The Micromechanics of Colloidal Dispersions

Transcription

1 The Micromechanics of Colloidal Dispersions John F. Brady Divisions of Chemistry & Chemical Engineering and Engineering & Applied Science California Institute of Technology Pasadena, CA 91125, USA Multiscale Modeling and Simulation of Complex Fluids University of Maryland 13 April 2007 Complex Fluids/Complex Flows S. Quake Page #

2 Some Examples/Applications Food stuffs & additives Personal care products Biological fluids & cells Ceramics, colored glass MR/ER fluids Resins, catalysts Paints, coatings, inks Nanowriting Biological Fluids Actin network Weitz lab., Harvard. Red blood cells Microangela EM gallery Macrophage Microangela EM gallery Liz Jones (2002) Page #

3 Swimming Eutreptiella flagellate Molecular Biology of the Cell, 4th Edition, by Alerts, Johnson, Lewis. Raff, Roberts, Walter Propelling Listeria Bacteria Propels itself by enzymatic synthesis of actin -- the comet tail Page #

4 Colloid science & microfluidics Electrophoresis E Electrophoresis of DNA U U = E J. Han and H.G. Craighead, Cornell University Nonliving -- nanomotors Catalytic nanomotor The mechanism of selfpropulsion is unknown. Some candidates: surface tension gradients caused by the catalytic reaction on the Pt surface, electrochemical flows between Pt and Au, etc. Paxton et al. (2004) 10 mm Janus Page #

5 Autonomous Motion or Science Fiction? Design or construct objects at the micro-, nano- or molecular scales that can move themselves. Have truly portable devices (e.g. sensors, drug delivery, lab-on-a-chip). Learn something about biological systems. surgeon nanobot Erik Viktor Tirumkudulu et al (1999) Shinbrot & Muzzio (2000) Pattern Formation Zoueshtiagh & Thomas (2000) Fluidized bed (Jackson 2000) Page #

6 Length and Time Scales of Complex Fluids Method Time Scale Granular Dynamics Granular Media 1hr Stokesian Dynamics Photo by Y. Monovoukas Colloids Suspensions 1s Brownian Dynamics Polymers 1ns 1nm 1 μm 1mm Size Scale Particle Size Scale 1m mm μm nm Å } } Simulation Method Granular Dynamics (St >>1) Bubble Dynamics ( u = 0) Stokesian Dynamics (Re << 1) Re = Ua Re = Ua St = p Re <1, f Pe = arbitrary D Re Molecular Dynamics N s N p N s N p ~ a p a s 3, p ~ a p s a s 3, CPU~ a p a s 6 N p Page #

7 Characteristic Scales: A Simple Example Spherical particle of 0.5μm of specific gravity 2 falling in water. a g Particle Size : a = 1 2 μm Fall Speed : U = 1 2 μm/s inertial viscous advection diffusion Re = Ua Pe = Ua D Reynolds Number : Re = Diffusivity : D = 1 ( μm 2 )2 /s Peclet Number : Pe = 1 2 Stokes - Einstein - Sutherland Relation : D = ktr 1 = kt 6a Micromechanics Continuum Approximation: a p >> a s N s ~ a p ( a s ) 3 N p a p s ~ a s 3kT m s s ~ a s 2, = p s ~ ( a p a s ) 2 Therefore, the solvent can be treated as a continuum : Re = Ua <<1 Du Dt = p 2 u, u = 0 Page #

8 Micromechanics (Re << 1) Langevin equation for particle motion: m du dt = F H F B F P Hydrodynamic: F H =R FU U =6aU Stokes drag Multiparticle: F H =R FU ( x) ( U U ) p ~ O(m /6a) 10 8 s Fluid Motion: Stokes Equations 0 = p 2 u u = 0 u = U x no slip at particle surfaces Micromechanics (Re << 1) Langevin equation: Hydrodynamic: m du dt = F H F B F P ( ) F H =R FU ( x) U U Stokes drag p ~ O(m /6a) 10 8 s Brownian: F B = 0, F B 0 ()F B ()= t 2kTR FU x ( ) t O(10 13 s) () s << p x 2 2D 1 D = ktr FU = kt 6a (D. Weitz) t Mean-square displacement Page #

9 Micromechanics (Re << 1) Particle Motion: Hydrodynamic: Brownian: Interparicle/ external: m du dt = F H F B F P ( ) F H =Rx ( ) U U Stokes drag F B = 0, F B 0 ()F B ()= t 2kTR( x)() t O(10 13 s) F P = V p g, electrostatic, etc. p ~ O(m /6a) 10 8 s Fluid Motion: Stokes Equations 0 = p 2 u u = 0 u = U x no slip at particle surfaces Micromechanics (Re << 1) Particle Motion: Hydrodynamic: Brownian: Displacement in momentum relaxation time m du dt = F H F B F P ( ) F H =Rx ( ) U U Stokes drag F B = 0, F B 0 ()F B ()= t 2kTR( x)() t O(10 13 s) x a = Re <<1 0 = F H F B F P p ~ O(m /6a) 10 8 s Only configurational degrees of freedom!! Page #

10 Interparticle forces Steric Stabilization V/kT b a b a hydrodynamic hard core excluded annulus interparticle hard core Electrostatic Stabilization 2a 2b r Nature of Hydrodynamic Forces: F H =-R(x) U Near Field Lubrication F/6a Total Far Field Many Body Interactions decay as 1 r (r-2)/a Page #

11 Stokesian Dynamics Method: O(N ln N) Split the hydrodynamic interactions into nearand far-field parts: F H = R U = R nf U R ff U F H = R U = R nf UR ff U F/6a Near Field Lubrication 2 Total 3 (r-2)/a 4 Far Field Many Body 5 6 Near field: Lubrication interactions are two-body effects and can be added pairwise. Calculations can be done in O(N) operations R nf = R nf R nf 2 =R 2B B nf Far field: Many-body effects are computed by representing the particles as force densities on a grid and using Fast Fourier Transforms (FFT) to compute the velocity field. The force is then computed via Faxen laws and determined iteratively (convergence is rapid after the initial time step). F ff H = R ff U F H ff = R ff U Hydrodynamic Interactions Lubrication: closely spaced particles move as a single (rigid) rod, whether you push or pull. Push Pull Many-body: point particles falling due to gravity have a negative fall speed at high concentrations. U/U Many-body/ multipole Point force Page #

12 Short-time self-diffusivity D s o /D o Sierou & Brady JFM (2001) ASD infinite limit Ladd (1990) infinite limit Segre et al. (1995) Van Megen & Underwood (1989) Vanveluwen & Lekkerkerker (1988) Ottewill & Williams (1987) Vanveluwen et al. (1987) Van Megen et al. (1986) Van Megen et al. (1985) Pusey & Van Megen (1983) = D s 0 ( ) = kt M eq Batchelor (ca. 1995) Dilute limit : 0 D s 0 ( ) ~ D 0 ( 11.83) Close packing : =1 rcp 0 ( ) ~ D 0 ln( 1 ) D 0 s Near Equilibrium Behavior: () Stokesian Dynamics (N=27-64) Phillips, et al (1989) Ladd (1990) Phung (1994) Accelerated Stokesian Dynamics N=125 N=343 N=512 N=1000 N=2000 Experimental Results van der Werff, et. al. (1989) Shikata & Pearson (1994) Asymptotic Form 6.5ln(1/) 0.17/ = (1 / rcp ) rcp = ª ! ~ as 0 ~ ln(1 m ) 1 as m 0 1 ª (1 = m ) 1 as! m Page #

13 High-frequency dynamic viscosity & short-time self-diffusivity () s () and 1/D High-frequency dynamic viscosity Phillips, Brady & Bossis (1989) Ladd (1990) Phung (1993) van der Werff, et. al. (1989) Shikata & Pearson (1994) Curve Fit Short-time self-diffusivity Phillips, Brady & Bossis (1988) Ladd(1990) Phung (1993) Pusey & van Megen (1983) van Megen et al. (1985) van Megen, Underwood & Snook (1985) Ottewill & Williams (1987) van Veluwen et al. (1987) van Veluwen & Lekkerkerker (1988) van Megen & Underwood (1989) Brownian Self-Diffusivity (long-time) D 0 = kt 6a D( ) D 0 Brady JFM (1994) g The selfdiffusivity decreases with increasing concentration as the diffusing particle must push past its neighbors to move. Fuchs et al (1992) ( ) 2.62 s Mode Coupling : D ( ) ~ D s 0 g Page #

14 Zero-shear Brownian viscosity (Pe =0) Banchio & Brady (2002) High Frequency Elastic Modulus G' a 3 /kt Stokesian Dynamics N=27 Shikata & Pearson (1994) ST1 ST3 ST5 = m Page #

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