Physics | Hydrodynamics » Gerhard Nägele - Colloidal Hydrodynamics

Colloidal Hydrodynamics Gerhard Nägele Institute of Complex Systems, Research Centre Jülich, Germany Enrico Fermi Summer School „Physics of Complex Systems“, Varenna, Italy, July 4 – 9, 2012 1 Literature 1. E. Guazelli and JF Morris, A Physical Introduction to Suspension Dynamics, Cambridge University Press (2012) 2. W.B Russel, DA Saville, and WR Schowalter, Colloidal Dispersions, Cambrigde University Press (1989) 3. J.KG Dhont, An Introduction to Dynamics of Colloids, Elsevier, Amsterdam (1996) 4. G. Nägele, The Physics of Colloid Soft Matter: Lecture Notes 14, Polish Academy of Sciences Publishing, Warsaw (2004) 5. G. Nägele, On The Dynamics and Structure of Charge-Stabilized Colloidal Suspensions, Physics Reports 272, pp. 215-372 (1996) 6. E. Guyon, JP Hulin, L Petit, and CD Mitecu, Physical Hydrodynamics, Oxford University Press (2001) 7. E.J Hinch, Hydrodynamics at Low Reynolds Number: A Brief and Elementary Introduction & Sedimentation of Small

Particles, in Disorder and Mixing, NATO ASI Series E, Vol. 151, pp. 43 – 55 & 153 – 161, Editors: E Guyon et al, Dordrecht, Kluver (1988) 8. E.M Purcell, Life at Low Reynolds Number, American Journal of Physics, Vol 45, pp 3 – 11 (1977) 9. E. Lauga und R Powers, The Hydrodynamics of Swimming Microorganisms, Rep Prog Phys, Vol. 72, 096601 (2009) 10. J. Mewis and NJ Wagner, Colloidal Suspension Rheology, Cambrigde University Press (2012) 11. G. Nägele, Colloidal Hydrodynamics, Manuscript in preparation (2012) 2 Content 1. Motivation - Low-Reynold‘s number flow - Hydrodynamic interaction (HI) - Inertia – free particle dynamics 2. Examples of hydrodynamic effects - Individual particle level - Macroscopic transport properties 3. Low Reynolds number flow 6. Smoluchowski & Stokes-Liouville dynamics - Many-particle diffusion equation - Dynamic simulations 7. Collective diffusion and sedimentation - Hydrodynamic function - Sedimentation - Intrinsic convection

8. Suspension viscosity - Colloidal time scales - General properties - Stokes equation - Examples - Point force solution - Shear thinning and thickening - Boundary layer method - Faxén laws for spheres 4. Active Microswimmers - Kinematic reversibility - Scallop theorem - Artificial swimmers 5. Many - spheres HI - General properties - Mobility matrices 9. Dynamics of permeable particles - Brinkman fluid model - Applications 10. Appendices - Derivation of point force solution - Proof of mean-value theorem - Symmetry of mobility and friction matrices - Multipole method by Chichocki and coworkers 1. Motivation - Low Reynolds number flow - Hydrodynamic Interaction (HI) - Inertia – free particle dynamics 4 1.1 Low - Reynolds number flow • Inertial forces tiny as compared to friction forces: Reynolds - # << 1 • Laminar fluid flow (kinematic reversibility): linear force – velocity relations • Quasi-inertia-free motion of colloids / microorganisms and

solvent (Stokesian dyn.) Re  inertial forces   f aVp viscous forces Vp wale: 0 108 human swimmer: 10000 in water: gold fish: 100 colloid particle: 0.0001 bacteria /cells: 0.00001 Re << 1: colloids / microbes in water or macroscopic bodies in glycerine 5 granular media Colloidal dispersions (including proteins & viruses) molecules bacteria, protozoa (0.1 – 800 m) atoms human cell: 10 m 1 nm 10 nm 100 nm 1 m radius non - Brownian (L > 5 m)  Colloids, proteins and most bacteria share common inertia - free hydrodynamics 6 Single sphere translating in quiescent fluid • Sphere with stick (no-slip) BC ín quiescent infinite fluid V   0 Fe  0t  translational mobility 1 V 60a Fe   F H (6  4 : perfect slip) D0  k BT  0t r u (r )  1 p(r )  1 u  dr  0  dx dy dz   ux uy uz r long-range decay r2 stream lines = path lines when stationary flow 7

 Sphere in quiescent, infinite fluid, stick BC  Sphere stationary (rest frame) u   V u  0 u(arˆ )  nˆ  0 u(arˆ )  tˆ  0 8  Flowlines inside and outside a stationary spherical droplet in a non-quiescent viscous fluid Fore – aft symmetry u  V ui (arˆ )  nˆ  ua (arˆ )  nˆ  0 tˆ  σ i (arˆ )  nˆ  tˆ  σ a (arˆ )  nˆ ui (arˆ )  tˆ  ua (arˆ )  tˆ Air bubble in fluid: zero tangential stress 9 1.2 Hydrodynamic Interaction  Vn in general non - parallel to Fn but linear relation Vn  Instantaneous restructuring of flow profile around moving particles  Long – range interaction non – pairwise additive 10 • N freely rotating spheres in quiescent fluid: generalized Stokes law 1 μi j (X)  F tt j V   t t ( X)  Fe mobility problem F  (X )  V friction problem  μit tj  X   μiRP j R i  R j  μi j  X 

far-field HI:  O(r 1 ) F2t R1 diffusivity matrix 4 near-field HI:  O(r ) • HI acts quasi – instantaneously on colloidal time scales • near - field part non pairwise additive V V1 Dit jt  k BTμit jt (X)  t t (X)  1  V3 translational 33 mobility tensors e j 2 Vi  N X  R1,., R N  V   V1,., VN  T tt tt  μ11  μ1N       tt     tt tt  μ N1  μ NN   3N x 3N matrix symmetric & pos. definite • Moblity tensors required as input in calculations of colloidal transport properties • Flow pattern u(r), p(r) itself not needed DS  k BT tt Tr μ 11 3 N tt Vsed   μ 1p (X) p 1 - self-diffusion coefficient  Fe - mean sedimentation velocity (renormalization required) tt μ 11  O(r 4 ) tt μ 12  O(r 1 ) •  average over homogeneous and isotropic particle ensemble 12 Inertia – free particle

dynamics • quasi - inertia free motion on coarse-grained colloidal time- and length scales M  108 0 c e s t   B  x  l B  D0  B  104  “stopping distance” Rhodospirillum bacteria (length   5 m) 13 Inertia – free particle dynamics Generalized Smoluchowski Eq. Overdamped Langevin-Eq.  (t)  0  F  F  K MV i I i H i R i For Brownian particles: Random force Dynamic simulations N  P(X, t)   i   v i (X, t) P(X, t)   0 t i 1 PDF coarse-grained velocity from force balance Theoretical calculations • pure configuration - space description for t  B 14 • Moblity tensors are input in N – particle diffusion equation (Smoluchowski) N  P(X, t)  k BT  i  μittj ( X)   j   FjI   Fje  P(X, t)   t i, j  1 pdf Irreversibility (Brownian motion  T) • Positive definiteness of mobility matrix implies for zero

external forces & flow P(X, t  )  Peq (X)  exp   VN (X) FiI  i VN (X) • Stokes – Liouville equation for driven system of non – Brownian particles (T – indep.): N  P(X, t)    i  μittj ( X)   FjI  Fje  P(X, t )   t i, j  1 P(X, t  )  Pstat (X) ???  non-linear coupling: HI – mictostructure  T - limit existent ? G. Nägele, Phys Rep 272 (1996); J Dhont, An Introduction to Dynamics of Colloids, Elsevier (1996) 15 2. Examples of hydrodynamic effects - Individual particle level - Macroscopic transport properties 16 2.1 Examples: Individual particle level 17 Sedimentation of two non - Brownian Spheres (Re << 1) V0  0 Fe 0t  1 gravity 60a V0 (buoyancycorrected) Fe 18 The sedimentation race: bet which pair settles fastest, and how ? r gravity constant sep. vector r pair 0 Vsed  Vsed  0t Fe 20 Sedimentation of a non

- Brownian rod (Re << 1) V  μF e Fe  ζ  V F||e ê given Fe 1     2 Fe Fe (transl. mobilities for L >> D) ˆ ˆ    1  ee ˆˆ μ   ee V   Fe     F e   eˆ eˆ   eˆ  eˆ  Fe    2    1 /  Experiment: needle in syrup no rotation Flagella bundle: HI synchronized 21 Apparent like - charge attraction of particles near boundary Squires & Brenner, PRL (2000) charged glass wall or liquid - air interface Fext  - Fext ? • Attractive wall: apparent repulsion 22 Sedimentation of 3 non-Brownian spheres: symmetric & planar start configuration  3 tt  R i (t)    μik  X(t)    Fe dt  k 1  d Fe • End of world: point particles collapse on single point after finite time, acquire infinite velocity. • Spheres: equilibrium configuration reached after infinite time, with pair

of two upper spheres following lower one. Courtesy: M. Ekiel-Jezewska & E Wajnryb, Arch Mech 58, 489 (2006) 23 Five non - Brownian spheres in equidistant start configuration - pair distances are varying - dimer formation: „kissing“ - sensitive dependence on initial conditions - end - lagging of polymers Simulation by: G. Kneller, Centre de Biophysique Moleculaire, Orleans 24 Three sedimenting non - Brownian Spheres: non - symmetric start configuration (-1.1 ,0 ,116) (-1.1, 0, 120) d X  t t  X   Fe dt z/ non - linear in X μit tj  1 0 | R i - R j | gravity x/ x/ • Sensitive dependence on initial configuration for N > 2  chaotic trajectories Courtesy: M. Ekiel-Jezewska & E Wajnryb, Phys Rev E 83, 067301 (2011) 25 Hydrodynamic „diffusion“ (mixing) due to many - body HI • Sedimentation of index-matched non-Brownian glass beads   0.30 gravity - driven, non-equilibrium system E. Guazzelli &

J Hinch meandering trajectories & strongly fluctuating velocities Annu. Rev Fluid Mech 43 (2011) particle migration across streamlines (mixing) existence & form of final stationary particle distribution Pstat(X) still under debate slowly sedimenting colloidal particles  Brownian trajectories & Pstat(X) existent 26 Sedimentation: spherical cloud of non - Brownian particles (radius R) leakage toroidal circulation in cloud rest frame (cf. liquid drop) 0 Vsed  0t Fe cloud 0 Vsed  Vsed  N 1 e F 5  0 R M.J Ekiel-Jezewska, Phys Fluids 18 (2006), B. Metzger et al, J Fluid Mech 580, 238 (2007)  Cloud sediments faster than single bead  Instability for large N and large settling time (chaotic fluctuations due to many-bead HI)  Point - particle simulation (N = 3000)  Glass spheres (a ≈ 70 m) (in silicon oil) taken from: B. Metzger, M Nicolas and E Guazzelli, J Fluid Mech 580, 238 (2007) 28  Evolution: spherical

cloud torus breakup in two clouds t *  t / sed R t*  0 t *  700 t *  400 t *  800 taken from: E. Guazzelli and JF Morris, A Physical Introduction to Suspension Dynamics, Cambridge Univ Press (2012) 29 30 taken from: B. Metzger, M Nicolas and E Guazzelli, J Fluid Mech 580, 238 (2007) 2.2 Macroscopic transport properties 31 Self - diffusion of colloids (Brownian particles) R (t  0) direct interactions only Single - sphere diffusion W(t)  R (t)  R (0)  / 6  D0 t 2 D0  W(t) D0 t k BT 60a t time 32 D0 t W(t) DS t t time k T tt DS  B Tr μ 11 3 HI HI & DI eq  Self – diffusion in general slowed down by hydrodynamic forces 33 Macroscopic shear viscosity of colloidal dispersions (Brownian) V y h v x u(y)   y xˆ rotational viscosimeter Fx Area du x η dy shear stress viscosity (zero frequency)   du x dy  V h shear rate (rate of strain) 34 Non –

Newtonian behavior of suspension viscosity η shear-Peclet # shearthinning shearthickening (Hydro-Cluster) a 2 / D0   D   a 3 Pe  shear 1 /  shear rate (Peclet - #) 1 10 Zero shear 1000 Shear - thinning Shear – thickening (lubrication) Wagner & Brady, Physics Today, October 2009 35 Colloidal shear-thickening: Protection against thrustings and fire arms Shear - thickening dispersion Kevlar (synthetic armor fibre) 36 “Ice spike test” with and without colloid impregnation without with Silica in polyethylene glycol (   0.52 ) Courtesy: N.J Wagner, Univ of Delaware, Newark 37 Lubrication (stick BC) • Dominating pressure effect in thin fluid layer between near-contact smooth surfaces Fe h Fe squeezing motion h  4 0t   Fe dt a shearing motion Vrel  dh h(t)  4 (0 /a) Fe t exponentially slow approach until roughness matters h(0)e h   r  2a   0.05  a

 Finite contact time due to vdW attraction: u(r )  - AH /(r - 2a)  Surface roughness matters at very small distances h squeezing motion No-slip: shearing motion a Fe  ln   Vrel h a Fe  Vrel h 1/2 Drop: a F    h Vrel slip past each other Bubble: a Fe  ln   Vrel h slip past each other e Hydrodynamic and direct interaction radii • Influence of stabilizing layers and porosity effects can suppress lubrication aH core -shell a Spherical annulus model aH  a 0t  60a H Porous hard-sphere model microgel  slip length  Non - zero relative squeezing mobility at contact 40 n̂ uslip   tˆ  u  nˆ   u uslip t̂ slip length  wall rest frame  Slip length : distance inside surface at which fluid velocity extrapolates to zero  Hydrophilic surfaces Electric double layer hydration shell (ion, protein) 41 • Shear thickening

ceases with increasing brush thickness a aH increasing little HI only Wagner & Brady, Physics Today, October 2009 Bergenholtz, Brady & Vivic, J. Fluid Mech 456 (2002) 42 3. Low-Reynolds number flow - Colloidal time scales - Stokes equation - Point force solution - Boundary layer method - Faxén laws for spheres Gerhard Nägele Enrico Fermi Summer School „Physics of Complex Systems“, Varenna, Italy, July 4 – 9, 2012 1 3.1 Colloidal time scales   u(r, t)  f   u(r, t)  u(r, t)    p(r, t)  02 u(r, t)  f e (r, t)  t    u(r, t)  0 Re  f inertial force viscous force t  sound  a / csound  1010 sec  f Vp 2 / a 0 Vp / a 2  f a Vp 0 t   vort f Vp / t 0 VP / a 2  p(r )  02 u(r )  f e (r )  0 volumetric force density on fluid by external fields (particle) Reynolds number 1  u(r, t)   p(r, t) 

02 u(r, t)  f e (r, t) t a 2f   109 sec 0 Navier - Stokes Eq. incompressible flow still includes vorticity diffusion  vort 1 a Stokes equation Inertia - free force balance 2 Overview: time scales (particles with a = 100 nm in water) 1013 1010 109 sound  mol 103  B   vort t sec  D  a 2 / D0 diffusional relaxation time molecular view Quasi - inertia free particles and fluid motion - momentum relax. resolved - unsteady solvent flow momentum relax. time B  106 2    p   vort  0 9  f  M - Stokes equation flow - Many - particles Smoluchowski eq. Stokes # St  B ext 2    p  Re 9  f  Advection time ext   Colloids and microswimmer: Re <<1 and St << 1  Dry powder granular dynamics: St >>1 (large & heavy particles in a gas) a Vp 3 Shear-Peclet # Pe  D shear 

  D   a 3 5 ms for a = 0.1 m   D  5 sec for a = 1.0 m  2 min for a = 5.0 m   Pe << 1: Brownian motion is dominating  Pe >> 1: Flow advection dominates y x 4 3.2 Stokes equation • Linear Stokes equation BVP for N rigid particles in infinite and unbounded fluid (no ext. forces) p(r )  02 u(r )  0   u(r )  0 zero total force fluid incompressibility u(r )  Vi  Ωi   r  R i  Ωi (r  Vfluid ) Vi for r on particle surface Si (stick inner BC) Ri u(r )  0 , | r |   u (r )  u  (r ) , | r |   p(r )  const , | r |   p(r )  p (r ) , | r |  outer BC for quiescent fluid ambient flow due to sources „at infinity“ Helmholtz (1868) : • Unique solution u(r) for given BC‘s on inner and outer fluid boundaries (see Appendix) • Of all u(r) with div u(r) = 0, Stokes flow has minimal dissipation 5 Important ambient

flow: simple linear shear flow Rheology u  (r )   y xˆ  ω  r  e  : r 1 1 ω (r )    u  (r )    zˆ 2 2 1 1 1 T e  (r )   u   (r )   u   (r )   Tr  u  (r )    xˆ yˆ  yˆ xˆ   3 2 2 y compression x (Simple shear flow) = dilation + (Pure rotation) (Extensional flow) Stress tensor of incompressible fluid 0   p(r )  02 u(r )    σ(r ) T σ(r )   p(r ) 1  0 u   u     fluid stress tensor nˆ (r ) σ(r, t)  nˆ (r )  (r )   p(r )   0   u (r )   u  (r )  FH   S dS σ(r;X)  nˆ (r )  Fe    fluid force / area on sphere surface element dS at r exerted by surrounding fluid layer V Fe  F H single sphere force balance 7 Mobility and friction matrices 

Hydrodynamic force and torque on surface of particle i Si FiH   dS σ(r;X)  nˆ (r )  Fie Si TiH   n̂ Ri dS  r  R i   σ(r;X)  nˆ (r )  Tie S*i Si 8 Point - force solution (Oseen) - Find solution of Stokes eq. for a point force F acting on quiescent & unbound fluid at r‘ : p(r )  02 u(r )   f (r )   u (r )  0 f (r )  Fe (r  r ) - Solution for outer BC u(r  ) = 0 and p(r  ) = 0 (Appendix) : p(r )  Q0 (r  r )  F u(r )  T0 (r  r )  F   u(r )  0    T0 (r )  0 Q 0 (r )  1 4 r 2 rˆ 1 T0 (r )  1  rˆ rˆ  80 r volumetric force density on fluid Oseen tensor x  x   ( T0 ) (r )     2  80 r  r  1 including r = 0 u(r )   d r T0 (r  r )  f (r ) for localized force density acting on fluid 49 First application: friction coefficients

of thin rod of length L and thickness d 0 L  2n  1 (n  1) d  Approximate by sum of Stokeslets: V  u( 0 )  1  bead n d e d F   T0 (i eˆ d)  Fe L L i  n i 0 F    eˆ eˆ    1  eˆ eˆ    V e Fe   2  0 L ln (L / d)    2   End caps correction: ln (L / d)  ln(L / d)  0.12 • HI of segments lowers friction and renders it anisotropic (convection - along effect) • Friction ratio 1:2 is due to Oseen approximation: 1  ee ˆ ˆ  1  ee ˆ ˆ   2 ee ˆ ˆ  Pˆ   2Pˆ 9 3.4 Boundary layer method r u(r )   d r T0 (r  r )  f (r ) zero ambient flow e Ωp  Rigid particle p of arbitrary shape with stick (no-slip) BC: u D (r )  u (r )  u  ( r )  (s) dS T ( r  r )  f (r ) 0  r Vp Sp Rp Sp disturbance flow f (s) (r )   σ(r )  n(r )  Insertion of no - slip BC single - layer

„potential“ Surface traction on fluid at surface point r‘ two-dimensional integral equation for traction: Vp  Ω p   r  R p   u  (r )  (s) dS T ( r  r )  f (r ) 0  (r  S p ) Sp 10 Vp , Ωp , u   f (s) (r )  FpH , TpH , u  Particle with complex shape: Discretization / Triangularization N Vp  Ω p   ri  R p   u  (ri )   T0 (ri  rj )  f (s) (rj )  j1 (i 1,., N) 3N 3N inversion    Frequently only relations Vp ,  p   FpH ,  pH are required  „Rapid prototypeing“: form complex shapes (proteins) by connecting spherical beads 11 Bead modeling of complex-shaped particles B. Carrasco & J Garcia de la Torre, Biophysical J. 75 (1999) HYDROPRO packages T2 – bacteriophage model (V. Bloomfield, Biopolymers 5 (1967))  Preserve original volume  Rotne-Prager-Yamakawa HI approximation!  Intrinisic

viscosity  Mobilities & friction coefficients etc. 12 Far – distance flow field around a particle r  Expand around point inside particle: |r | |r | (s) dS T ( r )  r   T ( r )  .  f (r )   0 0  u D (r )  r Sp Sp  Split in symmetric and anti-symmetric parts: u D (r )   T0 (r )  F H  1 80 r 2 rˆ  T H  1 80 r  Freely mobile particle (force- and torque-free): F H ˆ ˆ ˆ ˆ r1  r r r S 3 :   2 H  0  T H active microswimmer  Freely mobile particle creates O(r-2) flow disturbance by its symmetric force dipole SH      f (s) (r ) r  r f (s) (r )  2 1 Tr r f (s) (r )  dS   2 3 S 1 p - rigid - no - slip Example: symmetric force dipole in y - direction (pusher: p > 0) d d u(r )   T0 (r  yˆ )  T0 (r  yˆ )   Fe yˆ 2 2   1 S H  p  yˆ yˆ  1  3   p  Fe d dipole

moment  Far – field flow : u (r )  cos   yˆ  rˆ p 3cos2   1 rˆ 2  80 r   Swimmer describeable as static force dipole for distances >> d , and when time – averaged over strokes (non-reciprocal cycle, friction-asymmetric) 14  Pusher: p > 0  Puller: p < 0 Production stroke  Algae Chlamydomonas,  E. Coli, salmonella, sperm,  Propelling part on head side  Propelling part at rear  Tend to repel each other („asocial“).  Tend to attract each other. 15 Faxén laws for spheres Vp  Ω p   r  R p   u  (r )  (s) dS T ( r  r )  f (r ) 0  (r  S p ) Sp  p (r )  0 u (r )  0  p  ( r )  0  u  (r ) SP  1   u  (r )  0   u  (r )  0 (bi - harmonic  mean - value property: Appendix)  dS u (r )  u (R p )  4 a 2 S (homog. Stokes eq) 2  p  Integrate over

Sp w/r to r, use mean-value theorem and 1  dS T0 (r  r )  4 a S i 2 1 1 60 a  a 2 u  ( R p ) 6 Sp Rp r r | r  R p |  a 16  Translational Faxén law for single sphere in ambient flow FH p     a2 2   60a  Vp   1    u  (r  R p )  6     Rotational Faxén law: - translational Faxén law - Stokes friction law when u∞ = 0 extra non - linear flow contribution 1  3 TH Ω u R 8 a ( )        p 0  p p  2   FH  0  TH :  a2 2  Vi   1    u ( R i ) 6   1 2 Ωi    u  ( R i )  Freely mobile particle advects with (surface-averaged) ambient flow at its center  No cross – streamline migration for Re 0  Tubular pinch or Segré-Silberberg effect in pipe flow for Re > 0 t=0 after long time Lift force drives particles towards ring at r / R ≈ 0.6 (inertia effect) F.

Feuillebois, Perturbation problems at low Reynolds numbers, Institute of Fundamental Technological Research Lectures, Warsaw (2004)  Shear-induced migration from high-shear to low-shear region (pipe center) for non-Brownian spheres even at Re 0, provided: - high concentration (many-particle HI effect) - sufficiently strong shear 18 4. Active Microswimmers - Kinematic reversibility - Scallop theorem - Artificial swimmers 19 4.1 Kinematic reversibility  p(r )  02 u(r )  find (r )  f e (r )  0 u(r ) S  V  Ω   r  R p  Ω V S   u (r )  0 u(r   )  u  (r ) ambient given flow (flow w/o particles) find  find ( V, Ω,  u ) linear relation S induced force density on fluid localized on S (rigid particle) • Since linear boundary value problem (BVP): V, Ω, u , f e   V, Ω, u , f e  u, p, find   u,  p, find   Motion reversal of

boundaries, external force density and ambient flow reverses sign of flow pattern only, not its shape. 20 Application: motion in highly symmetric systems u , V, Ω u ,  V, Ω V  0 • Lift forces arise when non- zero inertial contributions: du/dt 0 • V  0 also for flexible particles (polymers, drops) 21 -g Ω0 g • Three symmetry planes: symmetric rod sediments sidewise w/o rotation 22 Ω0 V Ω     μt t μt r    rt r r μ μ     Fe  M*g     Te  0    geometry - dep. BC - dep. g Ω0 • Boomerang-shaped body rotates while sedimenting until symmetric orientation reached • No rotation, when line of gravity through c.om intersects hydrodynamic center of friction 23 Highly viscous fluid Low - viscosity fluid - Laminar flow at Re << 1: kinematic reversibility and reciprocal history - Rotation speed

irrelevant - Irreversible motion of dye across circular stream lines for Re > 1 - Diffusion causes cross-streamlines particle motion G.I Taylor, Cambridge 24 ink t1 t2 t3 • Particle motion across circular stream lines induced by: - Brownian motion or external noise source - Inertia effects (Re  1) - Many - particle HI in sufficiently dense systems (chaotic hydrodynamic motion) - Reversibility - breaking direct particle interactions such as surface roughness D. Pine et al, Nature 438 (2005) J. Gollub and D Pine, Physics Today, August 2006 25 Ideal spheres, HI only spheres with surface roughness (lateral displacement after „collision“) 26 4.2 Scallop theorem • Internal forces and energy sources only: F H  0   Fe T H  0   Te - swimmers act as force dipoles in far – field flow „time“ • Purcell‘s scallop theorem: For net displacement after one shape cycle: - non - reciprocal sequence of body deformations: - at least 2 -

parametric deformations (two hinges) - skew – symmetric motion for example min max E.M Purcell, Life at low Reynold‘s number, Am. J Phys 45, 3 (1977) 27 G.I Taylor, Cambridge  Non-reciprocal periodic motion is required: - helical flagellum motion - two degrees of freedom for motion Re << 1: microrganism in water or macroscopic swimmer in glycerin 28 4.3 Artificial swimmers lL Purcell swimmer 3 - spheres swimmer artificial amoeba lR lL 4 1 3 2 VL  VR  1  Fint 0 1  pair (l R ) 0   pair H. Stark, Immer in Bewegung, Physik Journal 6, Nr 10 (2007) lR   Fint  5. Many- spheres HI - General properties - Mobility matrices Vi  N 1 μit jt (X)  F e j j Gerhard Nägele Enrico Fermi Summer School „Physics of Complex Systems“, Varenna, Italy, July 4 – 9, 2012 5.1 General properties  Identify ambient flow with incident flow on sphere i by N - 1 spheres (quiescent fluid)   0t

FiH  a2  N  Vi   1   i    dS T0 (r  R i )  fk(s) (r ) 6   k  i Sj  Consider dilute suspension where : Faxen law | Ri  R k |  a  fk(s) (r )  Fkh / 4  a 2  Ri  Use mean-value theorem for integral over the Sk  Rotne – Prager approximation for t - t mobilities : N   Vi    0 1 i j  (1  i j )TRP ( R i  R j ) j 1 3 TRP (r )  4 3  F Hj 1a a ˆ ˆ ˆ ˆ   1 rr   1  3rr r    2r    N   H    μiRP R  F j ij j j 1   0 for r    Rotne – Prager approximation i  μit tj (X)  0   (X)  (X)   F   V         r t rr H        (X)  (X)   T  0  tt tr H i iRP j (R i j )  0  Pros and cons: hydrodynamic drift part: from low to high mobility region Positive

definiteness of 3N x 3N matrix  t t (X) is preserved Easy to apply (theory & simulation) All flow reflections neglected Overestimates HI in general Multipole expansions including reflections / many - body HI & lubrication Appendix: Multipole method by Cichocki and coworkers 6 Hyrodynamic cluster expansion (3) μit tj (X)  0 1i j  (2) i j (X)   i j ( X )  .     2 - body HI  O(r -4 ) : i  j  -7 O(r ) : i = j 3 - body HI N   μ (X)   0  i j  ω11 ( R i p )  (1  i j ) ω12 ( R i j )   pi  ( 2) ij  Long - distance multipole expansion of 2 - body HI : pairwise - additive 3 3a  1a     O  r 7  ω12 (r )    1  r r     1  3 rr   2 r       4  r      Oseen term dipole term 

  back reflections Rotne - Prager part 4 ω11 (r )  15  a     r r  O  r 6  4  r   first self reflection  Rotne - Prager (RP) part suffices for dilute charge - stabilized dispersions ! 7 Two - spheres translational mobilities in infinite fluid • axial symmetry and isotropy ω12  ω21 ω12   F1   V1   1  ω11   0 V  F  ω 1 ω   2  21 22   2  r  R 2  R1 ω11  ω22 ˆ ˆ  yi j (r) 1  rr ˆ ˆ ωi j (r )  i j 1  x i j (r) rr • known recursion relations for (a / r) expansion & lubrication corrections F F F F V12  2  x11  x12  0 F F Vi  Vsed   y11  y12  0 F Vsed   x11  x12  0 F F V12  2  y11  y12  0 F Jeffrey & Onishi, J. Fluid Mech 139 (1984) Jones & Schmitz, Physica A 149 (1988) 8 • Drag

- along effect strongest for in - line sedimentation • Lubrication plays no role for motions considered here (Rotne – Prager o.k for r > 5a) 9 V12 V0   r /a  2  s 2 • Lubrication important for relative pair motion close to contact 10 Two – sphere „collisions“ A A B t1 B t2 A B t3 closed orbit trajectories (lubrication) y/a fore - aft symmetry frame origin x/a  Fore - aft symmetry can be broken, e.g, by hydrodynamic three – sphere collisions Ideal spheres, HI only spheres with surface roughness (lateral displacement after „collision“) 6. Smoluchowski & Stokes-Liouville dynamics - Many-particle diffusion equation - Dynamic simulations 13 6.1 Many- particle diffusion equation 14  Probability conservation of configurational pdf: Brownian force drives diffusion: FiB   k BT i ln P N  P(X, t)   i   Vi (X, t ) P(X, t)   0 t i 1  Inertia-free motion (zero total

force) for t  B 0  FiI  Fie  FiH  FiB Interaction forces: FiI  i VN (r N ) Hydrodynamic drag forces (for u∞= 0): N  Vi    μilt t (X)  FlH  FlI  Fle  FlB l 1  N – particle Smoluchowski equation N  P(X, t)  k BT  i  μittj (X)   j   FjI   Fje  P(X, t)   t i, j  1 Brownian motion  T P(X, t  )  Peq (X)  exp   VN (X) 15  6.2 Dynamic simulations  Discretized postional many – particle Langevin equation; N N R i  t 0     R i  t 0    μit tj  X 0   Fj  X 0   k BT  j  μit tj  X 0     2  d i j  X 0   n j  o      j 1 j 1 DI & external  Square - root mobility matrix d in random displacement: k BT μ Ri    central Gaussian displacement near-field HI (hydrodyn. drift part) tt  X0   d  X0 

 dT  X 0  N   μit tj  X0   Fj  X0   k BT  j  μit tj  X0    o      j 1 R i    R j    0  2 k BT μit tj  X 0    o    HI - coupling of displacements of i & j 16  Stokes – Liouville equation for non-Brownian driven spheres N  P(X, t)    i  μittj (X)   FjI  Fje  P(X, t)   t i, j  1 P(X, t  )  Pstat (X) ?  Forward integration schemes based on: N   R i  t 0     R i  t 0    μit tj  X 0   FjI (X 0 )  Fje ( X 0 )    o      j 1  Non-linear coupling: HI ↔ microstructure (PDF) 17 How to calculate correlation functions Cf g (t)  f (t) g * (0) eq      dX dX 0 P(X, t | X 0 ) Pin (X 0 )  Peq (X 0 ) f (X) g * ( X 0 ) Conditional pdf : X0 X during time t (from Smoluchowski eq.) N 

Density fluctuations: f (X)  g(X)   d r exp  iq  r     r  R l  3 l 1  Dynamic structure factor measured in dynamic light scattering ( Roberto Piazza) S(q, t)   exp iq   R l (t)  R p (0)  N l, p  1 eq 7. Collective diffusion & sedimentation - Hydrodynamic function - Sedimentation - Intrinsic Convection 19 7.1 Hydrodynamic function • Dynamic structure factor S(q,t) is measured in dynamic scattering experiment: ki S(q, t   D )  S(q) exp  q 2 D(q)t  q laser kf D(q)  D 0 H(q )  H(q) S(q) 1 N 0 short - time diffusion function photomultiplier N  qˆ  μ pt tj (X)  qˆ exp[i q  (R p, j  1 p  R j )] eq 2 / q H(q) = 1 without HI  Physical meaning: generalized sedimentation coefficient • Homogeneous system with spatially periodic force acting on each sphere:  Fj  q F e exp  - i q  R j  V(q)  V(q)

1 N weak external force on sphere j N  q   Vj exp i q  R j  mean (short - time) response j1  H(q)  0 Fe Vse0 d   0 Fe  q lim V(q)  Vsed  u susp (r; X) q0  mean suspension velocity  Lab frame = zero volume flux frame  Sedimentation velocity in zero - volume flux reference system 21 DS / D 0 related to cage diffusion self - diffusion 0 Vsed / Vsed Vsed  V 0 sed sedimentation HS :   0.33 22 Hard spheres – hydrodynamic & diffusion function D(q) / D0  H(q) / S(q) cage stiffening  Symbols: accelerated Stokesian dynamics simultion method  Lines: analytic theory (so-called uncorrected δγ – theory) A.J Banchio and G Nägele, J Chem Phys 128, 2008 23 Charged colloidal spheres: Experiment and theory SLS on charged silica spheres in 80:20 toluene/ethanol HI shielding: u (r )  H(q) 1 eff ( ) r Nägele et al. PRL 96, 108303 (2006) DS q good for low 

only, disregards HI shielding  Pairwise additive HI:  Corrected δγ - theory: close to exp. & simulation throughout liquid phase M. Heinen, P Holmqvist, A Banchio & G Nägele, J Appl Cryst 43 (2010) & J Chem Phys 135 (2011) Microgels: Holmqvist, Mohanty, Nägele, Schurtenberger, Heinen: Phys. Rev Lett, in press (2012) 7.2 Sedimentation clear fluid (supernatant) gravity monodisperse mixture sediment Brownian sedimentation  Slow settling of colloidal Brownian spheres starting from homogeneous state  Sedimenation Peclet – #:: Pe  F  F qˆ e e 0 Vsed   0t Fe  0t  1 diffusion time sedimentation time  2  D   / D0 0 sed   / Vsed  Pst (X)  Peq (X) Pin (X, t  0)    t  D   Pst (X)  Peq (X) 1 zero or pairwise-additive HI 3 - and more-body HI equil. distribution for zero gravity force 6 0 a 26  Find stationary & homog. pdf solution of GSE

under constant external force F  ˆ sm (X) Pst (X)  0 Pst (X)  O t 1 N Vsed   vi (X, t) N i 1 st  1 N tt  I e   μ   Fl  k BT l ln Pst (X)  F  N i,l 1 il     st long-time (memory) part of Vsed vanishes for PA - HI where Pst = Peq  Linear response stationary solution:    Ô0Bu e 0 ˆ Pst (X)  Peq (X) 1   F   du e  O B R j   O((Fe )2 )   j1 0 N ˆ 0B  O ˆ B ( Fe  0) O ViI  Ô0B R i adjoint Smoluchowski operator without external force  Mean sedimentation velocity at small Peclet numbers: Vsed 0 Vsed  1 N  qˆ  μilt t  qˆ t  0 i,l 1  memory part eq H(q  0) : short-time part zero for PA-HI or small Reason: mean motion in same direction  Smoluchowski‘s paradoxon : RP H(q  0)  1  n  dr g(r) qˆ  12 (r )  qˆ  regular part 

 ? Resolution: macroscopic solvent backflow due to container bottom - consider container bottom explicitly (Mazur & van Saarlos, ´78) - locallity of transport coefficients (Felderhof, Cichocki, ´80) - use zero volume flux condition for planes  to Fe (Batchelor, ´72) g  u(r ), r  Vfluid ususp (r;X)    Vi , r  Vparticle  dS u(r; X)  p  n Fe sa st 0 rest frame of container drives backflow 0 0 - use that : Vsed  ususp (r;X)  Vsed lim H(q)  Vsed H(q  0) q 0 relative to ensemble averaged volume flow velocity, then container-shape & position independent (c.f intrinsic convection) 29 • (Short-time) sedimentation on Rotne - Prager level (zero volume velocity frame) short Vsed 0 Vsed V   V short sed 0 sed       15 g( x; )    1 -  5  12  dx x (1 - g(x; ))  dx 8 1 x2  1     known analytically   for

hard spheres in PY      HS x  r 2a 1 2 66 2  3 (1 ) / (1 2 ) + 1 5         in RP-PY  5 5   1  6.546   21918 2  ? exact (Cichocki et al., 02)  30 Sedimentation of Brownian hard spheres gravity Vsed 0 Vsed Cichocki et al., JCP, 2002 Banchio & Nägele, JCP 127, 2008 short 0 Vsed / Vsed  1  6.546   21918  2  RP sed V 0 sed /V (1  ) 3 2   1 2 5   RP far - field HI PY - g(r) Sedimentation only weakly affected by near-field HI 31 Sedimentation of Brownian charged particles gravity Vsed 0 Vsed salt 0 Vsed / Vsed  1  1.8 1/3   Slower sedimentation of charged clay particles (river - delta) Watzlawek & Nägele; JCIS 214 (1997) 32  p sa  n Fe Pressure gradient drives homogeneous mean fluid backflow g(r) neutral. charged 1  Increased friction with backflowing fluid rm  1/3 r 33 7.3 Intrinsic

convection Vsed  Vsed (x )  u  p  u u 1 L st (x)   u st (x)   st  0 (x)  Vsed 1  6.55   O   st (x) g  0  0   9 / 4   Vsed near wall  a f (x)   st  p  1   f in bulk st 2 X: Z (x)  Vsed (x)  1   u B (x) st (x   L / 2)  0 wall rest frame L/2  u st  L/2 (x) dx  0 Z 0  9 / 8  Vsed  Particle - depleted buoyant layer near walls drives upward suspension flow  Lateral average of Vsed(x) gives the true (relative) sedimentation velocity D Bruneau, Phys. Fluids 8 (1995), and section 73 in book by Jan Dhont 34 Rest frame of spherical rigid container with radius R ∞    0 Vsed (o)  Vsed 1  3.55   O 2 gravity u   0 2 ( o )  3  V  O  sed st u B (o)   1 Z Vvess C.WJ Beenakker & P Mazur, Phys Fluids 28 (1985)    1  0

Vsed d 3r ususp (r, X)  0 Vvess In rest frame of vessel for incompresible fluid & particles: Weak intrinsic convection disappears at higher concentrations Y. Peysson & E Guazzelli, Phys Fluids 10 (1998) 35 7.4 Sedimentation of large non-Brownian particles  Stokes – Liouville equation for driven system of non - Brownian particles (T - indep.): N  P(X, t)    i  μittj ( X)   FjI  Fje  P(X, t )   t i, j  1 P(X, t  )  Pstat (X) ?  T - limit existent ?  Intricate non-linear coupling of microstructure (pdf) and HI, both dependent on X.  Instabilities at higher concentrations and strong driving (chaotic trajectories).  Simulations with large particle numbers are needed.  Locallity of Vsed presupposed: Vsed  u st   1  3.87   O   2  gstHS (r)    r  2a  transient closed pairs formed (cf. infinite-Peclet viscosity) n  dV  gstHS (r)  1 

1   B. Cichocki & K Sadlej, Europhys Lett 72 (2008) 36  Large - scale velocity fluctuations only transient  Quasi - steady state of smaller-scale fluctuations of about 20 mean-particle distances  Still open questions (stratification, side wall effects, locallity ) taken from: E. Guazzelii and J Hinch, Annu Rev Fluid Mech 43, 97 (2011) 37 taken from: E. Guazzelii and J Morris, A Physical Introduction to Suspension Dynamics (Cambridge, 2012) 7.5 Gradient diffusion  Gradient diffusion (Brownian particles): Condition: q << qm (r , t) j   D C  S(q, t)  S(q) exp  q 2 D C t  0 DC H(q) Vsed / Vsed = lim  q  0 D0 S(q) k BT   /  p osm  T, r s osmotic pressure 0 Hydrodynamic interaction : Vsed / Vsed Osmotic compressibility DC D0 little added salt a lot of added salt  (G.Nägele, Phys Reports 1995) 39 8. Suspension viscosity - General properties - Examples - Shear

thinning and thickening 40 8.1 General properties 41 ■ Macroscopic steady-state shear stress y  xy   xy (r; X) x st Fx A  d  u  x dy    ■ Effective suspension viscosity (  )      u    y xˆ . st    dX Pst (X;  ). 1  H  xy    1 shear relaxation contribution ■ High - frequency viscosity part (HI only):   )  0    t t ( X)  t d (X)   F H  (FI  F B )   V  e  X         dt H dd   e  S       (X)  (X)     ■ Strain-flow part: e   r   y xˆ  x yˆ / 2 n H S (X)  xy I xy B   xy  DI st symmetric force dipole (stresslet) ■ Shear – relaxation viscosity part:  )    n Vic  FiI  FiB 2   st FiI  FiB  i VN (X)

 k BT i ln Pst (X)   Vic (X)  1  R i  μit d (X)  : e    Convective velocity (particle force- and torque free) HIs: 3rd rank shear mobility tensor of particle i ■ Shear – Péclet number: Pe  diffusion time flow time D a 2 / D0     a 3   1 /  G.K Batchelor, J Fluid Mech 83, 97 (1977) W.B Russel, J Chem Soc Faraday Trans 2, 80 (1984) G. Nägele and J Bergenholtz, J Chem Phys 108 (1998) 43 8.2 Examples  High-frequency viscosity of no - slip Brownian hard spheres    D  1 Pe  1     0  0 Einstein 1905: 1.0 1911: 2.5 (Hopf) η η0  1  2.5   50 2  91 3  Batchelor & Green (1972) Cichocki et al. (2003)   Virial expansion in volume fraction applicable to lower concentrations only  High frequency viscosity diverges at random closed packing:  rcp  0.64 High - frequency viscosity of

charged Brownian spheres    D  1  (   0) 0 charged spheres        Lower high – frequency viscosity for charged spheres (CS)  But: Δη(CS) > Δη(HS) Heinen, Banchio & Nägele, J. Chem Phys 135 (2011) 45 Steady-state versus high-frequency viscosity of hard spheres    0  0 Theory: w/o HI      Theory: with HI η / η0 HI      0  0  1  2.5   59 2  Batchelor (1972) B 0.9 from  xy  xyI  0 for HS  Theory: Banchio, Nägele, Bergenholtz, Phys. Rev Lett82 (1999) Exp.: Segrè et al., Phys Rev Lett 75 (1995) Viscosity of a 1 -1 electrolyte NaCl in water o T  25 C   3.6 A   0     Mode - Coupling theory with HI: C. Contreras-Aburto and G Nägele, J Phys: Condensed Matter, to appear (2012) 47 8.3 Shear-thickening and thinning 2D – Simulation:

relative shear rate = 0.1, 10, 1000 (Peclet-number) Without HI η With HI A  0.67 shearthinning shearthickening (Hydro-Cluster) shear rate (Peclet - #) 1 10 1000 rcp A  0.82 crys  0.907 A Foss und Brady J. Fluid Mech 407 (2000) Steady-state pair distribution function g(x,y): Brownian hard spheres Taken from: J. Brady, Chem Engineering Science 56, 2921 (2001) y x Pe = 0.1 10 g cont  O(Pe) 1000  Tight boundary layer (BD – convection) around compression axis  Shear-thinning & thickening even to O(Ф2)       / 0  1  2.5   60 2        / 0  1  2.5   695 2   pure strain  Batchelor (1972) Brownian hard spheres: simulation results η η0 η = η H + ηB η ηB ηH   0.45  a2 /D0 Pe = γ  Shear – thickening explained here by transient hydrocluster formation (lubrication)  Brownian stress contribution ceases with increasing

shear rate taken from: Foss und Brady, Journal of Fluid Mech. 407, 167 (2000) 50 9. Dynamics of permeable particles - Brinkman fluid model - Applications Enrico Fermi Summer School „Physics of Complex Systems“, Varenna, Italy, July 4 – 9, 2012 1 Hydrodynamic model: - uniformly porous spheres - rigid skeleton. Extension to core-shell structure: Abade, Cichocki, Ekiel, Nägele, Wajnryb; J. Chem Phys 136, 104902 (2012) Microgels: Holmqvist, Mohanty, Nägele, Schurtenberger, Heinen: Phys. Rev Lett, in press (2012) 2 9.1 Brinkman fluid model (of porous hard spheres) Brinkman equation of flow u inside a sphere j: 0  u(r )  p(r)   0 / k   u(r )  w j (r )   0   force density due to fluid - skeleton friction x = a / k  a / rpore w j ( r )  Vj  Ω j   r  R j  small k: zero relative flow (0/k) [u – wj]  0 (stick BC) inverse fluid penetration depth large k: Stokes

flow also inside spheres relative to radius a  Spheres of uniform porosity k  (mean pore size)2  Model characterized by:  and inverse penetration depth x  k depends on size, shape and distribution of interstices (model - dep.) 3 Boundary conditions and method of calculation   in (r )   0 / k   u(r )  w j (r )   0   Sj  Vj out Interpretation: induced force density inside Sj σ (r )   p(r )1  0  u(r )   T u(r )   continuity of u on Sj   u (r )  0  continuity of  on Sj FjH   0 / k  also on S dr  u(r )  w (r )     dr f (r )   dS σ (r )  n(r )  V V S j j j j j  Use induced force picture (Stokes flow inside & outside spheres)  Determine many - sphere mobility tensors using spherical multipole expansion method by Cichocki and coworkers (HYDROMULTIPOLE program) 4 9.2

Applications: Generalized sedimentation coefficient increasing permeability g(r) 1 x  a / rpore neutral. charged r Solvent permeability less influential for charged particles 5 Abade, Cichocki, Ekiel, Nägele, Wajnryb; J. Chem Phys 136, 104902 (2012) Application: high - frequency viscosity    diff  1 ω zero porosity  0   1    (x)   O( 2 ) 0 5 / 2  x          1 slip & bubble intrinsic viscosity Abade, Cichocki, Ekiel, Nägele & Wajnryb, JCP 132 (2010) & JPCM 22 (2010) 6 Short - time properties of porous spheres 1 1.2 DS ( x ) D0 ( x ) H(q m ) 0.8 0.6 porosity increase zero porosity non - porous 0.4 0.2 1 Vsed ( x ) 0 Vsed (x) 0 0.1 0.2 0.3 0.4  0.8 particles with long grafted polymers: 0.6 0.4 x = 20 - 30 0.2 0 0 0.1 0.2  0.3 0.4 Abade, Cichocki, Ekiel, Nägele, Wajnryb: 7 JCP 128 (2010) & PRE 81 (2010) Various

definitions of dynamic effective radii  Effective radii agree practically for x > 30  Then no distinction between different flow profiles 8 Abade, Cichocki, Ekiel, Nägele & Wajnryb, JCP 133 (2010) Generalized Stokes - Einstein (GSE) relations  translational self-diffusion : ? DS (  )   rotational self-diffusion : ? k BT D R ( )  6   ( ) a k BT 8  ( ) a 3 S(q)  cage diffusion coefficient: D(q m ; )  D0 H(q m ) S(q m ) ?  k BT 1 6  ( ) a q qm 9 Generalized Stokes-Einstein relations for permeable spheres Abade, Cichocki, Ekiel, Nägele & Wajnryb, JPCM 22 (2010) 10 Test of validity of generalized Stokes-Einstein relations Abade, Cichocki, Ekiel, Nägele & Wajnryb, JPCM 22 (2010) 11 Collective diffusion coefficient DC D0 12 10. Appendices - Derivation of point force solution - Proof of Mean-Value Theorem - Multipole method by Chichocki and coworkers Enrico

Fermi Summer School „Physics of Complex Systems“, Varenna, Italy, July 4 – 9, 2012 10.1 Derivation of point force solution  Unbounded infinite fluid:: p(r )  02 u(r )  f (r )   u(r )  0 iq p(q )  0q 2 u(q )   f (q ) q  u( q )  0 u(q )   dr eiqr u(r ) et cetera    iq  dq e  iqr A(q )B(q )    2  3  dr A(r  r ) B(r ) iq  q p(q )  0q 2 q  u(q )   q  f (q )   0 p( q )  i q  f (q ) q 2  u( q )  u(r )   dr T0 (r  r )  f (r ) T0 (r )  1  2  Q0 (r )   iqr d e q 3  1 0q 1 0q 2 1  qˆ qˆ   f (q)  use convolution theorem p(r )   dr Q0 (r  r )  f (r ) 1  qˆ qˆ   2 1 1  rˆ rˆ  80 r 1 1  iqr  q   iqr 1 d e i d e    q q r r  2 2 3 3  3 q 4 r  2   q   2

 1 2 ‐ have employed that :  2 d q  dq q   0  d q 4  iqr d  e 1  4 j0 (q r) 1 q   j1 (q r)   iqr ˆ ˆ ˆ ˆ     d e 4 j (q r) q q 1 r r 2  qr   q   j2 (x)  3 j1 (x) x  j0 (x) 3 10.2 Proof of mean - value theorem  Expand around center of sphere i: u  (r )    r  R i n  n n 0 n!  1 u  (r ) r R  u ( R i )   r  R i   u ( R i )   r  R i  r  R i  : u  ( R i )  . i 2 2 2 u   0  x a   u  0 dS x. x x  0  since odd integrands ( x  r - R i ) odd factors 1   dS u (r )  4a u (R i )  0   2  dS x x  : u (R i )  0 Si Si   2 2 4 4 a 1 6 a2 2   dS u (r )  4a  1  6   u (R i )   Si Generalizations: R. Courant and D Hilbert,

Introduction to Mathematical Physics II, Springer (1968) 4 10. 3 Symmetry of mobility and friction matrices ‐ Consider two solutions (uA, pA, fA) and (uB, pB, fB) of Stokes equation (linearity!)   σ A  u B  σ B  u A   f B  u A  f A  u B Lorentz reciprocity relation ‐ Use for A and B the N ‐ sphere BVP with stick BC in quiescent fluid (u = 0 at ) ‐ Integrate over all space (super ‐ vector notation): 0   dr   .   FB  VA  FA  VB    TB   A  TA   B  dr fi  dr  r  R i   f i Vi    tt    Vi Ti  VA   tt  FA & VB   tt  FB when TA  0  TB FB  tt  FA  FA  tt  FB Fi  tt Tr ‐ More generally since forces and torques can be selected arbitrarily:   tt    rt   tr    tt  rr   rt     tr   rr    Tr 

 tt    rt   tr    tt  rr   rt     tr   rr    Tr 5 10.4 Multipole method by Cichocki and coworkers • N spheres with stick BC in unbound fluid with ambient flow u0(r) w i (r )  Vi  Ω i   r  R i  For r inside and on sphere i: Vi r  Vi N wi (r )  u0 (r )   dr T0 (r, r )  fi (r )    dr T0 (r, r )  f j (r ) j i Define Green integral operator (propagator) G(ij) for i  j by localized on surface Sj (for stick & slip‐stick BC )  G(i j) f j  (r)   dr T0 (r, r )  f j (r ) r  Vi  G(ij) depends on solvent properties & outer BC only (here Oseen tensor) Incident flow inside sphere i volume (non ‐ singular therein): N uinc (r )  u0 (r )    G(i j) f j  (r ) j i 6 By linearity of Stokes flow is: fi (r;X)   dr ζ 0 (r, r )   w i (r ;X)  uinc (r ;X)  Single – sphere

friction kernel 0 (localized on Si) depends on hydrodynamic model & BC of a single sphere only, and not on X Use again short – hand notation by omitting field variables and integral w / r r´: fi  Z0 (i)  w i  uinc   Z0  0 ( r, r )  w i  uinc  Z01 (i) fi Single ‐ sphere friction operator has obviously an inverse: Insertion of incident flow gives: w i  u0  Z01 (i) fi N   G(i j) f j integral equations for induced force densities j i Introduce super – vector / matrix notation: w  u 0   w1  u0 ,., w N  u0  et cetera   Ĝ   j 1- i j G(i j) Ẑ01   j i j Z01(i) 7 Allows to write integral equations more compactly as:   ˆ f  Zˆ 01  G ˆ f w  u 0  Zˆ 01 f  G Again inversion is allowed because of 1‐1 correspondence: ˆ  1 w  u 0   Zˆ 0 1  G ˆ Zˆ 0  f   Zˆ 01  G     1  w

 u0  (formal solution for f) Formally expand in multiple scattering series (friction problem): ˆ Zˆ 0  G ˆ Zˆ 0G ˆ Zˆ 0  .  w  u 0  f  Zˆ 0 1  G   fi  Z0 (i)  w i  uinc  uinc  w i G(i k) u0 Z0 (i) G( j i) G( j i) fi 8  Obtain coresponding multiple scattering series for mobility problem  Project on basic set of spherical tensor multipoles  ilm  ,   0,1   Infinite set of linear algebraic equations relating spherical force and velocity multipoles  Truncate set at multipolar order L (L  3)  Lubrication correction: modified version of method by Durlofsky, Brady and Bossis  Positive definiteness and symmetry of exact mobility matrix is preserved  Advantage: only Z0 must be changed when changing particle model B. Cichocki et al, J Chem Phys 112 (2000) L. Durlofsky, JF Brady, and G Bossis, J Fluid Mech 180 (1987) 9 Acknowledgment I am very grateful to Dr. Claudio

Contreras‐Aburto (ICS‐3, Research Centre Jülich) for his help in the preparation of a number of figures