1. Phase Connectivity and Homogeneity
goals
percolation concepts
role of grain or particle size
conductor and insulator illustration
bimodal mixtures
mixing and evaluation of homogeneity

2. Ordered and Random Packing

3. Difference
coordination – number of touching neighbors
ordered packing –repeated position, simple translation vector, each has same coordination number
random packing –assembled without pattern, differing distances, coordination number distributed

4. Packing Coordination

5. Phase Connectivity
experiment; randomly mix conductor and nonconductor particles
with abundance of insulator, composite is nonconductive
with abundance of conductor, composite is conductive
critical point is percolation limit
depends on three-dimensional contacts

6. Consider Subtractive Experiment
idealized 2D conductor structure

7. Several Random Removal Steps
partial removal of conductors, still circuit exists

8. Non Conductive
no continuous path (2D) for conduction

9. Early Demonstration
silver particles in epoxy, composite conductivity

10. Problem is Classic in Physics
some variants
loose particles, same size, porous
loose particles different in size
conductor is larger
insulator is larger
compacted particles, same size, dense
compacted particles different in size
nonspherical variants on above

11. Dispersed Phase Fracture
crack denied
easy fracture
path, because
hard grains
are dispersed

12. Porous Copper Powder Compact
at 69 % dense
mode coordination
number is 7

13. Coordination Change with Density
best fit
NC = f2

14. Coordination Distribution

15. Homogeneity Role

16. Some Findings
loose random spheres conductor + insulator, same size, 60 % packing, both phases percolated from 28 to 72 vol %
loose ordered spheres conductor + insulator, same size, percolation depends on coordination, but 25 to 75 vol % both connected
loose random mixed spheres of differing sizes, conductor + insulator see chart

17. Full Density Grains, Polygonal Shape
typical
12 to 14
faces on each
grain or
coordination
number

18. Expected Grain Shapes

19. Full Density Grains
depend on size ratio and concentration for conductivity
packing same size spheres, coordination 13, percolated from 19 to 81 vol %
size differs, can extend range
rule of thumb is 20 vol % gives connected system
elongated grains give percolation down to 1 vol %

20. Three Dimensional Spheres

21. Relation for Long-Range Contacts
𝑁 𝐶 𝑝 >1.5
NC is coordination number (3D)
p is probability of contacting a conductor
for loose monosized spheres gives 12 %
for dense monosized grains gives 21 %

22. Full Density Percolation
fully dense
condition
randomly
mixed powders

23. Example Concerns filters, pores must be connected fluid plugging pores
plastic is to be made conductivity
seeking long range stress transfer
avoid easy fracture in hard phase
(tough ligaments)
wear resistance without low toughness
high temperature creep, keep connected
spark sintering requires conduction

24. Particle Shape Effect
random dense packing

25. Particle Size Effect nanoscale range
data for tungsten, note low density of nanoscale powder

26. Spheres and Whiskers

27. Whisker Packing

28. Bimodal Packing

29. Size Ratio Role

30. Example Mixing

31. Maximum Density Random Bimodal
optimal large content
large and small powders fL and fS
  𝑋 𝐿 ∗ = 𝑓 𝐿 𝑓 ∗
packing density at optimal f*
𝑓 ∗ = 𝑓 𝐿 + 𝑓 𝑆 (1− 𝑓 𝐿 )

32. Rule of Thumb
monosized spheres 60 – 64 % dense
7-fold size difference
optimal 73 vol % large
expected peak at 87 % density
multiple modes
demonstrated 5 modes, 95 % dense

33. Composite Whiskers + Spheres

34. Ideal Bimodal Ordered Packing

35. Discrete Element Analysis
each element is a single particle
allow computer interactions – gravity, sticking, size, shape
simulate problems, such as segregation in handling
simulations rely on basic rules
need > 10,000 particles

36. Stability Criterion

37. Narrow Size Range, Random Packing

38. Bimodal, Different Sizes
note some
segregation

39. Bimodal 5 wt.% Very Small small particles percolate through
gaps between large
particles

40. Mixing & Blending Technologies
goal is to
BLEND = combine two powders same composition
MIX = combine two different powders
homogenize
additives
involves shear, short time motion

41. Homogeneity Parameter
rely on multiple sample variance in composition
ranges from 0 to 1
composition fluctuation variance S2
compare to variance perfectly mixed samples SR2
variance for the unmixed condition SI2
𝑀= 𝑆 𝐼 2 − 𝑆 2 𝑆 𝐼 2 − 𝑆 𝑅 2
𝑆 𝐼 2 = 𝑋 𝑃 (1 − 𝑋 𝑃 )
XP = concentration of the major component

42. Mixer Types Wet
wet = shear, twin screw, double planetary

43. Segregation - Homogeneity

44. Wet Mixing – Double Planetary

45. Dry Mixing

46. Production Scale
typical mixing time
10 to 20 min
bottom discharge

47. Dry Mixing Mechanisms

48. Process Control Agents
additives
stearic acid
paraffin wax
mineral oil
kerosene
polyacetal
polyethylene glycol

49. Fluid Bed Coating

50. Glued Particles

51. Time Based Homogeneity
mixture homogeneity M increases with mixing time t
𝑀= 𝑀 𝐼 + 𝛼 𝑒𝑥𝑝 𝐶+𝐾 𝑡
MI = initial mixture homogeneity
α, C and K = constants for specific conditions

52. Summary Comments random versus ordered packing percolation behavior
coordination number
several “handling” aspects
mixing
homogeneity
bimodal
discrete element analysis