luis a. pugnaloni - pasi2014.njit.edupasi2014.njit.edu/lectures/pugnaloni.pdf · cm carlevaro et...
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Structural description of packed particulates
Luis A. Pugnaloni
Departamento de Ingeniería Mecánica, Facultad Regional La Plata, Universidad Tecnológica Nacional
PASI 2014
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 1 / 34
Outline
Packing fraction, φCoordination number, z, and kissing numberPair correlation function, g(r)
Autocovariance function, ξ(r)
Bond order parameters, Ql and Wl
Fabric tensor, FVoronoi cellsContact network polygonsArchesInterplaysExamplesConclusionsChallenge
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 2 / 34
Packing fraction
Packing fraction, volume(area)fraction: φ, c, η
φ =real volume of particles
apparent volume of assembly.
Not to confuse with number density
ρ =NV,
V : container volume.If all particles are the same
φ =Nvg
V= ρvg ,
vg : volume of one particle.
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 3 / 34
Packing fraction limits
3D: 0 < φ < π
3√
2≈ 0.74 Kepler-Hales (fcc or hcp)
2D: 0 < c < π√
36 ≈ 0.91 (hexagonal)
Packing!!Random Close Packing ≈ 0.64(3D) or 0.82(2D)JG Berryman, PRA (1983)S Torquato et al., PRL (2000) → Maximally jammed packing
Random Loose Packing ≈ 0.55(3D) or 0.77(2D)GY Onoda et al., PRL (1990)LE Silbert, Soft Matter (2010)
Really Loose Packings!D Bi et al., Nature (2011) → Shear jammed packingM Pica Ciamarra et al., PRL (2008) → zero entropy packing
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 4 / 34
Packing fraction limits
Polygons (2D) CM Carlevaro et al., JSTAT (2011)Non-tiling: 0.75 < c < 0.86Tiling: 0.77 < c < 1.0
Cohesive powders (3D)0.15 < c < 0.48
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 5 / 34
Packing fraction - Measuring it
Do basic nastygeometry
Do somephotographyφ =
Nblack pixelsNpixels
Do probabilisticgeometryφ = Nhits
Nshots
Calibrate withelectrostaticcapacity
Use heightφ =
NvgAh
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 6 / 34
Coordination number - kissing number
Coordination number
Kissing number
If only point contacts: 〈z〉 = 2Total number of contactsNumber of particles
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 7 / 34
Kissing number - limits
Maximum (mono-sized)Spheres: z = 12Disks: z = 6
Sequential deposition(any convex shape / hard particles)2D: 〈z〉 = 2 (2N)
N = 4 (add 2 contacts per particle)3D: 〈z〉 = 2 (3N)
N = 6 (add 3 contacts per particle)
Isostatic (point contacts) Constraints = Forces2D: 3N scalar constraints (2 translational, 1 rotational) and 2Nc scalar forces→ Nc = 3N
2 → 〈z〉 = 2 NcN = 3
3D: 6N scalar constraints (3 translational, 3 rotational) and 3Nc scalar forces→ Nc = 2N → 〈z〉 = 2 Nc
N = 4
Some values (confocal): φ = 0.4→ 〈z〉 = 4; φ = 0.65→ 〈z〉 = 8
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 8 / 34
Kissing number - measuring it
The kissing number is difficult to measure since is difficult to tell apart realcontacts from near contacts.In simulations, simply saving position requires high precision. Is safer totell from the forces.Poeschel rule for ending contact forces impedes to define contacts fromgeometry!Use of paint, oxidation, fluorescent markers, photoelastic particles, liquidbridges, conductivity, etc.
Behringer Kudroli
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 9 / 34
Pair correlation function
4πr2ρg(r)dr = Probability of finding the center of a particle in a shell of radiusr and thickness dr centered on another particle.
Palombo, Sci. Rep. (2013)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 10 / 34
Pair correlation function - measuring it∫∞0 4πr2ρg(r)dr = N − 1
h(r) = g(r)− 1 can be integrated if no long ranged correlations exist
S(k) = F{h(r)} = h̃(r) (static structure factor, scattering experiments)Detailed positionsScan throughout all pairs ofparticles and put thecenter-to-center distance in ahistogram with bin width dr .Then normalize with 4πr2ρdr toget g(r). You have to know thenumber density ρ beforehand.
maxbin; L; Ndr = float(L/maxbin)hist = [0]*(maxbin+1)rdf = {}
#### READ COORDINATES IN ATOMS = [] ####for i in range(npart):
xi = (atoms[i])[0]; ...for j in range(i+1, npart):xx = xi - (atoms[j])[0]; ...rij = sqrt(xx*xx + yy*yy + zz*zz)bin = int(ceil(rij/dr))if (bin <= maxbin):hist[bin] += 1
phi = N/(L*L*L) # NORMALIZATIONnorm = 2.0 * pi * dr * phi * Nfor i in range(1, maxbin+1):
r = (i - 0.5) * drval = hist[i]/norm/((r*r)+(dr*dr)/12.0)rdf.update({rrr:val})
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 11 / 34
Autocovariance function
SI2(r̄I, r̄2) = 〈AI(r̄1)AI(r̄2)〉 with AI(r̄) =
{1 if r̄ ∈ I0 if r̄ /∈ I
If the medium is statistical homogeneous and isotropic: SI2(r̄I, r̄2) = SI
2(r)ThenχI(r) = SI
2(r)− φI Autocovariance (integrable)
χI(r) = χII(r) since AI(r̄1) = 1− AII(r̄1) and φI = 1− φII, then χI(r) is unique.
Can be measured on experimental images or simulation data by drawingrandom positions uniformly distributed in the space. Does not need detailedinformation on contacts nor centers.
Y Jiao et al., PRE (2007)Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 12 / 34
Bond order parameter
Ql =
[4π
2l + 1
l∑m=−l
|Q̄lm|2]1/2
Q̄lm is the mean over all contacts ofQlm(r̄) = Ylm(θ(r̄ , φ(r̄)) the spherical harmonics.
Ql (and also other parameters like Wl ) can be calculated for a single grain orfor a entire packing.http://www.pas.rochester.edu/ wangyt/algorithms/bop/ (Yanting Wang)
fcc 0.57452hcp 0.48476
icosahedral 0.66332liquid 0 Yelow 0.740829
Green 0.759623PJ Steinhardt et al., PRB (1983)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 13 / 34
Fabric tensor
F =∑
contacts
n̄c ⊗ n̄c =∑
contacts
nxnx nxny nxnzny nx ny ny ny nznznx nzny nznz
F is symmetric and real→ Hermitian→ can be diagonalized and theeigenvalues are real: F1,F2,F3.Tr(F ) =
∑contacts |nc |2 = Nc
F = FI + FD (isotropic + deviator) with FI = 13 Tr(F )δij (3D)
[2 00 2
] [2 00 0
] [0 00 2
] [2 00 2
] [ 32
12
12
32
]Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 14 / 34
Voronoi cells
Voronoi cell: Area or volumeclosest to each point in the set(see white polygons). Thereciprocal network is Delaunay(see black network).For sphere and disk packingseach particle is contained in eachcell (not in other systems).Neighbors may or may not be incontact.The distribution of voronoivolumes seem to show a universaldistribution (for packings and forunjammed systems). T Aste et al.(various publications).
Voro++:http://math.lbl.gov/voro++/
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 15 / 34
Contact network polygons + quadrons
The contact network does notinclude neighbors not in contact.The indivisible polygons formed bythe network may be very distorted.The polygons tile the space, butare not assigned to any particle
R Arevalo et al., (Various publications)
The quadron tile the space and can beassigned to a particle.
R Blumenfeld (various publications)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 16 / 34
Arches
Supporting contactsMutually stabilizing contacts(MSC)Aggregates of particles with MSCHistory dependentni : number of arches of i particles(includes i = 1)ni (x): span of the arches of size i
PLAY MOVIE
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 18 / 34
Arches - Samples
Most arches are chain like even in3DTermination criteria
LA Pugnaloni (various publications)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 19 / 34
Interplays
φ ↑ → z ↑
Arches↑ → z ↓: zsupport = 2[1 + 1
N
∑Ns=1 ni
](2D)
Arches↑ → φ ↓ in general but not always〈z〉 = Tr(F )/NVoronoi cells can be used to assign a local volume to each particle and socalculate local φ
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 26 / 34
Sample results - Arches in pseudo-dynamics and DEM
R Arevalo et al., PRE (2006); LA Pugnaloni et al., PRE(2006); Adv. Complex. Syst. (2001);Physica A (2004)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 27 / 34
Sample results - Tapping of disks and spheres
LA Pugnaloni et al., PRE (2008)
States distinguishable by stress.I Sanchez et al., PRE (2010); LA Pugnaloni et al., Pap Phys. (2011).
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 28 / 34
Sample results - Tapping of disks revisited
R Arevalo et al., PRE (2013)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 29 / 34
Sample results - Tapping of polygons
CM Carlevaro et al., JSTAT (2011)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 30 / 34
Sample results - Tapping of narrow columns
RM Irastorza et al., JSTAT (2013)
Bowles-Ashwin model Bowles et al., PRE (2011)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 31 / 34
Sample results - Force chains and arches
CM Carlevaro et al., EPJE (2012)
Constraints force the PDF decay faster than exponential.BP Tighe et al., PRL (2008)
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 32 / 34
Conclusions
The structural description of a packing is the most used approach tosurvey changes of state.However, some features cannot be captures without taking into accountthe contact forces.Some seemingly structural descriptors (such as arches) containinformation on the history of the packing.There exist still a need for formal relations between different structuraldescriptors. Particularly to go from the mesoscale (arches) to themacroscale (φ).Newer structural descriptors are being proposed (see Lou’s lecture).
Luis A. Pugnaloni (Mecánica - UTN-FRLP) Packing structure 2014 33 / 34