1. 1 Particle Packing 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%. Che5700 陶瓷粉末處理

2. 2 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)

3. 3 Packing Density and Pore Size Che5700 陶瓷粉末處理

4. 4 Packing Characteristics Tortuosity  o : for cubic packing  o = 1.0; tetrahedral packing  o = 1.3 Number of particle contact Nc = 3 (PF) (CN)/(  a 3 ) PF = packing fraction; CN = coordination number for nonregular packing Nc = 3 (1-  )/(  a 3 ); since CN ~  /  (usually between 6 – 10) Container wall effect (on packing): insignificant when container dia./particle dia. > 10 Use two particle sizes, small one can fill into interstice, thus increase packing density Che5700 陶瓷粉末處理

5. 5 Furnas Model In theory, if three kinds particle in packing:  PF max = PFc + (1- PF c ) PF m + (1- PF c )(1- PF m ) PF f  f i, w = W i /W total  W c = PF c  c ; medium and fine the same The small particle size have to be small enough, size ratio > 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 Che5700 陶瓷粉末處理

6. 6 圖中直線代表粒子粒徑比 值無限大的理論值 ; 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.

7. 7

8. 8 Packing of Continuous Distribution 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, a min Che5700 陶瓷粉末處理

9. 9 Taken from JS Reed, 1995; often packing density 60-69%; In reality, particles not very spherical, will affect packing density

10. 10 Results from real particle size distributions, sample: calcined Bayer alumina; it is not very easy to rationalize

11. 11 Hindered Packing 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 Che5700 陶瓷粉末處理

12. 12 Ordered Structure in Suspension 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 Che5700 陶瓷粉末處理

13. 13 本圖取自 TA Ring, 1996; Measurement of ordered array structure: light diffraction (iridescence) n = 2 d sin   can estimate size of structure from diffraction peaks (d)

14. 14 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)

15. 15 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 R>Rc: pore surface convex; R<Rc: concave surface (those pores are able to disappear)

16. 16 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

17. 17 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: Al 2 O 3, 99.99% pure, 150-200 nm;

18. 18 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: 800 o C, very small shrinkage (< 0.2%), develop neck, provide strength for Hg intrusion analysis Mercury porosimetry better than SEM to measure pore size distribution No large pores (>75 nm): an indication of homogeneity

19. 19 Gel-casting versus uniaxial pressing

20. 20 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

21. 21 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

22. 22 Density – grain size trajectory of different processing

23. 23 (a) slip casting without binder, presintered at 1200 o C, then HIP 1170 o C, ave. grain size = 0.44 μm (b) gelcasting, presintered at 1240 o C, HIP 1200 o C, ave. grain size = 0.53 μm (both densities > 99.9%) All above data taken from J. Am. Cer. Soc. 89(6), 1985-1992, 2006.