Rheology of Slurries

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Rheology of Slurries
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• Review briefly interactions between polymer
  stabilized colloid systems

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Schematic Interaction Energy
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Schematic calculation, taken from J. Colloid
Interface Sci., 6492, 1951. Small size polymer,
less effective rigid better than flexible polymer
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Batch Consistency
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• Chapter 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)

• Factors
• Amount, distribution and properties of liquid
• Amount, size and packing of particles
• Types, 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
Plastic body
paste
slurry
granule
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More Comments
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• Plastic state often during extrusion, plastic
  pressing etc.

• Granule plastic body may rearrange due to
  applied force, to become more dense

• Paste 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
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• Mostly by weight sometimes by vol

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

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Some properties of suspension
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• 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

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Osmotic Pressure
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• Solute conc. produce chemical potential
  difference ?1o (T,P) ?1o (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 MW

• For non-ideal solutions, expressions for ? can be
  complex

• A simplified equation for polymer solution?11/2
  makes second virial coefficient zero called
  Flory point, or theta point ? theta temperature

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Osmotic Pressure in Colloidal Suspension
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• 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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Rheology
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• basically Newtonian fluid and non-Newtonian fluid

• Viscosity constant for Newtonian fluid for
  non-Newtonian power law fluid model, shown as
  follows

• Necessary 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.

Apparent viscosity
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Shear thinning
Shear thickening
??TA Ring, 1996
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Comparsion of Instruments
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• 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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Measurements
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• Double cylinder or cone-and-plate or capillary
  tube are three common methods Eq. derived to
  calculate viscosity from data T torque

Measuring shear rate should be close to shear
rate in use left figure shear rate varies with
position, hence often use narrow annulus
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Relative Indices
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• 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
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• Dilute suspension Einstein equation for
  spherical particles, ?2.5 limited to ?lt0.02
  (volume fraction) ?s solvent viscosity

• Electro-viscous effect by Smouluchowski? zeta
  potential is included

Generalized Casson eq.
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Effect of Polymers on Viscosity
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• 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)

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Dilute, Slightly Aggregated Suspension
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• 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

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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
• ??TA Ring, 1996

Two limiting viscosities
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Percolation Threshold
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• This 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 (??????)
  because one bond involves two sites only if site
  percolation, then each site can have z
  coordination

• One can estimate percolation threshold for
  specific structures

• Critical percolation volume fraction 16

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Theoretical prediction of percolation threshold
for various geometries ??TA 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 ??JS
Reed, 1995
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Concentrated Slurries
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• Can 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
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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
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• 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

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Taken from TA Ring, 1996
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Shear thinning ? 3 body interaction
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General Equation
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• Cross equation both low shear or high shear
  viscosity can be represented by following
  equation where?m 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

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Doughtery-Krieger equation ??JS Reed, 1995 ?cr
KH are two fiited parameters
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Anisotropic Particles
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• 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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??TA Ring, 1996
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Different particle structure, different rheology