Luis A. Pugnaloni - pasi2014.njit.edupasi2014.njit.edu/Lectures/pugnaloni.pdf · CM Carlevaro et...

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


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.


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


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


Packing fraction - Measuring it

Do basic nastygeometry

Do somephotographyφ =

Nblack pixelsNpixels

Do probabilisticgeometryφ = Nhits

Nshots

Calibrate withelectrostaticcapacity

Use heightφ =

NvgAh


Coordination number - kissing number

Coordination number

Kissing number

If only point contacts: 〈z〉 = 2Total number of contactsNumber of particles


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


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


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)


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})


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)


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++/


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)


Arches


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


Arches - Samples

Most arches are chain like even in3DTermination criteria

LA Pugnaloni (various publications)


Arches - History


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


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)


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).


Sample results - Tapping of disks revisited

R Arevalo et al., PRE (2013)


Sample results - Tapping of polygons

CM Carlevaro et al., JSTAT (2011)


Sample results - Tapping of narrow columns

RM Irastorza et al., JSTAT (2013)

Bowles-Ashwin model Bowles et al., PRE (2011)


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)


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).


Challenge

The unpacking challenge (Playing pick-up sticks )


Luis A. Pugnaloni - pasi2014.njit.edupasi2014.njit.edu/Lectures/pugnaloni.pdf · CM Carlevaro et...

Documents

Transcript of Luis A. Pugnaloni - pasi2014.njit.edupasi2014.njit.edu/Lectures/pugnaloni.pdf · CM Carlevaro et...