35:, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container. In other words, shaking increases the density of packed objects. But shaking cannot increase the density indefinitely, a limit is reached, and if this is reached without obvious packing into an ordered structure, such as a regular crystal lattice, this is the empirical random close-packed density for this particular procedure of packing. The random close packing is the highest possible volume fraction out of all possible packing procedures.
71:
state is RCP. The definition of packing fraction can be given as: "the volume taken by number of particles in a given space of volume". In other words, packing fraction defines the packing density. It has been shown that the filling fraction increases with the number of taps until the saturation density is reached. Also, the saturation density increases as the tapping
55:
of growth of ordered clusters to be exponentially small and relating it to the distribution of `cells', which are the smallest voids surrounded by connected discs. The derived maximum volume fraction is 85.3542%, if only hexagonal lattice clusters are disallowed, and 85.2514% if one disallows also deformed square lattice clusters.
234:
Products containing loosely packed items are often labeled with this message: 'Contents May Settle During
Shipping'. Usually during shipping, the container will be bumped numerous times, which will increase the packing density. The message is added to assure the consumer that the container is full
54:
The random close packing fraction of discs in the plane has also been considered a theoretically unsolved problem because of similar difficulties. An analytical, though not in closed form, solution to this problem was found in 2021 by R. Blumenfeld. The solution was found by limiting the probability
38:
Experiments and computer simulations have shown that the most compact way to pack hard perfect same-size spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. The problem of predicting theoretically the random
70:
Random close packing of spheres does not have yet a precise geometric definition. It is defined statistically, and results are empirical. A container is randomly filled with objects, and then the container is shaken or tapped until the objects do not compact any further, at this point the packing
58:
An analytical and closed-form solution for both 2D and 3D, mechanically stable, random packings of spheres has been found by A. Zaccone in 2022 using the assumption that the most random branch of jammed states (maximally random jammed packings, extending up to the fcc closest packing) undergo
95:
then the volume fraction depends non-trivially on the size-distribution and can be arbitrarily close to 1. Still for (relatively) monodisperse objects the value for RCP depends on the object shape; for spheres it is 0.64, for
755:
Donev, A.; Cisse, Ibrahim; Sachs, David; Variano, Evan A.; Stillinger, Frank H.; Connelly, Robert; Torquato, Salvatore; Chaikin, P. M. (2004). "Improving the
Density of Jammed Disordered Packings using Ellipsoids".
625:
Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaikin, P. M. (2004). "Improving the
Density of Jammed Disordered Packings Using Ellipsoids".
39:
close pack of spheres is difficult mainly because of the absence of a unique definition of randomness or disorder. The random close packing value is significantly below the maximum possible
590:
Ratnaswamy, V.; Rosato, A.D.; Blackmore, D.; Tricoche, X.; Ching, Luo; Zuo, L. (2012). "Evolution of Solids
Fraction Surfaces in Tapping: Simulation and Dynamical Systems Analysis".
547:
Rosato, Anthony D.; Dybenko, Oleksandr; Horntrop, David J.; Ratnaswamy, Vishagan; Kondic, Lou (2010). "Microstructure
Evolution in Density Relaxation by Tapping".
59:
crowding in a way qualitatively similar to an equilibrium liquid. The reasons for the effectiveness of this solution are the object of ongoing debate.
47:
and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of
235:
on a mass basis, even though the container appears slightly empty. Systems of packed particles are also used as a basic model of
695:
820:
248:
40:
278:
Torquato, S.; Truskett, T.M.; Debenedetti, P.G. (2000). "Is Random Close
Packing of Spheres Well Defined?".
48:
778:
647:
31:
objects obtained when they are packed randomly. For example, when a solid container is filled with
339:
424:
Zaccone, Alessio (2022). "Explicit
Analytical Solution for Random Close Packing in d=2 and d=3".
258:
773:
642:
91:
The particle volume fraction at RCP depends on the objects being packed. If the objects are
765:
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305:
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as the tapping amplitude goes to zero, and the limit as the number of taps goes to
731:
127:
24:
524:
511:
309:
568:
253:
41:
close-packing of same-size hard spheres into a regular crystalline arrangements
603:
353:"Disorder Criterion and Explicit Solution for the Disc Random Packing Problem"
814:
394:
214:
137:
92:
787:
656:
795:
739:
664:
576:
463:
402:
317:
236:
292:
97:
23:) of spheres is an empirical parameter used to characterize the maximum
338:
Modes of wall induced granular crystallisation in vibrational packing.
494:
72:
110:
Comparison of various models of close sphere packing (monodispersed)
438:
369:
122:
80:
710:
Jaeger, H. M.; Nagel, S. R. (1992). "Physics of
Granular States".
589:
546:
484:
32:
28:
277:
75:
decreases. Thus, RCP is the packing fraction given by the
512:"Maximizing space efficiency without order, analytically"
754:
624:
812:
688:Porous Media: Fluid Transport and Pore Structure
709:
350:
777:
646:
523:
516:Journal Club for Condensed Matter Physics
437:
368:
291:
86:
172:E.g., dropped into bed or packed by hand
685:
423:
813:
509:
485:
13:
351:Blumenfeld, Raphael (2021-09-09).
14:
832:
690:(2nd ed.). Academic Press.
679:
618:
583:
540:
503:
478:
456:10.1103/PhysRevLett.128.028002
417:
387:10.1103/physrevlett.127.118002
344:
332:
271:
249:Close-packing of equal spheres
103:
1:
732:10.1126/science.255.5051.1523
264:
65:
158:E.g., spheres slowly settled
43:, which is 74.04%. Both the
7:
525:10.36471/JCCM_March_2022_02
310:10.1103/PhysRevLett.84.2064
242:
10:
837:
686:Dullien, F. A. L. (1992).
569:10.1103/physreve.81.061301
340:Granular Matter, 21(2), 26
229:
604:10.1007/s10035-012-0343-2
155:Very loose random packing
821:Granularity of materials
510:Likos, Christos (2022).
217:(Coordination number 12)
134:Thinnest regular packing
49:granular crystallisation
45:face-centred cubic (fcc)
788:10.1126/science.1093010
657:10.1126/science.1093010
426:Physical Review Letters
357:Physical Review Letters
280:Physical Review Letters
259:Cylinder sphere packing
211:Densest regular packing
186:Spheres poured into bed
490:"Random Close Packing"
200:E.g., the bed vibrated
87:Effect of object shape
183:Poured random packing
197:Close random packing
169:Loose random packing
17:Random close packing
770:2004Sci...303..990D
724:1992Sci...255.1523J
639:2004Sci...303..990D
561:2010PhRvE..81f1301R
448:2022PhRvL.128b8002Z
379:2021PhRvL.127k8002B
302:2000PhRvL..84.2064T
142:Coordination number
112:
487:Weisstein, Eric W.
215:fcc or hcp lattice
108:
100:candy it is 0.68.
764:(5660): 990–993.
718:(5051): 1523–31.
697:978-0-12-223651-8
633:(5660): 990–993.
549:Physical Review E
286:(10): 2064–2067.
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592:Granular Matter
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349:
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206:0.625 to 0.641
192:0.609 to 0.625
128:Packing density
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25:volume fraction
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5:
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477:
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363:(11): 118002.
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254:Sphere packing
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203:0.359 to 0.375
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9:
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2:
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598:(2): 163–68.
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555:(6): 061301.
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432:(2): 028002.
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138:cubic lattice
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123:Void fraction
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93:polydispersed
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42:
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26:
22:
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237:porous media
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175:0.40 to 0.41
109:
90:
69:
61:
57:
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20:
16:
15:
119:Description
104:For spheres
439:2201.04541
370:2106.11774
265:References
66:Definition
774:CiteSeerX
643:CiteSeerX
612:254114944
534:247914694
495:MathWorld
472:245877616
411:237617506
395:0031-9007
98:M&M's
73:amplitude
815:Category
804:33409855
796:14963324
748:44568820
740:17820163
673:33409855
665:14963324
577:20866410
464:35089741
403:34558936
326:13149645
318:11017210
243:See also
81:infinity
766:Bibcode
758:Science
720:Bibcode
712:Science
635:Bibcode
627:Science
557:Bibcode
444:Bibcode
375:Bibcode
298:Bibcode
230:Example
223:0.7405
150:0.5236
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220:0.2595
147:0.4764
800:S2CID
744:S2CID
669:S2CID
608:S2CID
530:S2CID
468:S2CID
434:arXiv
407:S2CID
365:arXiv
322:S2CID
288:arXiv
164:0.56
116:Model
77:limit
33:grain
29:solid
792:PMID
736:PMID
692:ISBN
661:PMID
573:PMID
460:PMID
399:PMID
391:ISSN
314:PMID
161:0.44
784:doi
762:303
728:doi
716:255
653:doi
631:303
600:doi
565:doi
520:doi
452:doi
430:128
383:doi
361:127
306:doi
27:of
21:RCP
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