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Assume that the particles are spherical in shape with a diameter of 1 micron; that is, a radius of 0.5 microns. Each particle therefore has a volume of 0.5236 cubic microns, approx.
Now the volume to be filled is 1 cm3 = (104 microns)3 = 1012 cubic microns.
The highest density that can be achieved for sphere packing is pi/sqrt(18) so the maximum volume that the spheres can occupy is pi/sqrt(18)*1012 cubic microns = 7.405*1011 cubic microns.
At 0.5236 cubic microns each, this represents 1.4142*1012 particles or sqrt(2)*1012 of them.
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