rheology of slurries

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Rheology of Slurries

Review briefly interactions between polymer stabilized colloid systems:

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Schematic Interaction Energy

Schematic calculation, taken from J. Colloid Interface Sci., 6:492, 1951.

Small size polymer, less effective; rigid better than flexible polymer

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Batch ConsistencyChapter 14 in JS Reed book

5 consistency state: Bulk powder (no liquid) Agglomerates (granules) Plastic body Paste Slurry (dilute solution called suspension; slip:

slurry containing clay)

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Factors:Amount, distribution and properties of liquidAmount, size and packing of particlesTypes, amount and distribution of additives Interparticles forces: attractive or repulsive

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DPS = degree of pore saturation = volume of liquid / volume of pore

granule

Plastic body

paste

slurry

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More Comments

Plastic state : often during extrusion, plastic pressing etc.Granule & plastic body may rearrange due to applied force, to become more densePaste : often used in printing (thick films in electronic ceramics)Slip or slurry: for casting

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Springback

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For plastic material, DPS = 1, on decompression, due to small compressibi-lity of liquid, volume expansion accom-panying slight particle rearrange-ment occur springback SB

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Batch Calculation

Mostly by weight%; sometimes by vol%

Mostly based on total weight, sometimes based on weight of major ceramic powders

))((i

i

i

ivi

MMf

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Some properties of suspension

Some related to solute conc. only, unrelated to its chemistry: vapor pressure lowering, freezing point depression, boiling point elevation

a1 = activity; TBP = normal boiling point

)]11

(exp[]1[&ln 11

11 TTRg

Hatmp

p

pa

BP

ovapo

o

]11

[ln 1 TTRg

Ha

BP

ovap

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Osmotic PressureSolute conc. produce chemical potential difference: 1

o (T,P) = 1

o (T, P+) + Rg T ln(a1); : osmotic pressure (membrane is capable to separate solvent and solute)

thermodynamics: = c2 Rg T (similar to ideal gas law; osmotic pressure exerted by solute concentration c2)Since c2 = w2/M2 can be used to determine MWFor non-ideal solutions, expressions for can be complexA simplified equation for polymer solution:1=1/2 makes second virial coefficient zero; called Flory point, or theta point theta temperature

})2/1(/{ 2212^ x

V

RgT

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Osmotic Pressure in Colloidal Suspension

One of source: electrical double layer of colloids; many complex equations, results as the right figure (TA Ring, 1996);Affected by zeta

potential, double layer thickness, solid volume fraction etc. a,b,.. Different

particle packing models

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Rheologybasically: Newtonian fluid and non-Newtonian fluid

Viscosity = constant for Newtonian fluid; for non-Newtonian power law fluid model, shown as followsNecessary to know rheology to predict flow of suspension into mold; predict velocity distribution, shear stress on wall, pressure distribution in mold, etc

Rheology important to – transport, mixing, forming etc.

dy

dvxxy

dy

dv

dy

dv x

n

xpxy

1

Apparent viscosity

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取自 TA Ring, 1996;

Shear thinning

Shear thickening

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Comparsion of Instruments

Capillary viscometer: simple to use, easy to change temp. and shear rate, similar to real fluid condition, can study extrudate behavior at the same time; drawback: rate of shear is not constant across capillary

Coaxial cylinder viscometer: all region under constant shear rate, easy to calibrate; drawback: high viscous material difficult to fill in, polymer may creep up along shaft

Cone and plate viscometer: also constant shear rate in all region, small sample, less heat build up; easy to fill in, easy to clean up; drawback: rate of shear limited to low rates

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MeasurementsDouble cylinder or cone-and-plate or capillary

tube are three common methods; Eq. derived to calculate viscosity from data; T = torque;

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Lba

abT

aa

422

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Measuring shear rate should be close to shear rate in use; left figure: shear rate varies with position, hence often use narrow annulus

22

22

2

2

ab

ba

ra

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Relative Indices

Some simple relative index for viscosity: e.g. time of fluid to pass a small hole

Gel strength – related to history of sample, need to stir with high shear for some time, settled, then measurement

Index of structural buildup – B gel = (Y2 - Y1)/ln(t2/t1) t2, t1 = time to wait

Index of structural breakdown B thix = (Y2 - Y1)/ ln(t2/t1); or (p1 - p2)/ln(t1/t2) [after constant shear rate different time; or different shear rate, same time]Elastic nature: memory effect, not ideal

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Four regimes of uniform rigid-sphere system: (I) Newtonian fluid; (II) shear thinning regime; (III) high shear Newtonian regime; (IV) shear thickening regime

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Equations

Dilute suspension: Einstein equation – for spherical particles, =2.5; limited to <0.02 (volume fraction); s = solvent viscosity

Electro-viscous effect by Smouluchowski: zeta potential is included

...1

s

)]]

2[

(1[12

2

aos

mY

mma )/(

Generalized Casson eq.

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Effect of Polymers on Viscosity

Polymer effect: (a) increase viscosity of solution; (b)adsorb on particle surface to increase its effective volume c [1 + (Ls/a)3]; Ls = span of polymer layer on particle surface

P = polymer volume fraction soluble in solvent (after deduction of adsorption; + dilation effect)

cScps

aL

])/(1[1 3p

f

ps

5.21

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Dilute, Slightly Aggregated Suspension

Colloidally unstable suspension; memory effect over long time scales thixotropy

Cross equation: c and m are fitted parameters;o = low shear limit viscosity; = high shear limit viscosity

m

c

o

)(1

)(

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Cross equation characteristics, and its corresponding particle structure (in suspension); shear rate stopped, Brownian motion will bring particle back to its network•取自 TA Ring, 1996;

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Two limiting viscosities

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Percolation ThresholdThis concept occurs in many situations; here to unstable colloidal system, exist a minimum particle concentration, if higher than this value, particle form bridging network, showing yield strength; from Newtonian fluid to Cross equation or Bingham plastic fluid

percolation or bond percolation (後者數值較低 ) – because one bond involves two sites only; if site percolation, then each site can have z coordinationOne can estimate percolation threshold for specific structures

Critical percolation volume fraction ~ 16%

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Theoretical prediction of percolation threshold for various geometries: fromTA Ring, 1996

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For electro-statically stabilized suspensions: when close to PZC, viscosity of suspension increase quickly; away from pzc, like a Newtonian fluid; but for much higher or lower pH, due to ionic strength, double layer thickness decrease, system unstable again

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Around PZC, high viscosity; after adding HEC, pzc shift highest viscosity point also shift; due to HEC, value of viscosity also increase;取自 JS Reed, 1995

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Concentrated SlurriesCan be sub-divided into different systems, e.g. stable or unstable; polymer or not; mono-modal particle size distribution

Polymer may entangle together pseudo-plastic flow Cross equation; some of parameters may be estimated from theory, e.g. m = (Mn/Mw) 1/5 [Mn = number averaged MW; Mw = weight averaged MW; ratio of these two values = width of MW distribution]Concentrated suspension often time dependent rheology thioxtropy due to particle structure may change with shear stress different stress lead to different steady state

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Time Dependent Behavior

After rest for a while, a gel strength developed due to particle structure formation;With yield stress, coating can resist creep flow (gravitation)

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Monodisperse System

Derivation rely on description of particle structure and their interaction

Still Cross equation, but for concentrated system, can be simplified to the following form: Pe = ratio between particle motion and diffusion; t for translational instead of rotational

81

)(t

o

Pe

Tk

a

D

aPe

B

s

tt

32 6

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Taken from TA Ring, 1996;

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22 )71.0

1(&)63.0

1(

ss

o

Shear thinning 3 body interaction

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General Equation

Cross equation: both low shear or high shear viscosity can be represented by following equation: wherem =maximum volume fraction often a fitted value from experimental data 0.5 – 0.74; n = 2 – 3; often 2

Doughtery-Krieger eq. similar; others include Mooney equation, Chong equation etc

n

ms

)1(

][)1( DKeqcrKh

crf

]))[1/(exp( 2 MooneykKHf

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Doughtery-Krieger equation: 取自 JS Reed, 1995cr & KH are two fiited parameters

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Anisotropic Particles

E.g. rod, plate-like particles (clay) and its rheology; still use Cross equation to describe rheology; with one extra parameter r = b/a (aspect ratio)

For clay: different face, different charge, hence different behavior (structure) under different pH

For clay particles

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取自 TA Ring, 1996

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Different particle structure, different rheology