Dynamics of charged-particles dispersions:

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(1)Dynamics of charged-particles dispersions: From large colloids to nano-sized bioparticles and electrolyte ions Gerhard Nägele Institute of Complex Systems, Research Centre Jülich, Germany. 5th Warsaw School of Statistical Physics, Kazimierz Dolny, Poland, June 22 – 29, 2013. 1.

(2) Books and Lecture Notes 1.. W.B. Russel, D.A. Saville, and W.R. Schowalter, Colloidal Dispersions, Cambridge University Press (1989). 2.. E. Guazelli and J.F. Morris, A Physical Introduction to Suspension Dynamics, Cambridge University Press (2012). 3.. G. Nägele, Colloidal Hydrodynamics, in Proceedings of the International School of Physics, "Enrico Fermi", Course 184 "Physics of Complex Colloids", ed. by C. Bechinger, F. Sciortino and P. Ziherl, (IOS, Amsterdam; SIF, Bologna) pp. 507 – 601 (2013). 4.. J.-L. Barrat and J.-P. Hansen, Basic Concepts for Simple and Complex Fluids, Cambridge University Press (2003). 5.. J.H. Masliyah and S. Bharracharjee, Electrokinetic and Colloid Transport Phenomena, Wiley - Interscience (2006). 6.. H. Ohshima, Theory of Colloid and Interfacial Electric Phenomena, Interface Science and Technology Volume 12, Elsevier (2006). 7.. R.J. Hunter, Foundations of Colloid Science, Oxford University Press (1989). 8.. R.J. Hunter, Zeta Potential in Colloid Science, Academic Press (1988). 9.. J.K.G. Dhont, An Introduction to Dynamics of Colloids, Elsevier, Amsterdam (1996). 10.. G. Nägele, The Physics of Colloid Soft Matter: Lecture Notes 14, Polish Academy of Sciences Publishing, Warsaw (2004). 11.. G. Nägele, On The Dynamics and Structure of Charge - Stabilized Colloidal Suspensions, Physics Reports 272, pp. 215-372 (1996) 2.

(3) Content 1.. Introduction & Motivation. - Henry formula. - Examples of dispersions. - Strongly charged macroion. - Direct particle interactions - Brownian forces - Inertia – free dynamics - Low-Reynolds number flow examples. - Extensions to concentrated systems. 2. Low Reynolds number flow - Colloidal time scales - Stokes equation - Point force solution - Faxén laws for spheres - Many-spheres HIs. 3. Salient static properties. 5. Dynamics of interacting Brownian particles - Many-particle diffusion equation - Dynamic simulations. 6. Short-time colloidal dynamics - Hydrodynamic function - Sedimentation - High-frequency viscosity - A simple BSA solution model - Generalized SE relations. 7. Long-time colloidal dynamics. - Pair distribution function - Methods of calculation. - Memory equation and MCT. - Ionic mixtures. - Self-diffusion of DNA fragments. - Effective colloid interactions - Force on colloidal particle in electrolyte. - HI enhancement of self-diffusion. 8. Primitive model electrokinetics - Macroion long-time self-diffusion. 4. Electrophoresis of macroions. - Electrolyte viscosity and conductivity. - Hückel and Smoluchowski limits 3.

(4) Content 1.. Introduction & Motivation. 2.. Low Reynolds number flow. 3.. Salient static properties. 4.. Electrophoresis of macroions. 5.. Dynamics of interacting Brownian particles. 6.. Short - time colloidal dynamics. 7.. Long - time dynamics. 8.. Primitive model electrokinretics. 4.

(5) 1. Introduction & Motivation - Examples of dispersions - Direct particle interactions - Brownian forces - Inertia - free dynamics - Hydrodynamic effects: examples. 5.

(6) 1.1 Examples of dispersions •. Micron – sized charge-stabilized colloidal particles: 40 nm < ∅ < 5 µm |Qbare| ≈ 10 – 20 000 e. Temp. pH Conc.. - silica - goldsol - PMMA. - Ionic microgel particles (solvent: water, polar fluid). … Products / Applications: dispersion paints, pharmaceuticals, food stuff, cosmetics, waste water, drug delivery (microgels), … 6.

(7) •. Nanometer - sized globular proteins in (salty) water (|Qbare| ≈ 8 – 30 e). Apoferritin. Hen-egg lyzozyme. Bovine serum albumin (BSA). 6 nm 13 nm. 7.

(8) •. Strong electrolyte solution (e.g., NaCl in water). ε. ∅(Cl-) = 0.36 nm. ∅(Na+) = 0.55 nm. u αβ (r) = u Sαβ (r) + u Cαβ (r). E∞ u Cαβ (r) k BT. = LB. z αzβ r. (. , r > a α + aβ. Λ ≈ Λ 0 − const × n T. ). L B =e 2 /( ε k B T) ≈ 0.7nm (Bjerrum length). Falkenhagen - Onsager limiting law for conductivity valid only for nT = n+ + n- < 0.01 mol /litre 8.

(9) 1.2 Direct particles interactions •. Dressed macroion model and DLVO theory. u(r). u el (r) repulsion. 40. a. 20. r. 0. u vdW (r). -20. 1. 2.  exp[ κ a]  exp[ −κ r] βu(r) = L B Z2   r  1+ κa . effective charge number Z < |Qbare |/ e. 2. r [µm ]. r/σ. 3vdW-attraction 4. u(r) ≈ u el (r) for |Z | >> 1 κ 2 = 4πL B ( n | Z | +2n s ) / (1 − φ).  Tuning of range and strength of effective pair potential by changing salt content & solvent  Microion electrokinetic effects disregarded in this model ! 9.

(10) •. For comparison: neutral colloidal hard spheres. u(r). Sterically stabilized particles:. „hard spheres“. 1. 2.  ∞, u(r) =   0,. 3. r/σ. r < σ =2a r>σ. • Pairwise additive N-particle eff. potential energy (approximation for CS, not for HS). ( ) ∑ u ( Ri − R= j ) ∑ u (Ri j ) i< j i< j. VN R = N. N. N. R N= X=. {R1 ,..., R N }. 10.

(11) Phase behavior of colloidal hard - sphere dispersion. Nv sphere 4 π 3 = φ = na Vsystem 3 π φfcc cp =. 18. Kepler: 1611 Hales: 1998. Pusey & van Megen, Nature 320, 1986 11.

(12) Phase behaviour of a charged - spheres system (silica spheres). Sirota et al. PRL 62, 1989. fcc – bcc coexistence. • •. BCC phase at low salinity (8 next neigbours), FCC phase at higher salinity (12 n.n.) Metastable glass - like phase at higher volume fractions (polydispersity) 12.

(13) 1.3 Brownian forces. R (t) R (0) 10 µm. R (t) − R (0) = 0 = W(t). 1 [R (t) − R (0)]2 6. Mean-squared displacement. Movie: E.R. Weeks, Austin 13.

(14) Self - diffusion of colloids (Brownian particles). R (t = 0) direct interactions only. Single - sphere diffusion. W(t) = D0 t [R (t) − R (0)] / 6 = 2. k T = D0 = B 6πη0a ζ0 k BT. D0 t. W(t). Stokes Einstein Sutherland diffusion relaxation time. τD = a / D0 2. time. t. 14.

(15) D0 t W(t). DLt. DS t. t. time HI. HI & DI. D L < DS < D0.  Self – diffusion in general slowed down by hydrodynamic interaction (HIs). 15.

(16) 1.4 Inertia - free dynamics granular media Colloidal dispersions (including proteins & viruses) Electrolyte Ions (hyd.). bacteria, protozoa (0.1 – 800 µm). molecules. human cell: ∼10 µm. atoms. 1 nm. 10 nm. 100 nm. 1 µm. radius non - Brownian. (L > 5 µm).  Colloids, proteins and most bacteria share common inertia - free hydro-dynamics 16.

(17) Inertia – free particle dynamics • quasi - inertia free motion on coarse-grained colloidal time- and length scales d V (t) M ≈ −ζ 0 V (t) dt. • momentium relaxation time M −8 10 ∆t >> τ= ≈ sec B ζ0. ∆x >>= lB. D0 τ B ≈ 10−4 σ. “stopping distance” Rhodospirillum bacteria (length σ ≈ 5 µm) 17.

(18) Low-Reynolds number solvent flow •. Inertial forces tiny as compared to friction forces: Reynolds - # << 1. •. Instantaneous flow profile restructuring around moving particle. Re =. V. wale: in water:. 𝛻 ⋅ 𝐮(𝐫) = 0. inertial forces ρf aV = viscous forces η0 108. human swimmer: 10000 colloid particle:. 0.0001. bacteria /cells:. 0.00001. incompressibility. −𝛻p(𝐫) + η0 Δ𝐮(𝐫) + 𝐟 e (𝐫) = 𝟎. linear quasi-staticStokes equation Inertia-free force balance. 18.

(19) Implication: linear particle hyd. force – velocity relations •. Sphere with stick (no-slip) BC translating ín quiescent infinite fluid. V =µ F t 0. µ 0t =. e. 1. V. 6πη0a. translational mobility. Fe. (6π → 4π : perfect slip). = D0 k BT µ 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 19.

(20)  Sphere in quiescent, infinite fluid, stick BC. u∞ = 0.  Sphere stationary (rest frame). u∞ = − V. u(arˆ ) ⋅ nˆ = 0. u(arˆ ) ⋅ tˆ = 0 20.

(21) N freely rotating spheres: hydrodynamic interactions (HIs) = Vi. N. ∑=1 μi j (X) ⋅ F tt. e j. translational 3×3 mobility tensors. j. V = µ t t ( X) ⋅ Fe. V3. V2 F2e. V1. mobility problem. R1. Dittj = k BTμittj X = {R1,..., R N }. diffusivity matrix. (. V = ( V1,..., VN ). ). tt RP μ= X μ ( ) i j R i − R j + ∆μ i j ( X ) ij. far-field HI:  O(r −1 ). • HI acts quasi – instantaneously on colloidal time scales • near - field part non pairwise additive. −4. near-field HI:  O(r ). T. tt tt  μ11  μ1N      µ tt =    tt tt  μ N1  μ NN  . 3N x 3N matrix symmetric & pos. definite 21.

(22) • Moblity tensors required as input in calculations of colloidal, protein, and electrolyte transport properties • Flow pattern of u(r), p(r) itself not needed. DS =. k BT tt Tr μ 11 3. - self-diffusion coefficient μ 11  O(r −4 ) tt. eq. N. Vsed. tt −1 ⋅ Fe - mean sedimentation velocity μ 12  O(r ). tt ∑ μ 1p (X). p =1. eq,ren. (renormalization required). • ⟨…⟩ average over homogeneous and isotropic particle ensemble. 22.

(23) Pure configuration - space decription for t >> τB. 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. 23.

(24) • Moblity tensors are input in N – particle diffusion equation (Smoluchowski) N ∂ P(X, t) k BT ∑ ∇i ⋅ μittj ( X) ⋅ ∇ j − β FjI − β 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 [ −β VN (X)]. FjI = −∇ jVN (X). G. Nägele, Phys. Rep. 272 (1996) and J. Dhont, An Introduction to Dynamics of Colloids, Elsevier (1996). 24.

(25) 1.5 Low – Reynolds - number flow examples •. Kinemaric reversibility (ignore Brownian motion for time being). ∇ ⋅ u (r ) = 0. − ∇p(r ) + η0∇2 u(r ) + f e (r ) = 0 u(r ) S = V + Ω × ( r − R p ). u (r → ∞ ) = u ∞ (r ). r ∈ Vfl. Ω V. S. ambient given flow (flow w/o particles). S. Since linear boundary value problem (BVP):. {. } {. V, Ω, u∞ , f e ⇒ − V, −Ω, − u∞ , −f e. }. {u, p} ⇒ {−u, − p}.  Motion reversal of boundaries, external force density and ambient flow reverses sign of flow pattern only, but not its shape. 25.

(26) 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. 26.

(27) 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 27.

(28) Application: motion in highly symmetric systems. {u∞ , V, Ω}. {−u∞ , − V, −Ω}. ∆V = 0 • Lift forces arise only when non- zero inertial contributions: du/dt ≠ 0 • ∆V ≠ 0 also for flexible particles (polymers, drops). 28.

(29) Purcell‘s scallop theorem for microswimmers • Internal forces and torques 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. φmin. φmax. E.M. Purcell, Life at low Reynold‘s number, Am. J. Phys. 45, 3 (1977). 29.

(30)  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 30.

(31) Simple artificial microswimmers lL Purcell swimmer. 3 - spheres swimmer. artificial amoeba. lR. lL 1. 4. 2. 3. lR non-zero area swept out. G. Nägele, Colloidal Hydrodynamics, in Proceedings of the International School of Physics, "Enrico Fermi", Course 184 "Physics of Complex Colloids", ed. by C. Bechinger, F. Sciortino and P. Ziherl, (IOS, Amsterdam; SIF, Bologna) (2013) 31.

(32) Sedimentation of two non - Brownian Spheres (Re << 1) V0 = µ0 Fe. µ0t =. 1. gravity. 6πη0a. V0. (buoyancycorrected). Fe. 32.

(33) The sedimentation race: bet which pair settles fastest, and how ?. 33.

(34) r gravity. constant sep. vector r pair 0 Vsed > Vsed = µ0t Fe. 34.

(35) • Drag - along effect strongest for in - line sedimentation • Lubrication plays no role for motions considered here (Rotne – Prager o.k. for r > 5a) 35.

(36) Sedimentation of a non - Brownian rod (Re << 1). V= μ ⋅ Fe. Fe = ζ ⋅ V F||e. ê. given. F⊥e 1. µ ⊥ ≈ µ 2. Fe Fe. (transl. mobilities for L >> D). ˆ ˆ + µ ⊥ (1 − ee ˆˆ) μ = µ ee V = µ Fe  + µ ⊥ Fe ⊥. ( eˆ eˆ )αβ = eˆ α eˆ β. F⊥e. ζ ⊥ ≈ 2 ζ ζ⊥ = 1 / µ⊥. Experiment: needle in syrup no rotation. Flagella bundle: HI synchronized 36.

(37) Apparent like - charge attraction of particles near boundary. figure taken from: Squires & Brenner, PRL (2000). charged glass wall or liquid - air interface. Fext → - Fext ?. • Attractive wall: apparent repulsion. 37.

(38) 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 38.

(39) Three sedimenting non - Brownian Spheres: non - symmetric start configuration (-1.1 ,0 ,1.16). (-1.1, 0, 1.20). d X= µ t t ( X ) ⋅ Fe dt. z/σ. non - linear in X. μit tj . 1 η0 | R i - R j |. 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, 067301 (2011). 39.

(40) Sedimentation: spherical cloud of non - Brownian particles (radius R) leakage. toroidal circulation in cloud rest frame (cf. liquid drop). 0 Vsed = µ0t Fe. cloud 0 Vsed ≈ Vsed +. N −1 e F 5 π η0 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) 40.

(41) . Evolution: spherical cloud → torus → breakup in two clouds → …. * t= t / τsed R. t* = 0. t * = 700. t * = 400. t * = 800. taken from: E. Guazzelli and J.F. Morris, A Physical Introduction to Suspension Dynamics, Cambridge Univ. Press (2012) 41.

(42) . Point - particle simulation (N = 3000). . Glass spheres (a ≈ 70 µm) (in silicon oil). taken from: B. Metzger, M. Nicolas and E. Guazzelli, J. Fluid Mech. 580, 238 (2007) 42.

(43) 2. Low-Reynolds number flow - Colloidal time scales - Stokes equation - Point force solution - Boundary layer method - Faxén laws for spheres. 43.

(44) 2.1 Colloidal time scales  ∂ u(r, t)  ρf  + u(r, t) ⋅ ∇u(r, t)  = − ∇p(r, t) + η0∇2 u(r, t) + f e (r, t)  ∂t  ∇ ⋅ u(r, t) = 0. ∆t  τsound = a / csound  10−10 sec. ρf Vp 2 / a ρf a Vp = = = Re 1 2 η0 viscous force η0 Vp / a inertial force. ρf. ∆t  τ vort. ρf Vp / ∆t η0 VP / a 2. − ∇p(r ) + η0∇2 u(r ) + f e (r ) = 0. volumetric force density on fluid by external fields (particle) Reynolds number. ∂ u(r, t) = − ∇p(r, t) + η0∇2 u(r, t) + f e (r, t) ∂t a 2ρf =  10−9 sec η0. Navier - Stokes Eq. incompressible flow. still includes vorticity diffusion. τ vort. 1 a. Stokes equation Inertia - free force balance 44.

(45) Overview: time scales (particles with a = 100 nm in water) 10−13. 10−10. τsound. τ mol. 10−9. τ B ≈ τ vort. t [sec]. 2 τD = a / D0 diffusional relaxation time. molecular view. Quasi - inertia free particles and fluid motion - momentum relax. resolved - unsteady solvent flow. momentum relax. time. τB =. 10−3. 10−6. M ζ0. =. 2  ρp.    τ vort 9  ρf . - Stokes equation flow - Many - particles Smoluchowski eq. Stokes #. St =. τB. =. τext. Advection time 2  ρp.    Re 9  ρf .  Colloids and microswimmer:. Re <<1 and St << 1.  Dry powder granular dynamics:. St >>1 (large & heavy particles in a gas). τext =. a Vp. 45.

(46) 2.2 Stokes equation • Linear Stokes equation BVP for N rigid particles in infinite and unbounded fluid (no ext. forces). −∇p(r ) + η0∇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). 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“. Ri. Helmholtz (1868) : • Unique solution u(r) for given BC‘s on inner and outer fluid boundaries • Of all u(r) with div u(r) = 0, Stokes flow has minimal dissipation. 46.

(47) 0 = − ∇p(r ) + η0∇2 u(r ) = ∇ ⋅ σ h (r ) T σ (r ) = − p(r ) 1 + η0 ∇u + ( ∇u )    h. fluid stress tensor. n (r ). σ(r, t) ⋅ n(r ). h σαβ (r ) = − p(r ) δαβ + η0 ∂ α uβ (r ) + ∂β u α (r )  h e Fh = σ r ⋅ n r = − F dS ( ;X) ( ) ∫+  S. fluid force / area on sphere surface element dS at r exerted by surrounding fluid layer. V. Fe = − F h. single sphere force balance. 47.

(48) Mobility and friction matrices  Hydrodynamic force and torque on surface of particle i. Si h h e Fih = dS σ ( r ;X) ⋅ n ( r ) = dS σ ( r ;X) ⋅ n ( r ) = − F i ∫ ∫ S+i. S*i. h e Tih = dS r − R × σ ( r ;X) ⋅ n ( r ) = − T ( ) i i ∫ Si+. n̂ Ri. ∇ ⋅ σh = 0. S*i. ( r ∈ Vfl ). 48.

(49) 2.3 Point-force solution (Oseen) - Find solution of Stokes eq. for a point force F acting on quiescent & unbound fluid at r :. −∇p(r ) + η0∇2 u(r ) = −f (r ). ∇ ⋅ u(r ) = 0. - The solution for outer BC u(r → ∞) = 0 and p(r → ∞) = 0 is :. = p( r ) Q 0 ( r ) ⋅ F = u(r ) T0 (r ) ⋅ F. Q0 (r ) = = T0 (r ). ∇ ⋅ u(r )= 0 ⇒ ∇ ⋅ T0 (r )= 0. = u (r ). 1 4π r. 2. rˆ. 1 (1 + rˆ rˆ ) 8πη0 r. f (r ) = F δ(r − 0) volumetric force density on fluid Oseen tensor. ( T0 )αβ = (r ). x α xβ   δ +  αβ  8πη0 r  r2  1. including r = 0. ∫ dr ' T0 (r − r ') ⋅ f (r ') 49.

(50) Boundary layer method = u (r ). ∫ d r 'T0 (r − r ') ⋅ f. e. r. (r ') zero ambient flow Ωp.  Rigid particle p of arbitrary shape with stick (no-slip) BC:. ∫ dS' T0 (r − r ') ⋅ f. u D (r ) ≡ u (r ) − = u ∞ (r ). (s). (r '). r'. Vp. Sp. Rp. Sp. single - layer „potential“. disturbance flow. f (s) (r ') = − σ h (r ') ⋅ n(r '). Surface traction on fluid at surface point r‘.  Insertion of no - slip BC. Vp + Ω p × ( r − R p ) −= u ∞ (r ). t wo-dimensional integral equation for traction: (s) dS' T ( r − r ') ⋅ f (r ') 0 ∫. (r ∈ S p ). Sp 50.

(51) {Vp , Ωp , u∞ } ⇒ {f (s) (r ')} ⇒ {Fph , Tph , u}  Particle with complex shape: Discretization / Triangularization N. (s) Vp + Ω p × ( ri − R p ) − u ∞ (ri ) = ∑ T0(r i − r j ) ⋅ f (r j )  j=1. (i ∈ {1,..., N}). 3N ×3N inversion.  Frequently only relations. {Vp , Ωp } ⇔ {FpH , ΩpH } are required.  „Rapid prototypeing“: form complex shapes (proteins) by connecting spherical beads 51.

(52) Far – distance flow field around a neutral particle. r  Expand around point inside particle:. = u D (r ). |r' | |r |. (s) dS' ( ) − ' ⋅ ∇ ( ) − ... ⋅ (r ') T r r T r f [ ] 0 0 ∫. r'. Sp. Sp.  Split in symmetric and anti-symmetric parts:. u D (r ) ≈ − T0 (r ) ⋅ F h +. 1 8πη0 r. 2. rˆ × T h −.  Freely mobile particle (force- and torque-free):. 1 8πη0 r. h ˆ ˆ ˆ ˆ 3 : r1 − r r r S ( ) 2. F h= 0= T h. active microswimmer.  Freely mobile particle creates O(r-2) flow disturbance by its symmetric force dipole. (. ). 1 2 Sh = − ∫ dS' f (s) (r ') r '+ r ' f (s) (r ') − 1 Tr r ' f (s) (r ')  2 3   S. - rigid - no - slip. p. 52.

(53) 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 θ= yˆ ⋅ rˆ. p. 3cos2 θ − 1 rˆ 2  8πη0 r .  Swimmer describeable as static force dipole for distances >> d , and when time – averaged over strokes (non-reciprocal cycle, friction-asymmetric) 53.

(54)  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. 54.

(55) 2. 4 Faxén laws for spheres Vp + Ω p × ( r − R p ) −= u ∞ (r ). ∫ dS' T0 (r − r ') ⋅ f. (s). (r '). (r ∈ S p ). Sp. − ∇p∞ (r ) + η0 ∆u∞ (r ) = 0. ⇒ ∆p∞ (r ) = 0 ⇒ u ∞ (r ). SP. ≡. 1. ∇ ⋅ u ∞ (r ) = 0. ∆ ∆ u ∞ (r ) =0. (bi - harmonic → mean - value property:. r ) u∞ (R p ) + ∫ dS u∞ (=. 4 πa 2 S. (homog. Stokes eq.). 2. (. p.  Integrate over Sp w/r to r, use mean-value theorem and 1. 4 πa S i 2. 1 1 6πη0 a. ∫ dS T0 (r − r ') =. ). a ∇2 u ∞ ( R p ) 6. Sp Rp. r. r'. | r '− R p | ≤ a. 55.

(56) . Translational Faxén law for single sphere in ambient flow. F ph. .    a2 2  = −6πη0a  Vp −  1 + ∇  u ∞ (r = R p )  6    . extra flow contribution. Rotational Faxén law:. T ph. 3. - translational Faxén law - Stokes friction law when u∞ = 0. = −8πη0a Ω p − ∇ × u ∞ ( R p )  2  . F h= 0= T h :. 1.  a2 2  Vi =  1 + ∇  u ∞ ( R i ) 6   Ωi =. 1 ∇ × u∞ (R i ) 2.  Freely mobile particle advects with (surface-averaged) ambient flow at its center  No cross – streamline migration for Re → 0. 56.

(57)  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 57.

(58) 2.5 Many-spheres HIs: Rotne-Prager approximation  Identify ambient flow with incident flow on sphere i by N - 1 spheres (quiescent fluid).  a2  N Vi −  1 + ∆ i  ∑ ∫ dS' T0 (r '− R i ) ⋅ fk(s) (r ') 6   k ≠ i Sj. − µ0t Fih=. Faxén law. | Ri − R k |  a.  Consider dilute suspension where :. (. fk(s) (r ') ≈ − Fkh / 4 π a 2. ).  Use mean-value theorem for integral over the Sk. Ri.  Rotne – Prager approximation for t - t mobilities : N. N h Vi ≈ − µ0 1 δi j + (1 − δi j )TRP ( R i − R j ) ⋅ F j = − μiRP j Ri j j 1=j 1. ∑ {. 3 TRP= (r ) 4. }. 3. 1a a ˆ ˆ 1 + r r + )   (1 − 3rˆ rˆ )  r ( 2r   . ∑. ( ) ⋅ Fhj. ( → 0 for r → ∞ ) 58.

(59) Rotne – Prager approximation ∇i ⋅ μit tj (X) ≠ 0.  µ (X ) µ (X)   F  V − ⋅  Ω = rt rr h      0   µ (X) µ (X)   T = tt. tr. h. ∇i ⋅µiRP j (R i j ) = 0 hydrodynamic drift part: from low to high mobility region.  Pros and cons:. Positive definiteness of 3N x 3N matrix µ t t (X) is preserved Easy to apply (theory & simulation) Upper bound to exact µ t t (X) All flow reflections neglected, lubrication neglected Overestimates HI in general Multipole expansions including reflections / many - body HI & lubrication General method: Hydrodyn. multipoles method by Cichocki and collaborators 59.

(60) Hyrodynamic cluster expansion (3) μit tj (X) = µ0 1δi j + ∆µ(2) + ∆µ (X) i j ( X) + ... ij     2 - body HI. -4  O(r ) : i ≠ j  -7  O(r ) : i = j. 3 - body HI. N   ∆ μ (X) = µ 0  δi j ∑ ω11 ( R i p ) + (1 − δi j ) ω12 ( R i j )   p≠i  (2) ij.  Long - distance multipole expansion of 2 - body HI :. pairwise - additive. 3. 1a 3a     + O ( r −7 ) ω12 (r ) =   1 + r r  +   1 − 3 rr   2 r       r 4         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 semi-dilute charge - stabilized dispersions ! 60.

(61) Two - spheres translational mobilities in infinite fluid • axial symmetry and isotropy ω12 = ω21. ω12   F1   V1   1 + ω11 = µ 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 )= + δi j 1 x i j (r) rr • known recursion relations for (a / r) expansion & lubrication corrections −F F F. −F. ∆V12= 2 ( x11 − x12 ) µ0 F. F. Vi = Vsed = ( y11 + y12 ) µ0 F Vsed = ( x11 + x12 ) µ0 F. F. ∆V12= 2 ( y11 − y12 ) µ0 F. Jeffrey & Onishi, J. Fluid Mech. 139 (1984) Jones & Schmitz, Physica A 149 (1988) 61.

(62) ∆V12 V0. ξ = r /a − 2 = s− 2 • Lubrication important for relative pair motion close to contact • Not probed for electrically repelling colloids. 62.

(63) Content 1.. Introduction & Motivation. 2.. Low Reynolds number flow. 3.. Salient static properties. 4.. Electrophoresis of macroions. 5.. Dynamics of interacting Brownian particles. 6.. Short - time colloidal dynamics. 7.. Long - time colloidal dynamics. 8.. Primitive model electrokinetics. 1.

(64) 3. Salient static properties - Pair distribution function - Methods of calculation - Ionic mixtures - Effective colloid interactions - Poisson-Boltzmann theory of microions - Force on colloidal particle in electrolyte. 2.

(65) 3.1 Pair distribution function •. g(r) = cond. probability of finding another particle at distance r. Correlation length ξ(T). g (r ). 1. r/σ 1. neighbor shell. 2. neighbor shell. g (| r − = r ' |) lim V 2 δ ( r − R1 ) δ ( r '− R 2 ). g(r) = lim. N(r, r + ∆r). ∞. ideal gas. ρ 4 π r ∆r 2. ∞. eq. →1. homogeneous isotropic. fluid near-field order 3. eq.

(66) Relation to colloid scattering experiments: Static structure factor S(q) ki. laser. q. R j (t). ϑ. kf. 2π / q photomultiplier. I(q) ∝ N P(q)S(q ) : for single and quasi-elastic scattering q =. {. }. 1 N S(q) = lim ∑ exp i q ⋅ R l (0) − R j (0) ∞ N l , j =1. g(r) = 1 + FT −1  . 4π ϑ sin    2  λ. = 1 + n ∫ dr exp {i q ⋅ r}[ g(r) − 1] eq. S(q) − 1 n . 4.

(67) g(r) and S(q) for dispersion of Yukawa colloidal spheres q m ≈ 2π / rm. φ=. rm. NVsphere V.  ∂n  lim S ( q ) = k BT   ∂Π q →0  coll T,µs ,{µ α ≠ µcoll }. particle cage. rm 5.

(68) 3.2 Methods of calculation •. Introduce total correlation function : h(r = 12 ) : g(r12 ) − 1. •. Define direct correlation function c(r) through Ornstein-Zernike equation :. h= ( r12 ) c ( r12 ) + n ∫ dr3 c(r13 ) h ( r23 ) total correlations of 1 and 2. direct correlations. indirect correlations of 1 and 2 through particles 3,4,…. h(r12 ) = c(r12 ) + n ∫ dr3 c(r13 ) c(r23 ) + n 2 ∫ dr3 dr4 c(r13 ) c(r24 ) c(r34 ) + 0(c4 ). •. General properties of c(r) :. c(r ) = −β u(r), r → ∞. valid for all densities. 6.

(69) Important closure relations c(r) = F[u(r),h(r)] • Rescaled mean spherical approximation (RMSA) :. c(r) = −β u(r), r > σeff > σ. + g(r = )= 0 σeff. • Hypernetted chain approximation (HNC) (energy and virial routes give same pressure, g(r) > 0 is guaranteed). c(r) e − β u(r) ⋅ e γ (r) − γ (r) − 1 =. γ (r) := h(r) − c(r). • Percus - Yevick approximation (PY). = c(r) e − β u(r) ⋅ [1 + γ (r)] − γ (r) − 1 • Rogers - Young mixing scheme (RY): thermodynamically partially self-consistent.  exp {γ (r) f (r)} − 1 c(r ) e −β u(r) ⋅ 1 + =  − γ (r ) − 1 f (r)  . χTVirial = χTCompr. determines α. f (r) = 1 − e − α r. α→ ∞:. HNC. α→0 :. PY 7.

(70) Performance check versus MC simulation Screened Coulomb potential:. g(r).  RY hybrid scheme performs best but is numerically most costly 8.

(71) New analytic method to calculate static structure factor SLS on charged silica spheres in 80:20 toluene/ethanol charge-stabilized silica spheres. MPB - RMSA. S(q). qσ MPB - RMSA: scheme close to experiment, simulation & RY scheme highly efficient Heinen, Holmqvist, Banchio & Nägele, J. Chem. Phys. 134, 044532 & 129901 (2011). 9. 9.

(72) Universal phase diagram for Yukawa – type charge-stabilized spheres 1,2. r = n −1/3. 1,0. T =. k BT. u el ( r ). 0,8 0,6 0,4. fluid. fcc. Freezing line. Sf (q m ) ≈ 3.2. 0,2. Hansen - Verlet. bcc. 0,0 0. 5. 10. λ =λκ r.  λ ) single phase point  {Z, n, ns ,a, L B }→ ( T,.  Discuss dynamics in fluid regime only !. 15. 20. 25. Gapinski, Patkovski, Nägele J. Chem. Phys. 136 (2012). 10.

(73) 3.3 Ionic mixtures Primitive model of ionic systems. •. Macroions and microions treated as uniformly charged hard spheres. •. Colloidal electrokinetics & electrolyte transport HS u αβ (r) = u αβ (r) + u Cαβ (r). u Cαβ (r) k BT m. ∑n α=1. = LB. z αzβ r. (. , r > a α + aβ. ). z =0. α α. •. no polarization. •. structureless Newtonian solvent. water at 20o C : L B = 0.71nm. η0 = 1 × 10−3 Pa ⋅ s 11.

(74) PM equilibrium pair distribution functions •. Cond. probability of finding ion of type β at distance r from ion of type 𝛼. g αβ (| r − r ' |= 1 ) h αβ + =. •. )(. (. h α β (r) = cα β (r) + ∑ n γ ∫ dr cα γ (| r − r ' |) hβ γ (r ') γ =1. cα β (r  ζ(T)) = − u α β (r) / k BT. Fourier - transformed OZ m×m matrix equation:. 1 [1 + H(q)] ⋅ [1 − C(q)] =. •. Rβ2. R1α. m-component Ornstein - Zernike equations: m. •. ). 1 lim V 2 δ r − R1α δ r '− Rβ2 eq n α nβ ∞. (. Cαβ (q) = n α nβ. ). 1/2. c αβ (q). Symmetric matrix S(q) of partial static structure factors: S(q)=. (. Sαβ (q) = δαβ + n α nβ. (. H αβ (q) = n α nβ. ). 1/2. h αβ (q). [1 − C(q)] −1. ) ∫ dr exp{i q ⋅ r} h αβ (r) 1/2. 12.

(75) Exact local electroneutrallity condition. ( n α nβ ). 1/2. Cαβ= (q). C(s) αβ (q). − 4 πL B. q. z α zβ. 2. 1 4π FT 3D   = 2 r q. short-range correlations, regular function at q = 0. •. From regular expansion of H(q) and C(s)(q) at q = 0:. N (elα) ( R). m. R. γ=1. 0. α. = ∑ n γ z γ 4 π ∫ dr r 2  h αγ (r) + 1. N (elα) (R → ∞) = − z α. ( α =1, ..., m ). mean charge number in sphere of radius R. •. It follows for binary ionic mixture:. = z1 S11 (0) z= 2 S22 (0). ( z1z1 ). 1/2. S12 (0) 13.

(76) g αβ (r). r × ( nT ). 1/3. HNC calculations, courtesy by: Marco Heinen, Düsseldorf University 14.

(77) ( z1z1 ). 1/2. = z1 S11 (0) z= 2 S22 (0). S12 (0). Sαβ (q). q / ( nT ). 1/3 15.

(78) ( α =1, ..., m ). N (elα) (r → ∞) = − z α. (1) N el ( r ) / z1. r × ( nT ). 1/3. 16.

(79) •. Charge-charge global structure factor for m - component PM. = δρel ( r ). ∑ z α ∑ δ (r − R αj ) − Nα. m. = α 1 =j 1. = SZZ (q). SZZ (q). 1 nT z 2. 1 z. 2. ρel. eq. 3 ∫ d (r − r ' ) exp{i q ⋅ (r − r ' )} δρel (r ) δρel (r ' ). m. ∑ ( x α xβ ). α,β=1. 1/2. z α zβ Sαβ (q) →. q2 κ. 2. eq. ( ). + O q4. Long - wavelength charge density fluctuations are offset since bulit-up of macroscopic electric field by thermal fluctuations is energetically too costly. 17.

(80) 3.4 Effective colloid pair potential. βu α β (r) = L B. •. Zα Zβ r. 2.  exp[ κ a]  exp[ −κ r] β u eff (r) = L B Z2   r  1+ κa . Eff. macroion - macroion potential from averaging over microionic degrees of freedom. 18.

(81) Effective macroion potential m. h α β (r) = cα β (r) + ∑ n γ ∫ dr cα γ (| r − r ' |) hβ γ (r ') γ=1. a r. h= ceff (r) + n c ∫ dr ' ceff (| r − r ' |) h cc (r ') cc (r). u eff (r). ceff (r  a α + aβ ) = −. k BT. effective macroion direct corr. function. {gcc , gc + , gc − , g+ + , g− − , g+ − } g= h α β (r) + 1 α β (r). h cc (r;n c ) ⇔ u eff (r;n c ) exact correspondence cαMSA β (r) = −. u α β (r) kBT. , r > ( a α + a β ) MSA closure for all ionic components 2.  exp[ κ a]  exp[ −κ r] = L BZ   r k BT  1+ κa . u eff (r ). 2. ( r > 2a). ( nc → 0) 19.

(82) 3.5 Poisson-Boltzmann theory of microions • Colloids: N spatially fixed dielectric spheres in infinite. ε. m-component electrolyte act as static external potential for small mobile micoions (m components). εp. • Born – Oppenheimer picture of microions • Apply equil. density functional theory (DFT) to grand. n. free energy functional:. Ω {ρα };X = A {ρα } + A id. A. id. m. k BT ∑. α=1. A MF =. 1 3 d r ∫ 2. Vαex (r;X). MF. {ρα } + A. (. ). 3 ln Λ 3α ρα (r − 1 d r ( ) ρ r α ∫  . 3 d ∫ r'. z αe =ψ. ex. ρel (r )ρel (r ') ε r −r'. (r;X) + Vsex r (r;X ). corr. m. {ρα } − ∑. α=1. 3 ex µ ∞ d r ( ) V (r;X)  ρ − r α α α ∫  . • Microion charge density trial function:. ρel (r )=. m. ∑ z α e ρ α (r ). α=1. • electric potential due to fixed macroions:. ρex el (r '; X) ψ (r, X) = d r ' ∫ εp r − r ' ex. 3. colloid‘s excluded volumes 20.

(83) Mermin variational principle (1965): equilibrium microion profiles minimize Ω. •. Ω {n α + δn α } − Ω {n α } = 0 + ( δn α ). •. 2. Ω {n α };X  = Ωeq {n α };X . Neglect short-range microion electro-steric correlations: A. corr. =0. → Euler-Lagrange eqns. for „electro - chemical“ microion potentials: = µ∞ k BT ln  Λ 3α n α (r;X)  + z αeψ(r;X) + Vsrex (r;X) α  . = ψ(r;X). 3 ∫d r'. n el (r '; X) + ψex (r;X) • total equil. mean electric potential ε r −r'. n α (r;X) = n α∞ exp {− z αeψ(r;X) / (k BT)}. •. field-free electrolyte reservoir. ( r ∈ Vfl ). = n∞ n α (r → ∞ ) α. {. }. =µ exp α∞ / k BT / Λ 3α. Microions treated as inhomog. ideal gas of point ions (w/r to entropy) except for Coulomb forces. Each microion exists indep. in mean field of others 21.

(84) •. Using ∆ r (1 / r − r ' ) = −4 πδ ( r − r ' ) → MF Poisson-Boltzmann equation for ψ. 4π m ∞ ∆ψ(r;X) = − ∑ n α z αe exp{−z αeψ(r;X) / ( k BT )} ε α=1. •. ψ ( r → ∞ ) =0. Electric BCs for constant surface charge densities on N spheres:. ( ε∇ψ − εp∇ψ ) ⋅ n = •. ( r ∈ Vfl ( X ) ). −4 πσi. (r ∈ Si , εp  ε for water ). Standard case: q-q electrolyte:. ∆φ(r;X) = κ 2 sinh [φ(r;X) ]. { }. PB ( Ωeq. PB,res n ± ;X) − Ωeq =. φ(r ) =qeψ(r ) / (k BT). 1 N σi ∫ dS φ(r ) + k BTcs ∑ 2qe i =1 Si. PB,res Ωeq = −2k BTc × Vfl = − p res Vfl. ∫. κ 2 = 8 π L B cs q 2. d 3r [φ sinh ( φ) − 2cosh ( φ) − 2]. Vfl. linear Debye - Hückel expression 22.

(85) 3.6 Force on colloidal particle in electrolyte. S*i. ε. •. N-1 colloids fixed and microions equilibrated. •. Small displacement of colloid i:. dΩeq (X) = −SdT − pdVfl + ∑ N α dµ α − Fi ⋅ dR i α.  ∂ Ωeq (X)  Fi (X) = −    ∂ R i T,{µα }. Ri. •. Start alternatively from mesoscopic electrolyte solution Stokes equation:. − ∇p(r;X) + η0 ∆u(r;X) + ρel (r;X) E(r;X) = 0 ∆ψ(r;X) = −. •. 4π ρel (r;X) ε. ( r ∈ Vfl ). ( r ∈ Vfl and E = −∇ψ ) total local electric field microion mean charge density. Introduce hydrodynamic and Maxwell fluid stress tensors (no electrostriction) 23.

(86) •. Hydrodynamic and Maxwell fluid stress tensors of incompressible electrolyte fluid T σ h (r ) = − p(r ) 1 + η0 ∇u + ( ∇u )   . = σ el (r ). 1 2  ε  EE − E 1  4π  2 . h σαβ (r ) = − p(r ) δαβ + η0 ∂ α uβ (r ) + ∂β u α (r ) . = σel αβ (r ). ε  1 2  E ( r )E ( r ) − E ( r ) δ α β αβ  4 π  2. ∇ ⋅ σ h (r ) + σ el (r )  = −∇p(r ) + η0 ∆u(r ) + ρel (r )E(r ) = 0  . • Stokes equation. = σ(r ) σ h (r ) + σ el (r ) FT = (r;X) ⋅ n(r ) = dS σ(r;X) ⋅ n(r ) = − Fe ∫+ dS σ ∫   * Si. Si. fluid force / area on sphere surface element dS at r exerted by surrounding charged fluid. n(r ). σ (r ) ⋅ n(r ). Si+. 24.

(87) Force between two symmetric plates at static equilibrium (u = 0) n = − xˆ. • E(x) = −ψ '(x) xˆ. ∞ Fe. FR. •= − FL FR F= ˆ − Fe = Rx. n = xˆ. x -d. •. 0. d. • E(0) = 0 ( EN ). xˆ ⋅ σ(x) ⋅ xˆ =− p(x) +. ε ψ '(x)2 8π. FR = − σ xx (x) + σ res xx A ε =  p(x) − ψ '(x)2  − p res 8π  . For 1-dim. geometry only follows indeed from static Stokes and Poisson eqs:. p(x) −. ε Sum of hydrostatic (osmotic) and Maxwell pressures is E(x)2 = const 8π constant inside two plates = static equilibrium, 25.

(88) •. Hydrostatic pressure p in 3D PB approximation:.     ∞ −∇p −  ∑ z αe n α exp {− z αeψ} ∇ψ = −∇  p − k BT ∑ n α (r )  = 0     α α      ρel. p(r ) = k BT ∑ n α (r ) α. •.  ∞ p r→∞ = ) p= res k BT ∑ n α   ( α  . EDL force / area on right plane is midplane – reservoir osmotic pressure difference. FL = p(x= 0) − p res= k BT ∑ n ∞ α ( exp {− z αeβψ(0)} − 1)= A α. •. 2k BTcs {cosh [ φ(0) ] − 1}. Solution of linearized PB equation for q - q electrolyte for 0 < x < h/2: φ cosh ( κx ) sinh ( κh / 2 ). s φ(x) =. ε φ '(h / 2) = −ε φs κ = −4 π σ ×βqe φ '(0) = 0 26.

(89) FL (h) 8π σ 2 ≈ exp {−κ h} 2 A εq. •. • provided κ = h h / λ D >> 1 and φs  1. Associated effective pair potential energy of two plates:. h u(d) F (h ') Ωeq ( h ) − Ωeq ( ∞ ) = − ∫ dh ' R = A A A. change in grand free energy of two plates system. ∞. res Ωeq ( h ) − Ωeq ≈−. 1 [σ Lφ( − h / 2) + σ R φ(h / 2)] 2qe. in linearized PB approximation for q - q electrolyte. 27.

(90) 4. Electrophoresis of macroions - Hückel and Smoluchowski limits - Henry formula - Strongly charged macroion - Extension to concentrated systems. 28.

(91) 4.1 Hückel and Smoluchowski limits. 29.

(92) Non - conducting sphere with constant uniform surface potential or charge in electrolyte solution and exposed to constant external electric field. E∞. +. (. Vel = µel E∞ + O E∞. _ •. Vel. a. oppositely charged diffuse layer. ). Hückel limit κa ≪ 1. Fel0= + Fh0 Ze E ∞ − 6πη0= aVel0 0. Ze < 0. λD = 1 / κ. 2. Ze 6πη0a. µ0el =. •. electrokinetic effects for non-dilute diffuse layer only. •. Retarding electro-osmotic drag by cross-streaming of oppositely charged fluid near sphere surface. •. Retarding relaxation effect force due to distortion of EDL away from spherical symmetry. Restructuring by microion diffusion and conduction, and by solvent convection is non-instantaneous 30.

(93) Unperturbed spherical equilibrium EDL • Non-linear PB boundary value problem for infinite q-q electrolyte. ∆φ(r) = φ ''(r) + 2φ '(r) / r = κ 2 sinh ( φ(r) ) , φ '(a) = − E(a) = −LB. Zq a. Ze a ψs = εr r. εζ 6 π η0. µ0el =. κ 2 = 8πL Bcs q 2. φ(r → ∞) = 0 = φ '(r → ∞). 2. • Coulomb potential for κa → 0. ψ(r ≥ a) =. φ(r) = βqeψ(r). Ze εa. ψs =ψ(a) =. r. size –indep. Hückel mobility linear in zeta potential identified as. ζ = ψs. • Long-distance exponential decay for κa > 0:. ∆ψ(r → " ∞ ")  κ 2ψ(r). ψ(r → " ∞ ")  A exp {−κ r} / r. factor A(κa, LBZq/a) comes from numerical solution. 31.

(94) Helmholtz planes and zeta potential. Ze < 0. r. ζ − potential ψs0. •. ψ0i. ψ0d. Zeta potential = EDL potential at zero shear surface where fluid starts to move relative to particle surface Courtesy: Rafael Roa, ICS-3, Jülich 32.

(95) Debye – Hückel equilibrium EDL • Linearization is everywhere allowed only if |LBZq/a |<<1. One can then use inner BC:. a ψs ψ DH (r ≥ a;Z) =. exp {−κ ( r − a )} r. Ze =ε a (1 + κ a ) ψs + O(ψs ) 2.  Z e  exp {−κ ( r − a )} = 1+ κa  εr  . DH ρel (r). ε κ2 = − ψ DH (r) 4π. ( ). ψ '(a) = −charge / ε a 2. ψ DH ( r;Z ). - DH overestimates strength of electric repulsion. ψ ( r;Z ). - Effective macroion charge from long-distance matching (all derivatives). ψ DH ( r;Zeff ). a. r. Zeff ≤ Z. 33.

(96) Analytic example: ultra - thin EDL • Can use analytic 1-dim flat plane solution: Zeff.  a  = Zsat eff  L q  κa  B . φsat s = φ ( r = a;Z → ∞ ) = 4 nsat c − ions = n c − ions (r = a;Z → ∞ ) = ∞. Z. non - physical.  1  ( L Bq / a ) Z   L Bq Zeff ( κa  1) = 4 ( κa ) tanh  ar sinh   a 2 2 a κ     Zsat eff ( φ= 0; κa > 1)=. •. a   1  6 4 a O + κ +   L Bq   κa  . Effective macroion charge saturation is a mean-field feature for uncorrelated point-like (monovalent) microions. It fails when microion steric effects matter. 34.

(97) Smoluchowski limit and flat plane electro-osmosis. Hückel limit (1920‘s). Helmholtz – Smoluchowski limit (1879 & 1903). nn==rˆrˆ. E∞ = const. κ a << 1, λ D >> a • field within EDL basically undistorted. κ a >> 1, λ D << a. Ε0 (r ). • field within ultrathin EDL tangentially curved around non – conducting sphere. E0 ( r ∈ EDL ) ∝ (1 − n n ) ⋅ E∞. •. Electro-osmosis: flow of charged electrolyte fluid past stationary (particle) surface. 35.

(98) • Apply Stokes equation with el. body force inside ulltrathin EDL. All vectors aligned with x – axis. E0 = E0 xˆ. us. y. Sδ. •. (Locally) constant incident electric field. Φ 0 (x) = − x E0. δ. - No external pressure gradient. ∇p = 0 λD. - BCs outside EDL. ψ EDL = 0= ψ ' EDL ( y → ∞ ) ζ ψ EDL. ε ε ∆ [ψ EDL (y) + Φ 0 ] = − ∆ ψ EDL (y) 4π 4π. −η0 ∆u(y) − ∇ p(y) + ρel (y) E0 = 0. ρel (y) = −.   εψ EDL (y) ∆  u(y) + E0  = 0 ⇒ 4 π η0  . u(y) = −. • On length scales ≫ λD :. ∇ u ( y → ∞ ) =0. ε [ζ − ψ EDL (y)] E0 4 π η0. εζ us = u(y  λ D ) = − E0 4 π η0. u(y= 0) = 0. effective slip velocity (flat plate rest frame) 36.

(99) Electro - osmotic plug flow in an open capillary tube glass wall (negatively charged) − − − − Anode. +. +. +. +. +. +. +. +. −. •. −. −. −. U. −. +. −. plug flow +. −. + +. −. −. +. E∞. Cathode. 2h. +. −. Electroosmosis used in microfluid devices to drive aqueous media through narrow micro - channels where Low-Reynolds number fluid dynamics applies to. 37.

(100) Open (a) versus closed (b) electro-osmotic cells. Sketch (not to scale). ∫ dS u = 0 A. •. 0. •. •. h. y. h / 3 if κ h  1. •. Absolute electrophoretic velocity measured in zero flow plane (Malvern Zeta-sizer) 38.

(101) Electro - flow problem outside thin EDL: matched asymptotic expansion • Stokes equation w/o body force (ϱel = 0) and slip inner velocity BC. Laplace equation for Φ0 • Spherical surface Sδ of radius a + 𝛿 with δ ≫ λD. −η0 ∆ u(r ) − ∇ p(r= ) 0 , ∇ ⋅= u 0 u =us + Vel + Ωel × r on Sδ. u → 0 for r → 0. Lab rest frame. • Zero electric force / torque on sphere + EDL:. = FT. h dS σ = ⋅n 0 ∫ Sδ. T T=. (. ). h r × σ ⋅ n= 0 dS ∫ Sδ. ⟹. 1 u → O  2  at least, for r → 0 r . ∆ Φ 0 ( r ) = 0 (source of Φ0 is source of E∞ ) n ⋅ ∇Φ 0 ( r ) = 0 on Sδ. ∇Φ 0 ( r ) → − E∞ for r → ∞. Neutral & non-conducting sphere. ⟹.  1  a 3  ˆ ˆ 1 1 3 E0 ( r ) = + − r r )  ⋅ E∞    (  2  r   Es ≈ E0 ( r = a rˆ ) =. 3 (1 − rˆ rˆ ) ⋅ E∞ ⊥ rˆ 2. 3ε ζ us = − (1 − rˆ rˆ )⋅ E∞ 8π. BC on Sδ 39.

(102) • Inspired guess: assume that outer flow field is given by „extension of BC on Sδ. u (r ) = −. εζ E0 ( r ) + Vel 4 π η0. Check if ok:. ∆u = = ∇ ⋅u. ( r > a + δ). irrotational potential flow ansatz. εζ ∇ ( ∆Φ 0= p 0 ) 0 ⇒ ∇= 4 π η0 εζ = ∆Φ 0 0 4 π η0. constant pressure. 1 u ( r → " ∞ " ) =O  3  r . This is unique solution of outer BVP, giving the electrophoretic velocity for fixed zeta potential:. = VelSm. µSm el =. εζ εζ E= ( r → ∞ ) E∞ 0 4 π η0 4 π η0. E∞. εζ 3 0 = µel 4 π η0 2. • Tangential electric field strength at S is 1.5 × E∞. 40.

(103) Streamlines of dipolar potential flow outside the EDL. u ( r ) = −µSm el [ E∞ − E0 ( r )] 3. ( r > a + δ). 1a  u ( r ) = −µSm   [1 − 3rˆ rˆ ] ⋅ E∞ = −∇ Ψ p ( r ) el 2 r . Ψ= p (r ). Sm. E∞. Vel. 3. µ el  a    r ⋅ E∞ 2 r. ∇ × u =0. • Flow decays faster than the hydrodynamic Stokes dipole of an active swimmer. 41.

(104) Electrophoresis of arbitrarily shaped rigid macroion with ultrathin EDL. = VelSm. εζ 4 π η0. • Outside BV part:. = E∞ , ΩSm 0 el. ∆ u (r ) = 0 n ⋅ u (r ) = 0 on Sδ. E∞. ∆ Φ0 (r ) = 0 n ⋅ ∇Φ 0 ( r ) = 0 on Sδ. Vel Sδ • • •. Smooth surface with curvature radii everywhere ≫ λD Constant surface (i.e. zeta) potential. • Φ 0 more complex, but still:. u ( r ) = −µSm el [ E∞ − E0 ( r )] • shape-indep. asymptotic form. No perpendicular charge conduction inside thin diffuse layer. •. External el. field homogeneous on size scale of macroion. •. PB – based mean-field result Keh & Anderson, J. Fluid Mech. 153 (1985); Teubner, J. Chem. Phys. 76 (1982). 42.

(105) 4.2 Henry formula • Electrophoresis of charged sphere with extended EDL. r. Fel + F h = 0 macroion and fluid are inertia-free. λD. Fel =∫ dS σ el ⋅ n =− ∫ d 3r ρel E electric force on sphere = - force on diffuse layer Sa Vfl. a u ( r → ∞ ) = − Vel.  a2  F = 6πη0a 1 + ∆  uin ( r = 0 ) hyd. drag force on stationary neutral sphere with stick BC 6  in incident flow field created by sources outside the sphere  h. uin ( r ) = − Vel +. ∫. d 3r ' T0 ( r − r ' ) ⋅ρel ( r ' ) E ( r ' ). Vfl. = 6πη0a Vel. ∫ d r U (r ) − 1 ⋅ρel (r ' ) E (r ) 3. Vfl. St. ε ∆ψ − 4π. ⟹.  a2  0 USt ( r ) = 6πη0a 1 + ∆  T ( r ) 6   total local electric field 43.

(106) E = −∇ψ = −∇ ( ψ EDL + Φ ). Φ ( r → ∞ ) = −r ⋅ E∞. Φ ∝ E∞. • Double linear expansion, in E∞ and in normalized zeta potential. ζ=. qeζ / ( k BT ). ψ EDL = ψeq EDL + O ( E ∞ ). (. 2 ψ EDL = ψeq,DH + O E , ζ ∞ EDL. ). Φ (r ) = Φ 0 ( r ) + O( ζ ) potential of neutral , non-conducting sphere subjected to E∞ ε H 6πη0a Vel = − ∫ d 3r  USt ( r ) − 1 ⋅ ∆ψeq,D r ) E0 (r ) + O(E 2∞ , ζ 2 ) ( E DL   4 π   Vfl. κ 2ψeq,DH EDL. −∇Φ 0. • To linear order in (small) zeta potential: no relaxation effect contribution. 44.

(107) • Result after integration is the Henry formula (1931) εζ fH ( κ a ) = µ0el × f H ( κ a ) 4 π η0. H µel =. valid for small and constant zeta potential or charge. For which DH theory applies to.. 1 4. f H (x) = 1 + x 2 exp {x}[ E3 (x) − E5 (x)]. fH ( κ a ). 𝜀𝜁 4 𝜋 𝜂0 relaxation. f H (0) = 1. f H ( ∞) =3 / 2. effect. κa • Macroion charge increases with increasing salinity for fixed potential. The increase of the bare electric force nearly counterbalanced by increased counterion electro-osmotic flow 45.

(108) Constant potential versus constant surface charge electric BC. Ze =ε a (1 + κ a ) ψs + O(ψs 2 ) H µel. H µel. ζ. 0 =µel × fH ( κ a ). chemical charge regulation. µ0el. H µel. Z. =µ0el ×. fH ( κ a ) 1+ κa. κa • Fixed macroion charge increasingly screened from external field with increasing salinity 46.

(109) 4.3 Strongly charged macroion • Stokes equation with electric body force • Poisson equation for total mean electric potential • Continuity equation for microion currents (1-1 electrolyte):. ( ∂ / ∂ t ) n α = 0 = ∇ ⋅ jα (r ) ( α = 1, 2 = ± ) • Nernst-Planck MF convection - el. migration - diffusion currents:. u ( r → ∞ ) = − Vel. = j± (r ) n ± (r ) u(r ) + n ± (r ) β D0± (  e∇ψ ) − D0± ∇ n ± (r ). = j± (r ) n ± (r ) u(r ) − β D0± n ± (r ) ∇ µ ± (r )  n (r )  µ ± (r ) =± e ψ ( r ) + k BT ln  ± ∞   n±  = n ± (r ) n ∞ ± exp {−β [ ± e ψ ( r ) − µ ± (r ) ]}. • relation to DDFT • MF electro-chemical potential • constant for zero external field; gives then PB equation for microions • outside the sphere 47.

(110) • All currents are zero w/o external field, and electrochemical potential is constant • Boundary conditions:. v ± (a rˆ ) ⋅ n = 0 = u(a rˆ ) ⋅ n. • insulating and solvent-impermeable sphere. ⟹. ∇µ ± (a rˆ ) ⋅ n = 0. ⟹. 𝛻µ± (𝐫 → ∞) = ±e 𝛻𝛻 𝐫 → ∞ = ∓ e 𝐄∞. n ± (r → ∞ ) = n ∞ ±. • also in presence of external field. • Field of electro-chemical potential (ECP) trangential to sphere surface • ECP is independent of electric BC on sphere surface • Expansion of all potentials and densities to linear order in external field leads to set of differential equations which can be solved numerically to obtain the el. mobility • Ohshima provides analytic expressions valid in certain salinity ranges. 48.

(111) µel red µel =H µel ( a = L B , Z = 1) red µel. Coulomb force no electrokinetics. e H µel 1) = ( LB , Z = 6πη0 L B. µel red. 3  ζ for κa  1 = 2   ζ for κa  1. red µel ( κa  1 finite, ζ → ∞ ) =2 log ( 2 ). ζ = e ζ / ( k B T ) • Mobility maximum: inhomogeneous conduction of counterions in thin diffuse layer . Assoc. slowing relaxation force grows faster with zeta potential than bare Coulomb force. H. Ohshima, „Theory of Colloida and Interfacial Phenomena“, Elsevier (2006). 49.

(112) red µel. Hückel. ζ = e ζ / ( k B T ). Smol.. ζ = e ζ / ( k B T ). • With increasing reduced zeta potential > 3, increasing salinity required to approach the Smoluchowski limit. O‘Brien and White, J. Chem. Soc. Faraday Trans. II 74 (1987). 50.

(113) Relaxation effect: counter - ion concentration profile and eletric dipole moment. Vel. figure taken from: F. Carrique et al., Langmuir 24, 2395 (2008). 51.

(114) red µel. ζ = e ζ / ( k B T ). • MF Implication: no 1 -1 correspondence between electrophoretic velocity and zeta potential. 52.

(115) Inclusion of ion steric effects on level of activity coefficients (Bikerman).  n ± (r )  µ ± (r ) =± e ψ ( r ) + k BT ln  ∞  − k B T ln 1 − n + (r ) a s3 − n − (r )a s3     n±  Smol.. red µel. Inceasing microion volume fraction ν reduces. „surface conduction“. κa = 50 ζ •. A.S. Khair and T. Squires, J. Fluid Mech. 640, 343 (2009). •. Poster 25 by Rafael Roa. 53.

(116) 4.4 Extension to concentrated systems. a φ =  b. 3. • MF elektrokinetic equations with standard inner BCs • Specify outer BCs on outer cell boundary r = b :. 1. zero vorticity (Kuwabara):. ∇ × u (r ) = 0. or: zero tangential hydrodynamic shear stress (Brenner). 2. unperturbed EDL field is zero at outer cell boundary (cell overall electroneutral) 54.

(117) Cell model extension of Henry formula (small zeta potential). 2 red × µel 3. κa • Strong mobility reduction by electro-osmotic drag for extended EDLs and increasing colloid particles volume fraction •. For larger (fixed) zeta potential: mobility decreases with increasing volume fraction Levine and Neal, JCIS 47 (1974); Ohshima, JCIS 188 (1997). 55.

(118) Principal drawbacks of the cell model (despite its success in exp. applications):. • It disregards fluid-like near-field ordering of the colloids • Selection of outer BCs is to some extent arbitrary • Wrong low - concentration prediction. The correct one is for 𝜅𝑎 ≫ 1:. ( ). 1 − 3 φ + O φ2  µel = µSm el    2. Chen and Keh, AICHe 34 (1988); Ennis and White, J. Colloid Interface Sci. 185 (1997). 56.

(119) 1 3 d r u ( r;X ) = 0 ∫ V V. 1 3 d r [ E ( r;X ) − E∞ ] = 0 ∫ V V. • Define particle velocity in bounded suspension relative to frame where the (particle plus fluid) velocity, averaged over whole suspension volume, is zero. 57.

(120) Content 1.. Introduction & Motivation. 2.. Low Reynolds number flow. 3.. Salient static properties. 4.. Electrophoresis of macroions. 5.. Dynamics of interacting Brownian particles. 6.. Short-time colloidal dynamics. 7.. Long - time colloidal dynamics. 8.. Primitive model electrokinetics. 1.

(121) 5. Dynamics of interacting Brownian particles - Many-particle diffusion equation - Dynamic simulations. 2.

(122) 5.1 Many - particles diffusion equation. 3.

(123) Brownian force drives diffusion:.  Probability conservation of configurational pdf:. 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. 0 = FiI + Fie + Fih + FiB. t  τB. 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 generalized Smoluchowski equation (Kirkwood & Riseman) N ∂ = P(X, t) k BT ∑ ∇i ⋅ μittj (X) ⋅ ∇ j − β FjI − β Fje  P(X, t)   ∂t i, j = 1. Brownian motion ∝ T. P(X, t → ∞) → Peq (X) ∝ exp [ −β VN (X)] 4. ).

(124) 5.2 Dynamic simulations . Discretized postional many – particle Langevin equation; N. N. R i ( t= R i ( t 0 ) + ∑ μit tj ( X 0 ) ⋅ Fj ( X 0 ) + k BT ∇ j ⋅ μit tj ( X 0 )  τ + 2 τ ∑ d i j ( X 0 ) ⋅ n j + o ( τ ) 0 + τ)   j =1 j =1 DI & external. . Square - root mobility matrix d in random displacement:. ∆R i (= τ ) ∆R j ( τ ) 2 k BT μit tj ( X 0 ) τ + o ( τ ) 0. . near-field HI (hydrodyn. drift part). central Gaussian displacement. k BT μ t t= ( X 0 ) d ( X 0 ) ⋅ dT ( X 0 ). HI - coupling of displacements of i & j. (accelerated) Stokesian dynamics simulation method for Brownian particles. 5.

(125) 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. = ∫ d r exp ( iq ⋅ r ) ∑ δ ( r − R l ) f= (X) g(X) 3.  Microscopic density fluctuations:. l =1.  Dynamic structure factor measured in dynamic light scattering. S(q, t) lim = ∞. 1. ∑ exp {iq ⋅  R l (t) − R p (0) } N. N l, p = 1. eq. 6.

(126) 6. Short - time colloidal dynamics D0 t. - Hydrodynamic function. W(t). DS t. DLt. - Sedimentation - High - frequency viscosity t. - A simple BSA solution model - Generalized SE relations. τD = a 2 / D0 HI. HI & DI. short - time resolution: τB ≪ t ≪ τD. 7.

(127) 6.1 Hydrodynamic function •. Dynamic structure factor S(q,t) is measured in dynamic scattering experiment:. S(q, t  τ D ) ≈ S(q) exp  −q 2 D(q)t . ki laser. q kf. H(q) D(q) = D0 S(q). H(q ) = lim ∞. 1 N µ 0t. short - time diffusion function. N. ∑. p, j = 1. ϑ. photomultiplier. qˆ ⋅ μ pt tj (X) ⋅ qˆ exp[i q ⋅ ( R p − R j )] eq. 2π / q. H(q) = 1 without HI 8.

(128) DS / D 0. related to cage diffusion self - diffusion. 0 Vsed / Vsed. Vsed < V. 0 sed. sedimentation. HS : φ =0.33mm. 9.

(129) Physical meaning: generalized sedimentation coefficient •. Homogeneous system with spatially periodic force acting on each sphere (linear response):.  Fj = q F e exp  i q ⋅ R j  V(q). st. lim ∞. V(q) = st. 1 N.  q ∑ ⋅ Vj exp i q ⋅ R j  N. j=1. st. mean (short - time) response st. Vse0 d =. H(q) µ 0 Fe.  Vsed lim  V(q) = q→0  . weak external force on sphere j. − qˆ ⋅ lim ∞. µ 0 Fe.  q.  3 d r exp i ( ; X) q r u r ⋅ { } susp  V∫ 1. suspension velocity field. ∇ ⋅ ususp = 0 ⇒ q ⋅ ususp ( q; X= ) 0. . Lab frame = zero volume flux frame. . (Short-time) sedimentation velocity in zero - volume flux reference system 10.

(130) Employed methods of Calculation  Accelerated Stokesian Dynamics (ASD) simulations (Banchio & Brady, J. Chem. Phys., 2003) - extended to Yukawa – type charged colloids - lubrication effects disregarded  δγ method (Beenakker & Mazur, 1984) with self - part correction - truncated expansion in renormalized density fluctuations δγ - approximate inclusion of many-body HI only - lubrication effects disregarded  Pairwise - additive HI approximation for charged particles - full inclusion of 2-body HI (tables by Jeffrey) - positive definiteness of mobility matrix not guaranteed. only input: S(q). - „exact“ to first order in concentration only  Rotne - Prager far- field hydrodynamic mobility tensors - positive definiteness of hydrodynamic mobility matrix guaranteed - well-suited for charge-stabilized colloids at lower salinity 11.

(131) H(q) for Apoferritin protein solution (Yukawa - particle model). DS from ASD 13 nm symbols: ASD simulation dashed : δγ - theory solid: self-part corr. δγ -theory. = H(q) Ds / D0 + H d (q) Input in self-part corrected δγ - theory:. ASD for φ > 0.1 PA - theory for φ < 0.1. Nägele, Banchio, Pecora, Patkowski et al JCP 123 (2005). • self-part corrected Beenakker – Mazur theory works well for distinct part of H(q) 12.

(132) • self-part corrected Beenakker – Mazur theory works well for distinct part of H(q) (empirical finding for all ASD-simulated hard-sphere + Yukawa systems!) • Critical assessment and improvements over the Beenakker - Mazur theory: Karol Makuch and B. Cichocki, J. Chem. Phys. 137 (2012)  Poster 18. 13.

(133) Apoferritin : NSE experiment versus theory and simulation. d0D/0D(q) (NSE+SAXS) / D(q) (NSE) d0D/0 D(q) (ASD-theory) / D(q) (ASD-theory). D0 / D(q). D(q) = D0. H(q ) S(q ). • Reasonably good agreement between experiment and theory / simulation Nägele, Banchio, Pecora, Patkowski et al JCP 123 (2005). 14.

(134) Large charged colloidal spheres: Experiment and theory. H(q). DS. qσ good for low φ only, disregards HI shielding (µtt not positive definite). . Pairwise additive HI:. . Self-part 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. 109 (2012). 15.

(135) Hydrodynamic function peak height at freezing (Yukawa + HS potential) 1,4. H HS ( q m ) = 1 − 1.35 φ. limiting freezing line. 1,2. fluid phase. Η(qm). 1,0. S(q). hard-sphere Sf(qm) = 3.1 Sf(qm) = 2.85. 0,8 0,6 -4. 10. -3. 10. φ. -2. 10. 1. -1. 10. qm. . Universal fluid - phase area is insensitive to Hansen - Verlet criterion value. . Provides map of attainable peak values of H(q) in fluid phase state Gapinski, Patkowski & Nägele, J. Chem. Phys. 132, 054510 (2010). q.

(136) Comprehensive XPCS study of charge-stabilized poly-acrylate sphere suspensions. H(q m ). Westermeier, Heinen, Nägele et al.. J. Chem. Phys. 137 (2012) 17.

(137) 6.2 Sedimentation. gravity. Vsed 0 Vsed. HS. 0 Vsed / Vsed ≈ 1 − 1.8 φ1/3. • this is not a crystal !. salt CS. φ. . Slower sedimentation of charged clay particles (river - delta) ASD:: Banchio & Nägele, JCP 127, 2008. Watzlawek & Nägele; JCIS 214 (1997) 18.

(138) Plausibility argument. ∇ p. sa. = n Fe. Pressure gradient drives homogeneous mean fluid backflow. g(r) neutral.. charged. 1.  Increased friction with backflowing fluid. rm ∝ φ−1/3. r 19.

(139) 6.3 High – frequency viscosity ■ Macroscopic steady-state shear stress. Σ xy = σ xy (r; X). y. st. =. Fx. A ■ Effective suspension viscosity. =η. d ( u∞ )x dy. = γ η. x 1. H + η( γ ) =η∞ + ∆η = Σ xy. u ∞ = γ y xˆ = ... st. γ. ∫ dX P (X; γ )... st. (Σ γ 1. shear relaxation contribution. ■ High - frequency viscosity part (HI only):. η∞ (γ ) = η0 +.  µ t t ( X) µ t d (X)   Fh =  V − e∞ ⋅ X  −(FI + F B )  − ⋅   = H dt dd   − e − S µ (X) µ (X)    ∞    ■ Strain-flow part: e ∞ ⋅ r =γ [ y xˆ + x yˆ ] / 2. n H Sxy (X) γ. I xy. B + Σ xy ). DI. st,ren. symmetric force dipole (stresslet).

(140) ■ Shear – relaxation viscosity part:. ∆η(γ ) = −. (. n c I B V ⋅ F + F i i i γ 2. ) st,ren. =. Σ ( γ 1. I xy. B + Σ xy ). FiI + FiB = −∇i VN (X) − k BT ∇i ln Pst (X) ∝ γ. Vic (X)= 1 ⊗ R i + μit d (X)  : e ∞ ∝ γ. Convective velocity (particle force- and torque free). HIs: 3rd rank shear mobility tensor of particle i. ■ Shear – Péclet number:. Pe =. diffusion time. =. flow time. τD. a 2 / D0 = ∝ γ a 3 τ γ 1 / γ. 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) → Green-Kubo relation and MCT for mixtures 21.

(141) High-frequency viscosity of no - slip Brownian hard spheres. ω  ( τD ). −1. Pe = γ τ D << 1. η∞ ( γ → 0 ) η0 Einstein 1905: 1.0 1911: 2.5 (Hopf). η∞ η0. = 1 + 2.5 φ + 5.0 φ2 + 9.1 φ3 + ... Batchelor & Green (1972). Cichocki Ekiel-Jezewska Wajnryb. (2003). φ.  Virial expansion in volume fraction applicable to lower concentrations only.