hydrodynamic interactions and wall drag effect in colloidal
TRANSCRIPT
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 :-)
