Dispersion Rheology“Interlude'” Addition of a particulate phase to a liquid increases its viscosity in proportion to the volume fraction of the dispersed phase: dispersed-phase volume total volume However,the hydrodynamically-effective volume fraction,is often not the same as the formulated volume fraction. Dispersion Rheology"Interlude" Addition of a particulate phase to a liquid increases its viscosity in proportion to the yolume fraction o of the dispersed phase: 中= dispersed-phase volume total volume However,the hydrodynamically-effective volume fraction,is often not the same as the formulated volume fraction Effective Volume Fractione 中efr can be made greater than中by: +Increasing the hydrodynamic radius of the dispersed particle through .adsorption of polymeric stabilizer .particle surface electrical charge Flocculating the dispersion,trapping liquid phase within the flocs (volume of floc>volume of particles contained) 1
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The viscosity of suspensions Einstein showed that single particles increased the viscosity of a liquid(as a simple function of their phase volume () n=n.1+2.5) There is no effect of particle size,nor of particle position because the theory neglects the effects of other particles. This equation works for
2 The viscosity of suspensions Einstein showed that single particles increased the viscosity of a liquid (ηs) as a simple function of their phase volume (φ), η = ηs( ) 1+ 2.5φ There is no effect of particle size, nor of particle position, because the theory neglects the effects of other particles. This equation works for φ < 10% More concentrated suspensions Higher order terms : η =ηs 1+ 2.5φ + 6.2φ2 + O φ3 ( ) ( ) or the Krieger and Dougherty equation η =ηs 1− φ φ m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −[ ] η φ m where [ ] η is the intrisnic viscosity (2.5 for spheres) and φ m is the maximum packing fraction The maximum packing fraction of various Arrangements of monodisperse spheres Arrangement Maximum packing fraction Simple cubic 0.52 Hexagonally 0.605 Body-centered 0.68 Face-centered 0.74
perimental data to e (. 【m冷a Reference (40 gm) 061 0.69 Turian and Yuan (1977) Itanium dioxide .55 Turian and Yuan(97万 aterite 8 9对 03w2 3n Clarke(1967) 00×400m rtaIs 58 0371 2.15 Clarke (1967) (53-76m) o14 axial ratio 21 60 0233 14的 Giesekus (1983) Effect of Particle Shape 30 10 0 0 10 2030 40 50 Phase volume, Viscosity as a function of phase volume for various particle shapes. 4 月王下 3 2 0 10 20 30 0 Phase volume, Viscosity as a funetion of phase volume for various aspect ratio of fibres. 3
3 Effect of Particle Shape
Effect of the viscosity of the internal phase -) where n,=the viscoisty of the internal or the dispersed phase Several limiting cases: =2.5 the Einstein case (hard sphere) n,=n:[=1.75 =0,[n]=1 the situation for gas bubbles Effect of different sizes Effect of binary particle-size fraction on suspension 02 viscosity with total volume as parameter. The particle size ratio is 5:1. 40 30 Point 们 %large 20 601000 0 0 02040608 10 Q 60500 50 Froction of forge porticles 74100050 Effect of particle charge on particle size The electrical double layer K=03c% where e is the electrolyte concentration in mol/L and [a]is the valency of the This means that (assuming [z]"1)the following is true for aqueous electrolyte concentration,c =105,then double layer thickness 1/x-100 nm c-10 1/k-10nm c=10, 1/x-1 nm 4
4 [ ] η = 2.5 ηi + 0.4ηs ηi + ηo ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ where ηi = the viscoisty of the internal or the dispersed phase Several limiting cases: ηi = ∞; [ ] η = 2.5 the Einstein case (hard sphere) ηi = ηs ; [ ] η =1.75 ηi = 0; [ ] η =1 the situation for gas bubbles Effect of the viscosity of the internal phase Effect of binary particle-size fraction on suspension viscosity with total % volume as parameter. The particle size ratio is 5:1. Point φ η %large P 60 1000 0 Q 60 500 50 S 74 1000 50 Effect of different sizes Effect of particle charge on particle size The electrical double layer - - - - - - - - - + + + + + + + + + 1 κ= 0.3 c −1 2 [ ]z −1
Non-Newtonian Liquids Bingham Newtonian O 0: Dilatant Rheograms for Shear Thinning Shear Thickening 1= 2 Pseudoplastic 7 Newtonian Bingham Dilatant soft ice honey cheese cream syrup salad yoghurt margarine salad cream tomato dressing ketchup pate cream tomato gravy creamy juice soup oil consomme water "Non-Newtonianness" 5
5 Non-Newtonian Liquids σ γ& Pseudoplastic Newtonian Dilatant Bingham σ y Rheograms for Shear Thinning & Shear Thickening γ& Newtonian Pseudoplastic Dilatant Bingham γ η σ & = η
Complete Flow Curve (1)Sedimentation )2) Asphalt (2)Leveling (仔)Pouring h (4)Pumping logn Molasses (5)Rubbing Glycerol (6)Spraying Castor Oil 乡 5 Olive Oil 6) Water 7 logy 10 shampoo polymer 10 xanthan gum blood 10 10 103 102103 103 Shear rate.氵Is' Viscosity/shear-rate curves for blood.liquid crystallinc polymer.shampoo.yoghurt and an aqucous xanthan gum solution. 10 locust bean gum polysaccharide 102 guar gum modified cellulose 10° carrageenan 10 ou 103 10310 10 Shear rate,/s Viscosity/shear-rate curves for 1%by wt.various natural polymeric thickeners in water
6 Complete Flow Curve Asphalt Molasses Glycerol Castor Oil Olive Oil Water (1) Sedimentation (2) Leveling (3) Pouring (4) Pumping (5) Rubbing (6) Spraying (1) (2) (3) (4) (5) (6) η0 η∞ logγ& logη
Generalized Equilibrium Flow Curve* a b 2--n Logy General picture due to Hoffman,Choi and Krieger General Flow Curve(Dispersions) Four Flow Regimes I-(First Newtonian)Brownian diffusion keeps microstructure random;viscosity constant Il-(Power Law)Hydrodynamic forces impose order,particles align along flow streamlines: viscosity falls Ill-(Second Newtonian)Maximum order achieved;viscosity again constant IV-(Shear thickening)Ordered flow unstable; "log-jamming"of particles Why do fluids shear-thin? 1.Breakage of flocculates Flocs increase viscosity because: .Viscosity of a dispersion proportional to volume of particles in the dispersion .Volume of flocs volume of separate particles increasing shear stress 7
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Why do fluids shear-thin? 2.Hydrodynamic ordering .Brownian diffusion randomizes microstructure .Shear field aligns particles along streamlines .Viscosity (energy dissipation)inversely proportional to order ●● eeeeeee ● ●● 00000000 ● increasing shear stress (decreasing viscosity) Glass microsphere dispersion between glass plates II Note:Equilibrium flow curve represents only colloidally stable systems For unstable(flocculated)systems,better to use non-equilibrium flow curve methods. i.e.: Time-based ramp of shear rate/stress 8
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Particle Diffusional Relaxation Time (dilute dispersions) Time for particle to diffuse one-half its diameter kaT n=continuous-phase viscosity d a particle diameter ka=Boltzmann constant O T Absolute temperature The Peclet Number -大-g kgT When P>I thet>then slow diffusion and fast defomation then shear thinning occurs When P.a or n>n Logi or T
9 Particle Diffusional Relaxation Time (dilute dispersions) Time for particle to diffuse one - half its diameter ta = 6πηa3 kBT η = continuous - phase viscosity a = particle diameter kB = Boltzmann constant T = Absolute temperature The characteristic time for shear flow (tsr ) is the reciprocal of the shear rate. This is the time taken for a cubic element of material to be transformed to a parallelogram with angles of 45° (i.e. the time for unit strain to be applied). ta is the transition shear rate between and shear thinning 6 When P 1 then t t thus then a random Browian distribution dominates. When P 1 the t > t then slow diffusion and 6 P The Peclet Number tr 3 o e a sr e a sr e a sr 3 e η πη γ πη γ a k T k T a t t s B B s sr a ≈ ≈ ≈ = = & & ‘ Effect of particle diffusion on flow curve all of the above 6 ' ' ' ' 3 or T T or a a or a k T tr tr B tr > < = η η γ γ πη γ & & &
Generalized form of Cross Carreau Models 7o-1 +X） Cross a=1-n Carreau-B a=2 a curvature of transition from Newtonian to power law regimes(oc polydispersity) R>R....Because 'particle size distribution (psd)is broader than 门o Blue psd.ordering transition 110 encompasses a range of shear rates. Same effect holds for polymer MWD. R +Bi Logy COLLOIDAL FACTORS shape (aes 3oD u separation surface size(distribution) salt shape separation size (distribution) HYDRODYNAMIC FACTORS Shear rate,y (log scale) Flow curve of a suspension of colloidal particles. 10
10 Generalized form of Cross & Carreau Models ( ) ( ) ( ) to power law regimes( polydispersity) a = curvature of transition from Newtonian Cross a =1- n Carreau -B a = 2 1 1 0 ∝ + − = + − ∞ ∞ a n a Rt γ η η η η & ‘ ‘