Editing Sphere packing (section)

==Unequal sphere packing{{anchor|Unequal}}==
{{see also|Unequal circle packing}}
[[File:Binary sphere packing LS3.png|thumb|A dense packing of spheres with a radius ratio of 0.64799 and a density of 0.74786<ref name="doi10.1021/jp206115p">{{Cite journal | last1 = O'Toole | first1 = P. I. | last2 = Hudson | first2 = T. S. | doi = 10.1021/jp206115p | title = New High-Density Packings of Similarly Sized Binary Spheres | journal = The Journal of Physical Chemistry C | volume = 115 | issue = 39 | pages = 19037 | year = 2011 }}</ref>]]
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or [[interstitial compound|interstitial]] packing. When many sizes of spheres (or a [[particle size distribution|distribution]]) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are available.

When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a close-packed arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends sensitively on the radius ratio, but in the limit of extreme size ratios, the smaller spheres can fill the gaps with the same density as the larger spheres filled space.<ref>{{Cite journal | last1 = Hudson | first1 = D. R. | title = Density and Packing in an Aggregate of Mixed Spheres | doi = 10.1063/1.1698327 | journal = Journal of Applied Physics | volume = 20 | issue = 2 | pages = 154–162| year = 1949 |bibcode = 1949JAP....20..154H }}</ref> Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere.<ref>{{Cite journal | last1 = Zong | first1 = C. | title = From deep holes to free planes | doi = 10.1090/S0273-0979-02-00950-3 | journal = Bulletin of the American Mathematical Society | volume = 39 | issue = 4 | pages = 533–555 | year = 2002 | doi-access = free }}</ref>

When the smaller sphere has a radius greater than 0.41421 of the radius of the larger sphere, it is no longer possible to fit into even the octahedral holes of the close-packed structure. Thus, beyond this point, either the host structure must expand to accommodate the interstitials (which compromises the overall density), or rearrange into a more complex crystalline compound structure. Structures are known which exceed the close packing density for radius ratios up to 0.659786.<ref name="doi10.1021/jp206115p"/><ref>{{cite journal|first1=G. W.|last1=Marshall|first2=T. S.|last2=Hudson|journal=Contributions to Algebra and Geometry|title=Dense binary sphere packings|volume=51|issue=2|pages=337–344|year=2010|url=http://www.emis.de/journals/BAG/vol.51/no.2/3.html}}</ref>

Upper bounds for the density that can be obtained in such binary packings have also been obtained.<ref>{{cite journal |last1=de Laat |first1=David |last2=de Oliveira Filho |first2=Fernando Mário |last3=Vallentin |first3=Frank |title=Upper bounds for packings of spheres of several radii|arxiv=1206.2608|date=12 June 2012 |doi=10.1017/fms.2014.24 |volume=2 |journal=Forum of Mathematics, Sigma|s2cid=11082628 }}</ref>

In many chemical situations such as [[ionic crystal]]s, the [[stoichiometry]] is constrained by the charges of the constituent ions. This additional constraint on the packing, together with the need to minimize the [[Coulomb energy]] of interacting charges leads to a diversity of optimal packing arrangements.

The upper bound for the density of a strictly jammed sphere packing with any set of radii is 1{{snd}}an example of such a packing of spheres is the Apollonian sphere packing. The lower bound for such a sphere packing is 0{{snd}}an example is the Dionysian sphere packing.<ref>{{cite journal |last1=Dennis |first1=Robert |last2=Corwin |first2=Eric |title=Dionysian Hard Sphere Packings Are Mechanically Stable at Vanishingly Low Densities |arxiv=2006.11415|date=2 September 2021 | journal=[[Physical Review]] |volume=128 | issue = 1 | pages = 018002 | doi=10.1103/PhysRevLett.128.018002 }}</ref>