Maximum packing densities of basic 3D objects

Li, Shuixiang; Zhao, Jian; Lu, Peng; Xie, Yu

doi:10.1007/s11434-009-0650-0

Cited by 106 publications

(57 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…101,102 In a quite different context, investigations on jammed random packing of non-spherical granular matter have been also performed aimed at discussing the effect of shape and size of the particles on the packing density. [104][105][106][107][108][109][110][111] For mixtures of HS with HSC of the same diameter and with an HSC aspect ratio less than 1.3, it was shown that the mixture, which can be either in a segregated or in a dispersive state, always exhibits isochoric ideality. 109,110 The picture of the solvation in the mixture brought at the molecular level by spectroscopy may help in understanding the volumetric isothermal expansion ("swelling") of the CS 2 liquid phase.…”

Section: Discussion On the Non-ideal Behaviour Of The Mixturementioning

confidence: 99%

Assessing the non-ideality of the CO2-CS2 system at molecular level: A Raman scattering study

Besnard

Cabaço²,

Coutinho

et al. 2013

The Journal of Chemical Physics

View full text Add to dashboard Cite

The dense phase of CO 2 -CS 2 mixtures has been analysed by Raman spectroscopy as a function of the CO 2 concentration (0.02-0.95 mole fractions) by varying the pressure (0.5 MPa up to 7.7 MPa) at constant temperature (313 K). The polarised and depolarised spectra of the induced (ν 2 , ν 3 ) modes of CS 2 and of the ν 1 -2ν 2 Fermi resonance dyad of both CO 2 and CS 2 have been measured. Upon dilution with CO 2 , the evolution of the spectroscopic observables of all these modes displays a "plateau-like" region in the CO 2 mole fraction 0.3-0.7 never previously observed in CO 2 -organic liquids mixtures. The bandshape and intensity of the induced modes of CS 2 are similar to those of pure CS 2 up to equimolar concentration, after which variations occur. The preservation of the local ordering from pure CS 2 to equimolar concentration together with the non-linear evolution of the spectroscopic observables allows inferring that two solvation regimes exist with a transition occurring in the plateau domain. In the first regime, corresponding to CS 2 concentrated mixtures, the liquid phase is segregated with dominant CS 2 clusters, whereas, in the second one, CO 2 monomers and dimers and CO 2 -CS 2 hetero-dimers coexist dynamically on a picosecond time-scale. It is demonstrated that the subtle interplay between attractive and repulsive interactions which provides a molecular interpretation of the non-ideality of the CO 2 -CS 2 mixture allows rationalizing the volume expansion and the existence of the plateau-like region observed in the pressure-composition diagram previously ascribed to the proximity of an upper critical solution temperature at lower temperatures.

show abstract

Section: Discussion On the Non-ideal Behaviour Of The Mixturementioning

confidence: 99%

Assessing the non-ideality of the CO2-CS2 system at molecular level: A Raman scattering study

Besnard

Cabaço²,

Coutinho

et al. 2013

The Journal of Chemical Physics

View full text Add to dashboard Cite

show abstract

“…In 2005, Hales provided a definitive mathematical proof confirming the observed experimental maximum packing density of 74% (1). Packing of more complex structures is far more mathematically challenging and instead relies on empirical studies (2)(3)(4)(5)(6)(7)(8)(9). One of the most ubiquitous packing systems in the universe are rigid aggregates composed of a collection of monomeric units joined together into a fractal structure and subsequently densified through omnidirectional applied force, as shown in Fig.…”

mentioning

confidence: 84%

Packing density of rigid aggregates is independent of scale

Zangmeister

Radney

Dockery

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

Large planetary seedlings, comets, microscale pharmaceuticals, and nanoscale soot particles are made from rigid, aggregated subunits that are compacted under low compression into larger structures spanning over 10 orders of magnitude in dimensional space. Here, we demonstrate that the packing density (θ f ) of compacted rigid aggregates is independent of spatial scale for systems under weak compaction. The θ f of rigid aggregated structures across six orders of magnitude were measured using nanoscale spherical soot aerosol composed of aggregates with ∼17-nm monomeric subunits and aggregates made from uniform monomeric 6-mm spherical subunits at the macroscale. We find θ f = 0.36 ± 0.02 at both dimensions. These values are remarkably similar to θ f observed for comet nuclei and measured values of other rigid aggregated systems across a wide variety of spatial and formative conditions. We present a packing model that incorporates the aggregate morphology and show that θ f is independent of both monomer and aggregate size. These observations suggest that the θ f of rigid aggregates subject to weak compaction forces is independent of spatial dimension across varied formative conditions. M any systems are comprised of elementary subunits packed within a defined volume. The simplest 3D system consists of uniform spheres, and despite its apparent simplicity, a rigorous mathematical description eluded researchers for nearly four centuries dating back to Kepler's conjecture in 1611. In 2005, Hales provided a definitive mathematical proof confirming the observed experimental maximum packing density of 74% (1). Packing of more complex structures is far more mathematically challenging and instead relies on empirical studies (2-9). One of the most ubiquitous packing systems in the universe are rigid aggregates composed of a collection of monomeric units joined together into a fractal structure and subsequently densified through omnidirectional applied force, as shown in Fig. 1 (10).The formation and compaction mechanism of disordered aggregates is presumed to be independent of dimension, composition, and spatial scale, and has been observed in a diverse range of materials and conditions, such as the accretion of material in interstellar space and the formation and compaction of aerosol in the Earth's atmosphere. Many interstellar formations comprise nano-or microscale dust particles that begin as disordered monomeric subunits, which electrostatically aggregate to form fractal (lacey) agglomerates that serve as foundries for comets and planetary seedlings (10-16). Soot, ubiquitous in the Earth's troposphere, is also comprised of nanometer sized carbonaceous monomers aggregated in a disordered lacey structure. Compaction into spheres occurs after trace gas and/or liquid adsorption and evaporation. In both cases, the resulting structure is constrained by aggregate rigidity (17).The systems described above are similarly constructed from single-unit building blocks assembled into larger disordered structures. The final structure ...

show abstract

“…The composite moduli are constant because an increase in corresponds to r mineral form the EFM to the CMF, thus the overall composite properties do not change. The plots are shown from with the lower bound of corresponding to and the upper bound corresponding to the close packing of the fiber, 0.9069 for hexagonally close packed and ~0.72 for disordered fibers 14 . b,d) Show the stress concentration due to the CMF.…”

Section: Stress Concentration From Fibers: Cmf Modelmentioning

confidence: 99%

The nanocomposite nature of bone drives its strength and damage resistance

2016

View full text Add to dashboard Cite

Crystallization of ACP with timeTo probe the stability of the observed ACP in disordered phase with respect to the adjacent HA nanocrystals, we performed TEM analysis of the same region after a 30-day time lapse. During the 30 days, the TEM sample was stored at room temperature in a N2 environment. Figure S1a shows the initial arrangement of ACP (left) and HA nanocrystals (right), as well as the diffraction pattern inset showing the crystallinity of the HA. Figure S1b shows the same region after the 30 day incubation period; the HA nanocrystals on the right appear to have grown, confirmed by the increased (112), (211), (300) intensities in the diffraction pattern inset; we also observe the emergence of (002) reflections after one month, which further proves the formation of crystals. These results demonstrate suggest that biogenic ACP is a precursor for crystalline cHA formation in bone in the observed conditions, as similarly observed by Mahamid et al 1 . Figure S1. Time lapse micrographs of amorphous and nanocrystalline regions. Deprotonated mineral structure in disordered phase taken before (a) and after (b) a 30 day incubation at room temperature in a nitrogen environment. The electron diffraction pattern after the incubation (b, inset) shows an increase in crystallinity as evidenced by more intense HA reflections and the emergence of the (002) spots when compared to the initial diffraction pattern (a, inset). Size Dependent Model ConstructionWe propose that the yield strength, , decreases as the probability of having a flaw (i.e. a pore) on the pillar surface increases; that is, surface flaws serve as probabilistic stress concentrators, which initiate failure during compression. This is manifested most prevalently in the larger pillars, with diameters > 500nm. At these nano and micro length scales, it is reasonable to consider bone as a fiber-

show abstract

Maximum packing densities of basic 3D objects

Cited by 106 publications

References 27 publications

Assessing the non-ideality of the CO2-CS2 system at molecular level: A Raman scattering study

Assessing the non-ideality of the CO2-CS2 system at molecular level: A Raman scattering study

Packing density of rigid aggregates is independent of scale

The nanocomposite nature of bone drives its strength and damage resistance

Contact Info

Product

Resources

About