Rheology of Dispersions

Transcription

1 Rheology of Dispersions Outline BASF AG Hard particles Interactions among colloidal particles Repulsive particles Particle size distribution Shear thickening Attractive particles Prof. Dr. N. Willenbacher Institute for Mechanical Process Engineering and Mechanics

2 Textbooks Rheology H. Barnes, B. Hutton, K. Walters, Introduction to Rheology C. Macosko, Rheology: Principles, Measurements and Applications R. Larson, The Structure and Rheology of Complex Fluids Colloids D. Fennel Evans, H. Wennerström The Colloidal Domain W. B. Russel, D. A. Saville, W. R. Schowalter Colloidal Dispersions T. Tadros (Ed.) Colloid Stability Volume 1: The Role of Surface Forces Part I: Colloids and Interface Science N.J. Wagner and J. Mewis Colloidal Dispersion Rheology J. Goodwin Colloids and Interfaces with Surfactants and Polymers - An Introduction

3 Parameters Controlling Dispersion Rheology Particle volume fraction φ Particle size shape size distribution sphere radius a Brownian motion Particle interactions v hydrodynamic / shear forces thermodynamic electrostatic steric van der Waals depletion repulsion attraction d i r e c t e d f l o w

4 Brownian Motion s collisions with solvent molecules stochastic force, random motion prevents particles from settling kt D = 6πηa τ B 2 a = D Pe = τ γ B 3 a η = 6π γ kt 2 a = γ D Stokes Einstein no Brownian motion if viscosity is high and/or particles are large characteristic time scale Peclet number dimensionless shear rate Pe shear forces dominate Pe 0 Brownian motion dominates fat-droplets in skim-milk σ r σ a = kt 3 dimensionless stress

5 Hard Spheres s Ψ a 2 a d i s t a n c e Brownian motion Hydrodynamic interactions the term "hard" refers to the shape of the interaction potential, not only solid particles, but also liquid droplets or even gas bubbles can be treated as "hard" spheres

6 Maximum Packing Fraction random body centered cubic face centered cubic φ max = 0.63 = 0.68 = 0.74 monodisperse ellipsoids and other irregular shaped objects pack closer φ max = 0.74 for random packing Weitz, Science 2004

7 Hard Sphere Phase Diagram PMMA particles in organic solvent Pusey & van Megen Nature l i q u i d c r y s t a l l i n e l i q u i d c o e x i s t e n c e c r y s t a l - l i n e* g l a s s y r a n d o m c l o s e p a c k i n g s o l i d fcc crystal v o l u m e f r a c t i o n φ 0.74 *) crystallization takes place since gain of volume entropy dominates loss of configurational entropy!

8 Glass Transition in Dispersions freezing of diffusion processes / particle mobility analogous to glass transition in small molecule and polymeric glasses control parameter φ instead of T! cage effect φ << φ g φ φ g for colloidal suspensions viscosity diverges at φ g η = 0 A 1 ηs φ φ g γ γ = 1.6 Krieger-Dougherty = 2 Quemada = 2.55 Mode coupling theory

9 Zero Shear Viscosity vs. Volume Fraction η r = η 0 / η s E i n s t e i n η r = φ c o r r e c t f o r φ < B a t c h e l o r η r = φ φ 2 c o r r e c t f o r φ < K r i e g e r - D o u g h e r t y 0. 4 Q u e m a d a B a t c h e l o r E i n s t e i n φ / φ m a x K r i e g e r - D o u g h e r t y η r = 1 Q u e m a d a η r = φ φ m a x φ m a x - 2 φ φ m a x r e d u c e s t o E i n s t e i n a s φ 0

10 Hard Spheres Viscosity vs. Shear Stress η r φ = 0.45 η η 1 = η0 η 1 + b( φσ ) r σ = 0 η = η r σ η = η r particle radius 0 fluid viscosity φ = 0.40 PMMA particles in silicon oil φ = 0.30 φ = 0.20 φ = 0.10 Low shear viscosity increases stronger with φ than high shear viscosity I.M Krieger, in Polymer Colloids, 1988

11 Hard Spheres Viscosity & Particle Size η r η r η 0, r d e c r e a s i n g p a r t i c l e s i z e n m n m n m φ = η, r γ / s - 1 The viscosity of hard sphres is independent of particle size, C h o i + K r i e g e r, J C I S P e = the onset of shear thinning is shifted to higher shear rates as the particle size decreases a 3 η k T γ

12 Hard Hard-sphere Spheres Suspensions Effect Effect of Solvent of Viscosity viscosity η [Pa s] / a.u η r = η 0 / η s PS latex in water benzylalcohol m-cresol φ = increasing solvent viscosity increasing solvent viscosity shear shear rate γ [s -1 ] rate / a.u. 10 Krieger, Adv. Coll. Interface Sci Pe = a3 η s kt γ

13 Hard Rods and Disks - 1 s Axisymmetric particles 2b 2a axis ratio r P a = b prolate / rodlike (a>>b): oblate / disklike (a<<b): glass, graphite fibers, viruses, proteins, DNA red blood cells, mica flakes, clay Orientation distribution controlled by balance between hydrodynamic and Brownian forces random orientation at low shear rates for small particles and low fluid viscosity flow alignment at high shear rates for large particles and high fluid viscosity

14 Hard Rods and Disks 2 φ * 2 πb 2a b 4π 3 a a 3 2 rod * φ critical volume fraction at which particles start to interact much smaller for non-spherical particles than for spheres dilute * φ semidilute concentrated isotropic nematic φmax larger for non-spherical particles than for spheres

15 Non-Spherical Particles at High Volume Fraction Low shear viscosity increases with increasing anisotropy (at const. φ) glass plates 100 x 400 mm L D = 7 L D =14 glass rods 30 x 700 mm quartz grains 50 x 70 mm L D = 21 L D =1 spheres 40 mm glass fibres v particle volume fraction / % particle volume fraction / % Giesekus 1983 Clarke 1967

16 Anisotropic Particle Suspensions at High Shear Rates For anisotropic particles random orientation leads to a higher barrier to start flow, i.e. to an increase in low shear viscosity. However, under shear, these elongated particles can orient in the direction of flow, resulting in a lower high shear viscosity than for spherical particles with equivalent size.

17 Colloidal Interactions s 1 h/2a 2a h r 0,1 h φ max /2a = 3 1 φ φ = 0.5 2a = 100 nm h = 8 nm 2a = 10 µm h = 800 nm 0,01 0,1 0,2 0,3 0,4 0,5 0,6 φ at constant volume fraction distance between particle surface decreases with particle size

18 van der Waals Attraction van der Waals forces originate from electrostatic dipole-dipole interactions orientation averaged dipole-dipole interaction Keesom dipole-induced dipole interaction Debye fluctuating dipole-induced dipole interaction London summation and thermal averaging over all molecular dipoles two spherical particles with radius a and separation r 1 2a 2a r 4a Ψ vdw = AH + + ln r 4 a r r Ψ vdw Aa H 12h Derjaguin approximation valid for h<<a (small gap between particles) A H = Hamaker constant controlled by dielectric properties of particles + surrounding fluid dielectric constant ε and refractive index n A,B typical value A H» J typical range of vdw interaction 5 10 nm

19 Surface Charge & Electrostatic Double Layer colloidal particle particle surface solvent + electrostatic potential around a charged particle in a dispersion decays exponentially due to shielding effect of counter-ions 1 Ψel ( r) exp( κ r) r with κ kt e n z 1/ = εε r 0 / i i i charge density ρ Debye length κ -1 "range of electrostatic repulsion"

20 Steric Interaction s a L adsorbed, grafted or co-polymerized polymer chains on particle surface excess polymer concentration in the overlap region creates osmotic pressure repulsive interaction approximation for thin stabilizing layer (L<<a) Ψ kt steric = 0 2L h Ψsteric 4 a 1 h = L L h< 2L kt ν 2 2 π φ 2 p χ 1 Ψsteric a h h = L h< L kt L L 4π φ p χ ln ν L = thickness of stabilizing layer φ p = polymer concentration in stabilizing layer χ = Flory-Huggins parameter ν 1 = volume of a solvent molecule h = r-2a gap between particles 1 χ < 2 good solvent Ψ> 0 1 χ = 2 Q - solvent Ψ= 0 1 χ > 2 poor solvent Ψ< 0 repulsive attractive Napper, J Colloid Interface Sci, 1977

21 DLVO Potential s Ψ ( r) =Ψ ( r) +Ψ ( r) vdw el for steric interactions Ψ el is replaced by Ψ steric 2a Ψ (r) r energy barrier Ψ el (r) r 2a Ψ vdw (r)

22 Hard Sphere Mapping Electrostatic stabilization "Charged spheres" Steric stabilization a Ψ / kt a 1 φ eff L L a = φ (1 + ) 3 a a eff r behavior of repulsive sphere dispersions corresponds to that of hard spheres with φ eff effective volume fraction increases with increasing range of interaction

23 Charged Spheres Viscosity & Ionic Strength η 0, r PS200 mm [KCl] φ max,exp η 0,r 100 PS mm [KCl] PS mm [KCl] PS mm [KCl] φ eff φ = φ φ max max, exp Quemada K-D 1 0,1 0,2 0,3 0,4 φ 1 0,0 0,2 0,4 0,6 0,8 1,0 φ / φ max,exp Horn, Bergenholtz, Richtering, Wagner, Willenbacher, J Coll Interface Sci 2000

24 Viscosity and Particle Volume Fraction gel-like, crystalline η / Pas two phase, weak attraction? 10 3 styrene / acrylate dispersion up down φ = 0.49 φ = 0.47 φ = 0.45 φ = 0.44 φ = 0.43 φ = 0.42 φ = 0.40 φ = l i q u i d γ / s -1

25 Effect of Particle Size on Viscosity η / Pa s η increases with decreasing particle radius a, since φ eff increases at constant φ radius 35 nm 45 nm 65 nm 95 nm 125 nm at high shear rates hydrodynamic forces dominate over colloidal forces η independent of radius a 10-2 φ = γ / s -1

26 Viscosity of Bimodal Dispersions - 1 Non-Brownian Hard Spheres Colloidal Hard Spheres 10 3 η 0,r φ = η 0,r size ratio φ = φ = 0.58 φ = glass beads in PIB 30 µm < d < 230 µm Chong et al ξs 10 1 PMMA in bromoforme size ratio: 141 nm 84 nm = 1.7 Rodriguez et al ξ s 1.0

27 Viscosity of Bimodal Dispersions - 2 without colloidal interactions viscosity minimum for small particle fraction ξ s = 25-30% size ratio σ as large as possible when colloidal interactions get relevant φ φ eff increase in φ eff more pronounced for small particles optimum size ratio σ viscosity without colloidal interactions with colloidal interactions Dames & Willenbacher, Rheol Acta, size ratio

28 Viscosity of Bimodal Dispersions - 3 aqueous polymer dispersion φ = 0.62 σ = 2 σ = 4.3 Willenbacher et al., Adhesives & Sealants, 2003

29 Shear Thickening Occurs in Suspensions of... Non-Brownian particles quartz PVC CaCO 3 clay glass beads iron pigments starch blood cells Ribcap New Soft Helmet Turns Hard in Crash Colloidal particles polymer silica (SiO 2 ) ceramics (Al 2 O 3 ) iron oxide

30 Origin of Shear Thickening Shear thickening results from the flow-induced formation of transient particle clusters strong increase in turbidity supports cluster formation hypothesis viscosity / Pa s; turbidity / % silica particles in tetrahydrofurfural alcohol (index-matched) φ=0.65 so-called "Hydrocluster" Viscosity increase because of the anisotropic shape of the clusters and the increased effective particle volume fraction due to trapped solvent turbidity shear rate / s -1 viscosity Cluster formation controlled by the balance of hydrodynamic force needed to push particles together and the repulsive colloidal (often also called thermodynamic) forces Bender+Wagner, J Rheo 1996

31 Shear Thickening & Particle Interaction σ electrosteric repulsion increases with increasing ph shear thickening shifted to higher stresses and viscosity increase less pronounced low shear viscosity increases strongly Laun, Ang. Makromol Chem 1984

32 Attractive Particle Interactions - Outline Structure of suspensions containing attractive particles Mechanisms of aggregation / flocculation Rheological features of weakly and strongly flocculated dispersions yield stress and storage modulus Viscosity reduction due to weak attractive interactions Capillary forces in suspensions

33 Structure of Attractive Particle Suspensions fractal aggregate structure flocs immobilize water, f eff > f strong shear thinning above f c flocs form sample spanning network, percolation elastic, gel-like behaviour G > G², yield stress shear-induced break-down and recovery of floc structure thixotropy Weitz & Huang 1984 coagulation into compact solid aggregates and phase separation into solid and liquid fraction not considered here!

34 Flocculation of Charged Particles changing ionic strength by adding salt changing surface charge by varying ph (or other physico-chemical parameters) calculate critical ion concentration from DLVO-theory Y max = 0 and Y = 0 at Y max = kt B 4 ezψ s ncrit = tanh zl AH 4kT B n crit with l 63 b b e = 4πε ε kt r 58 nm ε r 1 4/ lb Qs n = crit ( εε A ) r 0 H 2/3 B 0,7nm Bjerrum length in water at room temperature 1 6 Schulze-Hardy rule, effectiveness of multivalent ions! z weak surface potential, symmetric electrolytes 2 1 z

35 Flocculation of Sterically Stabilized Particles aa stability criterion H Ψ vdw < kt hcrit 12kT sterically stabilizing layer must prevent particles surfaces to come closer than h crit 1 A a L H Poly hcrit with 2.5 LPoly 2 kt 10 L Poly varies strongly with temperature, especially around the q-temperature good solvent = repulsive steric force poor solvent = attractive steric force Ψ vdw h crit -kt r a L poly h crit

36 Flocculation by Addition of Polymers depletion flocculation asmall a center of polymer coil (or small particle) can not enter the shaded area = "excluded volume" osmotic pressure pushing large particles together reduction of excluded volume entropic phenomenon! non-adsorbing polymer needed attraction strength ~ polymer concentration attraction range ~ volume of polymer chain bridging flocculation dissolved polymer molecules attach to at least two particles requires affinity of polymer to particle surface long polymer chains needed in order to reduce loss of entropy

37 Rheology of Flocculated Suspensions Yield stress σ φ weak and strong flocculation γ» 3 y γ computer simulation chemically limited aggregation γ = 4.4 diffusion limited aggregation γ = 3.5 σ a δ strong flocculation δ» -2 y Storage Modulus G independent of frequency and G >> G² G ' α φ strong flocculation α» G ' a β strong flocculation β» 0 weak flocculation β < 0

38 Yield Stress of Flocculated Systems σ y φ a 3 2 φ 3 polystyrene particles a = 245, 480 and 1700 nm in water flocculated by adding BaCl 2 Buscall et al. 1998

39 Larson Flocculation S. 347 Induced Abb. by 7.18 ph "blocky"-shaped ZrO 2 particles a = 150 nm in water φ = φ = φ = φ = φ = i.e.p. surface charge changes with ph strong vdw attraction at isoeelectric point Leong et al., Trans Royal Soc Chem 1993

40 Larson S. 339 Abb. 7.8 Viscosity of Flocculated Systems Thermoreversible gelation of sterically stabilized suspensions η r octadecyl grafted SiO 2 particles in benzene φ = Depletion Flocculation η r 0.6 % 1.0 % 0.85 % acrylate particles a = 157 nm φ= 0.4 in "white spirit" with polyisobutene M w = g/mole K K K K K K 0.5 % 0.4 % 0.1 % / s -1 Woutersen & de Kruif, J. Chem. Phys Buscall et al., J Rheo, 1993