electrostatics - a short reminder F F Method of images: Introducing appropriate image charge(s) Force and field hard to calculate to satisfy the boundary condition on the wall, Unknown charge density we find the force and the field easily. " The same works for Stokes flows (but is a bit more complex). F

36 Point particle (stokeslet) near a free surface (I) (a) Stokeslet (b) Stokeslet Free surface Free surface Image stokeslet Image stokeslet Boundary condition: fluid velocity at the surface parallel to it For a free surface, the image is simply another stokeslet.

37 Point particle (stokeslet) near a free surface (II) Original stokeslet field Image stokeslet field Total flow field M. Ekiel-Jeżewska, R. Boniecki, ML

38 Point particle (stokeslet) near a hard wall (I) (a) Stokeslet (F) (b) Stokeslet (F) Hard wall Hard wall mage system: Stokeslet (-F) + + Stokesdoublet (2dF) Sourcedoublet (-2d 2 F) Image system: Stokeslet (-F) + + Stokesdoublet (-2dF) Sourcedoublet (2d 2 F) Boundary condition: fluid velocity at the surface vanishes (no-slip) For a hard wall, the image system contains higher order terms as well (dipolar, quadrupolar).

39 Point particle (stokeslet) near a hard wall (II) M. Ekiel-Jeżewska, M. R. Ekiel-Jeżewska, Boniecki, ML R.

40 Point particle (stokeslet) near a hard wall (III) Final results: Force parallel to the wall Force perpendicular to the wall M. Ekiel-Jeżewska, R. Boniecki, ML

41 Diffusion matrix of a spherical particle close to a wall Explicitly: Similar for rr t tt t t Due to the presence of the wall, diffusion becomes:! Anisotropic (parallel & perpendicular)! Position-dependent (distance to the wall matters)

42 Spherical particle close to a wall A problem over a century old - first works: Lorentz, Faxen U Lorentz (1907): valid for large distances, say, z/a > 4 Faxen (1923): Exact solution - Cox, Brenner, O Neill, Dean (1960s) - Brenner s formulas Interesting, but of limited practical value

43 Spherical particle close to a wall Theoretical prediction: H. Brenner Chem. Eng. Sci A.J. Goldman et al Chem. Eng. Sci " Experimental verification: Holmqvist, Dhont, Lang JCP 126, (2007). Perkins and Jones Physica A 189, 447 (1992). Cichocki and Jones Physica A 258, 273 (1998).

fixed time intervals Δt 6 mm Large particle, high viscosity low

44 How can this be measured? Large particles - microscopy & direct imaging" Idea: take photos at fixed time intervals Δt 6 mm Large particle, high viscosity low Reynolds number Changes in settling velocity = changes in friction coefficient Adamczyk et al J. Colloid Interface Sci. 1983

s lecture" Lab tour by Yi " " " Lan, Ostrowsky, Sornette,

45 How can this be measured? " Colloids - light scattering, e.g. Evanescent Wave Dynamic Light Scattering (EWDLS)" Peter s lecture" Lab tour by Yi " " " Lan, Ostrowsky, Sornette, PRL (1986)," Holmqvist, Dhont, Lang, PRE (2006,), JCP (2007),

46 What lies ahead? Near-wall dynamics of colloids is very rich in terms of phenomena to be understood." " This was just a very simple example (almost a spherical cow in vacuum)." " just the tip of the iceberg." Examples: Concentrated systems Charged systems " Non-spherical particles

47 This is not the whole story More in Berlin :-) Thank you for your attention!

48 Questions? Literature? Contact me at: Maciek Lisicki!

Colloidal Hydrodynamics and Interfacial Effects

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