1. Recent Work on Random Close Packing

2. 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

3. 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

4. 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

5. 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

6. 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

7. Theoretical Definition By Introducing the Concept of Maximally random jammed A schematic plot of the order parameter versus volume fraction for a system of identical spheres

8. 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?

9. 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

10. Quantify Order Using Polytetrahedral Nature In a tetrahedron, maximal edge length is e max, 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.

11. 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

12. Volume fraction of tetrahedra

13. Fraction of spheres involved in tetrahedra

14. 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.

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

16. Volume fraction of polytetrahedra

17. Comparison Notice the difference between the 2 graphs

18. 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

19. 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

20. Quantify Order But during the process, the value is chosen subjectively.

21. 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)

22. 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

23. THANK YOU!