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<10% More concentrated suspensions Higher order terms: n=n.0+2.50+620+0(0》 or the Krieger and Dougherty equation -发 where n]is the intrisnic viscosity (2.5 for spheres) and 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 2
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<Tor all of the above 9
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 & ‘ ‘
