Question

Please provide a worked solution for the following problem.

8. Given a large volume of metallic powders, all of which are perfectly spherical and have the same exact diameter, what is the maximum possible packing factor that the powders can take?

Answer 1

The maximum packing factor, is defined as the true volume of the particle divided by the apparent volume occupied by them when the packing is maximum. It is a dimensionless quantity and always less than unity. In atomic systems, by convention, the APF is determined by assuming that atoms are rigid spheres. The radius of the spheres is taken to be the maximum value such that the atoms do not overlap. For one-component crystals (those that contain only one type of particle), the packing fraction is represented mathematically by APF = Nparticle Vparticle Vunit cell The highest-density lattice arrangement of circles is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, like a honeycomb), and each circle is surrounded by 6 other circles. The density of this arrangement for circles of diameter D, is = *O20.9069. D is also the side of the hexagon in the first figure. The first term D 2 in the ratio above is the sum total of the area of all the circles and partial circles enclosed by the hexagon. The second term 3V3 D2 is the area of the hexagon itself. Hexagonal packing of equal circles was found to fill a fraction. * 0.9069 of area /12