Particle Packing

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Particle Packing
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• Forming strongly related to particle packing
  (science and technology)
• Results from packing packing density and
  porosity
• Factors particle size and distribution, particle
  shape, resistance of particles to pressure
  (deformation binder effect), flow resistance
  (friction between particles) For uniform spheres
  five different packing arrangements cubic,
  orthorhombic, tetragonal, pyramidal, tetrahedral
  etc.
• Different packing density higher coordination
  number to higher packing density, theoretical
  maximum 74.

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In theory, we can obtain ordered packing of
mono-disperse particles in reality, it is often
to get packing as shown above (small range of
ordering)
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Packing Density and Pore Size
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Packing Characteristics
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• Tortuosity ?o for cubic packing ?o 1.0
  tetrahedral packing ?o 1.3

• Number of particle contact Nc 3 (PF) (CN)/(? a3)

• PF packing fraction CN coordination number

• for nonregular packing Nc 3 (1-?)/(?a3) since
  CN ?/? (usually between 6 10)

• Container wall effect (on packing) insignificant
  when container dia./particle dia. gt 10

• Use two particle sizes, small one can fill into
  interstice, thus increase packing density

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Furnas Model
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• In theory, if three kinds particle in packing
• PFmax PFc (1- PFc) PFm (1- PFc)(1- PFm) PFf
• f i, w Wi/W total
• Wc PFc ?c medium and fine the same

• The small particle size have to be small enough,
  size ratio gt 7, to effectively increase packing
  density

• In industry, often mix two or more particles to
  get high density packing, to reach densification
  at lower sintering temperature

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• Highest density occurs when small particle fill
  completely porosity from large particles (volume
  fraction for fines 26 or porosity from large
  particles 26)
• In reality, since the size ratio will not be too
  large, the highest point of packing density
  usually moves toward the middle point.

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Packing of Continuous Distribution
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• E.g. log normal distribution theoretical
  calculation shows that, under random packing,
  larger geometric standard deviation , denser
  packing (spheres)

• Andreasen cumulative distribution (1) usually n
  0.33 0.5 experience 1/n increase, packing
  density increase

• Zheng modified distribution (2) one more
  parameter, amin

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Taken from JS Reed, 1995 often packing density
60-69 In reality, particles not very
spherical, will affect packing density
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Results from real particle size distributions,
sample calcined Bayer alumina it is not very
easy to rationalize
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Hindered Packing
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• Including external and internal factors
• Bridging of particles and agglomerates with rough
  surface of walls (mechanical vibration tap
  density, lubrication, large force causing
  particle fracture may improve somewhat)
• Coagulation , adhesion between particles also
  retard particle motion and hence packing into
  dense structure
• High aspect ratio often produce high porosity
• Adsorbed binder molecule also hinder particle
  movement

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Ordered Structure in Suspension
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• For monodisperse particle systems particle
  interaction gravity force ? ordered structure
  (so called order-disorder phase transition
  question a thermodynamic and mechanical
  equilibrium problem)

• Defects point defect (vacancy), line defect
  (dislocations), planar defects (grain boundary),
  volume defects (cracks)

• Point defect can be estimated from
  thermodynamics other defects related to
  processing

• Measurement of ordered domain size Scherer
  equation (peak broadening) ? FWHM k?/(L cos?)
  full width at half height k constant 0.9

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• ????TA Ring, 1996
• Measurement of ordered array structure light
  diffraction (iridescence) n? 2 d sin? ? can
  estimate size of structure from diffraction peaks
  (d)

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Sinterbility of Agglomerated Powders

• Source J. Am. Cer. Soc., 67(2), 83-89, 1984 (by
  FF Lange)
• A new concept Pore coordination number
  thermodynamic analysis pore will disappear only
  when its coordination number is less than a
  critical value

• Real system irregular particle size and shapes
  irregular arrangement (packing)
• Agglomerates hard (partially sintered) soft
  (held by van der Waals forces)

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• General experiences soft agglomerates produce
  better sintering results than hard agglomerates
• This author thinks particle arrangement is
  important
• A pore has its volume, shape and coordination
  number
• RgtRc pore surface convex RltRc concave surface
  (those pores are able to disappear)

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• Theoretical calculation equal-sized spheres,
  random packing, pore volume always 0.37 0.41
  (or density 059 0.63) for real powder tap
  density rarely over 30 of true density
• Theoretical calculation different sized sphere
  can produce bulk density up to 95
• Consolidation force to increase bulk density
  depend on resistance of particle packing unit to
  deformation (via particle rearrangement) as
  shear stress increase, agglomerate first to shear
  apart into their smaller domains, next domain
  deformation, finally, particle deform or
  fracture
• Grain growth a method to reduce pore
  coordination number grain growth from mass
  transport (temperature effect)
• If pore growth faster, we may get pores with
  higher coordination number

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Transparent Alumina

• Grain size ?500 nm residual porosity negligible
  (e.g. 0.03)
• Possible methods (a) Use high sintering
  temperature (grain growth problem) or (b)
  through special particle coordination and low
  temperature sintering (shaping technique or
  particle size distribution key homogeneity
  e.g. no agglomerates)
• Following data from J. Am. Cer. Soc. 89(6),
  1986-1992, 2006.
• Raw material Al2O3, 99.99 pure, 150-200 nm

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• Shaping methods (a) dry pressing (uniaxial
  pressing at 200 MPa cold isostatic pressing CIP
  at 700MPa (pre-shaped at 30-50 MPa) (b)
  gel-casting (4-5 wt monomer) (c) slip casting
  into porous alumina mold
• Binder burnout 800oC, very small shrinkage (lt
  0.2), develop neck, provide strength for Hg
  intrusion analysis
• Mercury porosimetry better than SEM to measure
  pore size distribution
• No large pores (gt75 nm) an indication of
  homogeneity

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• Gel-casting versus uniaxial pressing

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• Pore size distribution do not change much from
  green state to intermediate sintering stage
• Homogeneity poor for uniaxial pressing
• Pore size 50 nm 1/3 of particle size

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• Slip casting provides the best particle
  coordination pore size 35 nm 1/5 particle
  size
• Observation Smaller and larger pore are
  eliminated at similar rates

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• Density grain size trajectory of different
  processing

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• (a) slip casting without binder, presintered at
  1200oC, then HIP 1170oC, ave. grain size 0.44
  µm
• (b) gelcasting, presintered at 1240oC, HIP
  1200oC, ave. grain size 0.53 µm (both densities
  gt 99.9)
• All above data taken from J. Am. Cer. Soc. 89(6),
  1985-1992, 2006.