Recent Work on Random Close Packing

1. Polytetrahedral Nature of the Dense Disordered Packings of Hard Spheres

PRL 98, 235504 (2007)

A.V. Anikeenko and N. N. Medvedev

2. Is Random Close Packing of Spheres Well Defined?

VOLUME 84, NUMBER 10 (2000)

S. Torquato, T.M. Truskett, P. G. Debenedetti

The History

J D Bernal raised the question in 1960 from observation;

In the last 48 years, much research was done over this, but so far, few satisfactory conclusions have been reached

Why This Is An Important Question

• We can obtain a better insight of phase transition

An EXPERT should be able to show you more its importance,

but sorry, I am just a small potato

Definition (From Observation)

• The traditional experiments indicated an interesting limit 0.636

• One such experiment is done in 1969, limiting value 0.637 was obtained simply from experiments.

• Nowadays, 2 ways of simulations are mainly applied

Can We Have A Theoretical Definition?

• Jammed system means all particles are “jammed”

• Introducing a parameter that quantifies “order” (difficult one, may be subjective)

• Maximally Random Jammed （ MRJ ） system is the one we desire, and its volume fraction is the RCP limit

Theoretical Definition By Introducing the Concept ofMaximally random jammed

• A schematic plot of the order parameter versus volume fraction for a system of identical spheres

Independent of how we quantify order?

Can we use simulations to have a feeling for the probability distribution for RCP to answer the question above?

How Can We Quantify Order• In the paper Is Random Close Packing of

Spheres Well Defined, they didn’t come up with an exact and simple way to quantify the order

• The paper Polytetrahedral Nature of the Dense Disordered Packings of Hard Spheres showed us an interesting phenomenon, which inspired me a simple way to quantify order

Quantify Order Using Polytetrahedral Nature

• In a tetrahedron, maximal edge length is emax, minimal edge length is 1 (i.e. the diameter of the sphere)

• Define a value

• Small values of unambiguously indicate that the shape of the simplex is close to regular tetrahedron with unit edges.

Define tetrahedra

• By simply trying, they found that the best value may be

• Here, we realize that the choose of the value 0.225 may be subjective. But, as we go on, we saw phenomena independent of 0.225. And, 0.225 may just be a most obvious value

Volume fraction of tetrahedra

Fraction of spheres involved in tetrahedra

Polytetrahedral Aggregates

• Definition:

clusters built from three or more face adjacent tetrahedra

• Isolated tetrahedra and pairs of tetrahedra (bipyramids) are omitted as they are found in the fcc and hcp crystalline structures.

Polytetrahedral Aggregates

• In the general case polytetrahedra have the form of branching chains and five-member rings combining in various ‘‘animals’’ MOTIF?

Volume fraction of polytetrahedra

Comparison

• Notice the difference between the 2 graphs

• This gives me a hint for quantifying order

• I think, we should find the function for “volume fraction of isolated tetrahedra and bipyramids” against packing volume fraction, i.e. the difference between the 2 graph. We use L to denote it.

• From pure observation, we notice that L is approximately a constant at low density, but after 0.646, L has a sudden increase

Quantify Order

Quantify Order

• We may use K to denote the volume fraction of spheres within polytetrahedral aggregates.

• To quantify disorder, we use K/L.

• Thus we use 1/(1+K/L) to denote order

Quantify Order

But during the process, the value is chosen subjectively.

But from the left graph, we may expect that different choices of

may lead to the same MRJ result.

(This is only my naive expectation)

The Following Work May Be Done

• Obtain the - graph from simulation under different choices of

• Is the MRJ is independent of how we quantify order?

• Is the limit independent of

• Of course, the most important thing for me is to get into the problem

THANK YOU!