hydrodynamic interactions and wall drag effect in colloidal

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

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.

Part I. Crash Course of Suspensions Hydrodynamics

Emulsions, cosmetics, healthcare products, food, gels

Macromolecules, "micelles, lipid bilayers, proteins, engineered nanoparticles

Active (Janus) particles, "motile microorganisms,"bacteria

Colloidal soft matter

Colloidal sizes

granularmedia

atoms

1 nm 10 nm 100 nm 1 µm radius

bacteria, protozoa

Colloidal dispersions (including proteins & viruses)

molecules

non - Brownian

(0.1 – 800 µm)

human cell: ∼10 µm

(L > 5 µm)Brownian motion is important on these length scales!"

"

Mesoscopic description (N. van Kampen)

Macro

Meso

Micro

Thermodynamics

Positions and momenta of particles etc.

Brownian Motion Einstein’s theory of fluctuations

Separation of time scales!

molecular view

molτ 20D a / Dτ =

1310− 1010− 910− 610− 310−

[ ]sect

- momentum relax. resolved - unsteady solvent flow

quasi - inertia free particles and fluid motion " Stokes flow

diffusional relaxation time

Overview of time scales (particles with a = 100 nm in water)

B vortτ ≈ τsoundτ

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.

Microworld hydrodynamics

Volvox bacteria stirring the fluid (R. Goldstein, Cambridge)

Aim: description of the fluid flow field

Conservation/balance laws:

Continuity equation - incompressible flow

Momentum balance equation (Navier - Stokes)

+

Stokes Hydrodynamics

Reynolds number Characteristic length & flow velocity

G.G. Stokes

Stokes equations

Microworld: small U and l or: large viscosity

No inertia, no turbulence!

O. Reynolds

Low Reynolds Number Flows - the movie

G I Taylor, Cambridge U.""National Committee for Fluid Mechanics Films (1960s)"

http://web.mit.edu/hml/ncfmf.htmlAlso on YouTube:https://www.youtube.com/watch?v=51-6QCJTAjU

All movies were taken from:

1.Linearity - superposition principle!2.Instantaneity (stationarity) on relevant time scales!3.Kinematic reversibility

Properties of Stokes equations

World of Aristotle: ‘no force, no velocity’

Stokes Equations. Linearity: Two rods

Example: two sedimenting rods

Observation:

Stokes Equations. Linearity: Tilted rod

Settling angle ≠ Tilt angle "Superposition of motion in two perpendicular directions

How does it move?

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

Stokes Equations. Kinematic reversibility.

Reversal of forces reverses the trajectories!

Stokes Equations. Kinematic reversibility.

Reversibility + symmetry = useful tools

(a) (b) (c) (d)

The particle sediments near a wall at a constant distance due to symmetry!

Sedimentation of a sphere - Stokes law

G.G. Stokes

∞ =u 0 ∞ = −u V

A single sphere in an unbounded fluid settling under influence of a force (gravity)

Flow disturbance caused by the sphere

Long-ranged 1/r

Two particles - Hydrodynamic interactions

Single-particle term Contribution to velocity of particle 2 from the force acting on particle 1

Higher-order terms

Two particles - Hydrodynamic interactions

Single-particle term Contribution to velocity of particle 2 from the force acting on particle 1

Higher-order terms

C. W. Oseen

Oseen tensor T

Linear relation between forces and velocities;!dependence on all other particles

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!

Higher-order terms: Multiple reflections

Linear relation between forces and velocities;!dependence on all other particles

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

HI’s are complicated, yet crucial in dynamics of colloids

3V

1V

t2F

V2

Velocity ForceMobility matrix

Mobility matrix - Hydrodynamic interactions

25

(-1.1 ,0 ,1.16)

Three sedimenting non-Brownian Spheres: asymmetric starting configuration

gravity

Courtesy: M. Ekiel-Jezewska & E. Wajnryb, Phys. Rev. E 83, 067301 (2011)

• Sensitive dependence on initial configuration for N > 2 → chaotic trajectories

(-1.1, 0, 1.20)

x/σ

z/σ

x/σ

( )t t ed X X Fdt

=µ ⋅

non - linear in X

Complex system? Simplify!

Making a complex problem simple

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.

Vn

toroidal circulation in cloud rest frame (cf. liquid drop)

leakage

Particles of complex shape in flow

Particles of complex shape in flow

Translational and rotational degrees of freedom

P

Apply force and/or torque and find velocities

(mobility problem)

P

Apply velocity and/or ang. velocity and find force and torque

(friction problem)

Particles of complex shape in flow - mobility problem

P

Generalized mobility matrix

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

Different particle types and boundary conditions

aH

a

Spherical annulus"model

Porous hard-sphere"model

Ha a<

core -shell

microgel

t0 0 H6 aζ = πη

λ slip length

Solid particleStick boundary conditions"on the surface

Unifying description by effective hydrodynamic radius - Cichocki et al. (2014)

Part II. Walls, Walls Everywhere!

Gaurav Goel et al J. Stat. Mech. (2009)

Adamczyk et al J. Colloid Interface Sci.

6 mm

Cichocki & Jones, Physica A (1998)Hulme et al,Lab Chip (2008)

The wall drag effect

The 2003 Ig Nobel Prize

Wall hindrance effect

Wall = additional boundary = more viscous dissipation = hindrance of the flow = drag force

Measurement idea:"Apply force - measure velocity, e.g."

U

Smoluchowski — Einstein

measure D:!LS, optical methods

Measure diffusion coefficients and use the fluctuation-dissipation theorem

Point particle close to a wall

F

Solid hard wall

F

Free boundary (air-water, fluid-fluid)

A B

Detour - method of images

Electrostatics - a short reminder

F

⇔F

F

Force and field hard to calculateUnknown charge density

Method of images: Introducing appropriate image charge(s)

to satisfy the boundary condition on the wall, we find the force and the field easily.

"The same works for Stokes flows

(but is a bit more complex).

Point particle (stokeslet) near a free surface (I)

Free surface

Stokeslet

Image stokeslet

(b)

Free surface

Stokeslet

Image stokeslet

(a)

For a free surface, the image is simply another stokeslet.

Boundary condition: fluid velocity at the surface parallel to it

Point particle (stokeslet) near a free surface (II)

M. Ekiel-Jeżewska, R. Boniecki, ML

Original stokeslet field Image stokeslet field Total flow field

Point particle (stokeslet) near a hard wall (I)

+ +

Stokeslet (F)

Stokeslet (-F)Image system:

Stokes-doublet (-2dF)

Source-doublet (2d2F)

+ +

Stokeslet (F)

Stokeslet (-F)Image system:

Stokes-doublet (2dF)

Source-doublet (-2d2F)

Hard wall Hard wall

(a) (b)

For a hard wall, the image system contains higher order terms as well (dipolar, quadrupolar).

Boundary condition: fluid velocity at the surface vanishes (no-slip)

Point particle (stokeslet) near a hard wall (II)

M. Ekiel-Jeżewska, R. Boniecki, MLM. Ekiel-Jeżewska, R. Boniecki, ML

Point particle (stokeslet) near a hard wall (III)

M. Ekiel-Jeżewska, R. Boniecki, ML

Final results:

Force perpendicular to the wall

Force parallel to the wall

Diffusion matrix of a spherical particle close to a wall

tt

tttExplicitly:

Similar for rr

Due to the presence of the wall, diffusion becomes:!Anisotropic (parallel & perpendicular)!Position-dependent (distance to the wall matters)

Spherical particle close to a wall

A problem over a century old - first works: Lorentz, Faxen

U

Lorentz (1907):

Faxen (1923):

valid for large distances, say, z/a > 4

Exact solution - Cox, Brenner, O’Neill, Dean (1960s) - Brenner’s formulas

Interesting, but of limited practical value

Perkins and Jones Physica A 189, 447 (1992). Cichocki and Jones Physica A 258, 273 (1998).

Theoretical prediction: H. Brenner Chem. Eng. Sci 1961. A.J. Goldman et al Chem. Eng. Sci 1967. "

Experimental verification: Holmqvist, Dhont, Lang JCP 126, 044707 (2007).

Spherical particle close to a wall

How can this be measured?

Large particles - microscopy & direct imaging"

6 mm

Idea: take photos at fixed time intervals 𝚫t

Adamczyk et al J. Colloid Interface Sci. 1983

Large particle, high viscosity low Reynolds number

Changes in settling velocity = changes in friction coefficient

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),

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."

"

What lies ahead?

Concentrated systems Charged systems Non-spherical particles

Examples:

Thank you for your attention!

This is not the whole story… More in Berlin :-)

Questions? Literature? Contact me at:

Maciek Lisicki!mklis@fuw.edu.pl!

www.fuw.edu.pl/~mklis