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

Ordered and Random Packing

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

Packing Coordination

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

Consider Subtractive Experiment idealized 2D conductor structure

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

Non Conductive no continuous path (2D) for conduction

Early Demonstration silver particles in epoxy, composite conductivity

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

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

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

Coordination Change with Density best fit NC = 2 + 11 f2

Coordination Distribution

Homogeneity Role

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

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

Expected Grain Shapes

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 %

Three Dimensional Spheres

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 %

Full Density Percolation fully dense condition randomly mixed powders

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

Particle Shape Effect random dense packing

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

Spheres and Whiskers

Whisker Packing

Bimodal Packing

Size Ratio Role

Example Mixing

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 )

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

Composite Whiskers + Spheres

Ideal Bimodal Ordered Packing

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

Stability Criterion

Narrow Size Range, Random Packing

Bimodal, Different Sizes note some segregation

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

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

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

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

Segregation - Homogeneity

Wet Mixing – Double Planetary

Dry Mixing

Production Scale typical mixing time 10 to 20 min bottom discharge

Dry Mixing Mechanisms

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

Fluid Bed Coating

Glued Particles

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

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