Wednesday, June 11, 2014

Packing Density

In the particle theory of matter, all matter is made up of tiny particles. You've probably drawn circles in boxes to represent solids, liquids and gases. Each of those circles is a 2-dimensional representation on the paper of a 3-dimensional sphere. So, your representation of the particle theory of matter also implies that all matter is made up of particles which are spherical in shape.

If you draw the circles so close to each other that they are touching, you label the drawing "solid".
If the circles are little bit further apart, you label the diagram "liquid".
If the circles are much further apart, you label the drawing "gas".
But, have you ever wondered just how close you can pack those spheres together?

The study of soot is becoming very important because soot in the air relates to the balance of climate: heating from light absorption versus cooling from light reflection.
Soot is made up of small round particles of carbon about 10 or 20 nanometres across. The particles stick together randomly in short chains and clumps of a half dozen or more spheres. These, in turn, clump loosely together to form larger, loose aggregates of 10 or more which over a few hours will compact into a somewhat tighter ball which is atmospheric soot.
The closer you pack the soot spheres together, then the more dense the soot becomes. That is, since all the atoms have the same mass, if you pack them more tightly together then they will occupy a smaller volume, and, since density is mass per unit volume, the density of the soot will increase.

Mathematicians have been looking at sphere packing problems for more than a hundred years. In 1831, Carl Friedrich Gauss (one of the greatest mathematicians the world has known), determined that a close-packed arrangement in which 1 sphere is surrounded by 12 other identical spheres has an average density of π/√18 ≈ 0.74
The assumed density of soot in models of atmospheric soot has, until now, been 0.74.

A research group at the National Institute of Standards and Technology (NIST), have made measurements of actual soot particles and found the density of soot to be 0.36 not 0.74
So, the researchers set out to model soot particles using 6 mm plastic spheres glued together in thousands of random combinations forming clumps of from 1 to 12 spheres which were then used to fill every available size of graduated cylinder. When they graphed their results as a function of clump size,  they got a curve which levelled off at 0.36, the same as the density of soot particles they measured, but not the same as that predicted by the close-packing of spheres according to Gauss.

It now appears that, under normal conditions (not extreme temperature or pressure), the density of close-packed particles of any size, from nanometres to tens of metres is 0.36

C. D. Zangmeister, J. G. Radney, L. T. Dockery, J. T. Young, X. Ma, R. You, M. R. Zachariah. Packing density of rigid aggregates is independent of scale. Proceedings of the National Academy of Sciences, 2014; DOI: 10.1073/pnas.1403768111

Further Reading

Suggested Study Questions
  1. Draw a diagram to represent each of the following states of matter using the particle theory of matter:
    • solid
    • liquid
    • gas
  2. Calculate the volume of a spherical carbon particle that is 20 nanometres in diameter.
  3. Calculate the volume of a cube in which the length of each side is 20 nanometres.
  4. Calculate the ratio of the volume of the sphere to the volume of cube.
  5. Why is this ratio NOT 0.74?
  6. Draw a diagram showing the close-packing of just 2 carbon particles each with a diameter of 20 nanometres.
  7. Draw a square around each sphere to represent the volume of space occupied by the close-packed spheres and label the total volume occupied and the volume of carbon particles.
  8. For the close-packing of 2 carbon particles as drawn above, calculate the ratio of the volume occupied by the spheres to the total volume of the rectangular prism.
  9. Consider placing 1 more sphere above the 2 spheres you have already drawn. Draw a diagram of an arrangement that will occupy the
    • maximum total space
    • minimum total space
  10. Research the difference between hexagonal close packing and cubic close packing.

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