Ideality in Granular Mixtures: Random Packing of Non ...€¦ · Ideality in Granular Mixtures: Random Packing of Non-Spherical Particles. ... hard dumbbell systems. ... PRE, 2007

Andriy KyrylyukVan ‘t Hoff Lab for Physical and Colloid Chemistry, Utrecht University, The Netherlands

Ideality in Granular Mixtures:Ideality in Granular Mixtures: Random Packing of NonRandom Packing of Non--Spherical ParticlesSpherical Particles

OutlineOutline

• Motivation.

• Spheres: the Bernal packing

• Thin rods: the ideal gas in random packings

• Near-spheres: density maximum + ideality(packing surprise #1)

• Mixtures: universality + ideality(packing surprise #2)

MotivationMotivation

• Nature:- sand, gravel, etc.

• Science:- colloids- granular media

• Technology:- catalyst carriers - food technology- reinforced composites

Packings in

Ordered sphere packingOrdered sphere packing

Kepler’s

conjecture : you can’t pack spheres denser than to asolid volume fraction of = 0.7405./ 18

Disordered, Disordered, ‘‘randomrandom’’ sphere packingsphere packing

Disorded

spheres pack at a lower density of about 0.64 (the Bernal sphere packing).

Hard sphere phase diagramHard sphere phase diagram

Volume fraction

fluid fluid + crystal crystal

glass0.500.58 0.64

0.740

(RCP)

Sha

pe a

niso

tropy

?

fluid

a) Colloids

b) Granular matter

Shap

e anis

otrop

y?

A.J.

Liu

and

S.R

. Nag

el, N

atur

e,19

98

J

jamming

Classical reference system for amorphous matter, colloidal glasses, etc.

Bernal random sphere packingBernal random sphere packing

Bernal random sphere packingBernal random sphere packing

volume fraction = 0.63

S.R. Wiliams and A.P. Philipse, Phys. Rev. E, 2003A. Wouterse et al., J. Chem. Phys., 2006J.D. Bernal, Nature, 1960

radial distribution function

64.060.0

Spheres are exceptional Spheres are exceptional ……

Failure to analyse

these packings in terms of ‘effective spheres’

Colloidal silica ellipsoids Granular matter

S. S

acan

naet

al.,

J. P

hys.

: Co

nden

s. M

atte

r,20

07

Generalize Generalize ‘‘BernalBernal’’ to particles of any shapeto particles of any shape

Conjecture: any particle shape has a unique, size-invariantmaximum random packing density


S. S

acan

naet

al.,

J. P

hys.

: Co

nden

s. M

atte

r,20

07

Where and how to start?Where and how to start?

Is any of these (or other) random packings truly random, in the sense that all spatial and orientational correlations are absent?


S. S

acan

naet

al.,

J. P

hys.

: Co

nden

s. M

atte

r,20

07

Thermal gas: Reference is an ideal gas of uncorrelated thermal particles.

Granular matter: Reference: an ideal packing of uncorrelated mechanical contacts.

A. Philipse, Langmuir 12, 1127 (1996)

A. Wouterse, Thesis (2008)

The ideal packingThe ideal packing

Counting uncorrelated contacts:

exVr

inside Vex

outside Vex

Orientationally averaged exclude volume:

Particle contactsParticle contacts

TT

( )exV

V f r d r

( ) 1

( ) 0

f r

f r

Contact number ( ) ( ) ; ( )TV

c f r r d r r

local nr.density

average nr. density

exV

Ideal packing law for uncorrelated contacts:

ex

cV

V

rdrf ;)(~

Ideal packing lawIdeal packing law

Ideal packing law for uncorrelated contacts:

ex

cV

= average contact number on a particle

Particle volume fraction : pV

p

ex

Vc

V

pV

c

= particle volume

But do uncorrelated contacts exists in dense granular packings ?

p

ex

VV

= fixed by particle shape.

Ideal packing lawIdeal packing law

Long thin rodsLong thin rods

Simulations Experiments

Long thin rodsLong thin rods

Clearly, as a rule, packings are non-ideal :

In the Bernal sphere packing, contacts are highly correlated.

In the random disc packing, correlations do not vanish in the thin-disc limit.

NonNon--ideal packingsideal packings

Packing (spherocylinders)Packing (spherocylinders)

Aspect ratio

Vol

ume

fract

ion

S.R. W

illia

ms

and

A.P.

Phi

lipse

, Phy

s. R

ev. E

, 200

3

Random contact equation:

cDL2

for L/D >> 1; < c >

10

L

Dspherocylinder

Packing (spherocylinders)Packing (spherocylinders)

Aspect ratio

S.R. W

illia

ms

and

A.P.

Phi

lipse

, Phy

s. R

ev. E

, 200

3

Vol

ume

fract

ion

density maximum

Random contact equation:

cDL2

for L/D >> 1; < c >

10

L

Dspherocylinder

Packing (ellipsoids)Packing (ellipsoids)A.

Don

evet

al.,

Sci

ence

, 200

4

(triangles) Ellipsoids(circles) Spherocylinders

A. W

oute

rse

et a

l., J

. Phy

s.:

Cond

ens.

Mat

ter,

2007

Is there universality in the density maximum?

Colloidal rods (spheroids)Colloidal rods (spheroids)

S. S

acan

naet

al.,

J. P

hys.

: Co

nden

s. M

atte

r,20

07

Packing (rodPacking (rod--sphere mixture)sphere mixture)

spherocylinder

sphere

+rod/sphere mixture:

Mechanical contraction method (MCM)Mechanical contraction method (MCM)L

D

System:

(a) spheres (b) spherocylinders

Procedure:

…43 1010 V

75 1010 V

VV 3/1

1

VVs

Dilute system is mechanically contracted until overlaps cannot be removed anymore. Result is a reproducible random packing density.

ApproachApproach

ijcijiiij nrvt̂

C

j

ijiji t

s1

1 iiii Ivv

C

jijiji nv

1

̂

zyxrnrnI

C

jcijijcijijiji ,,,,,

1

1

)()()()()(

constraint:

rate of overlap changing:

overlap removal speed:

Lagrange multiplier method direction of overlap removal:

A. Wouterse et al., J. Phys.: Condens. Matter, 2007

Packing (binary sphere mixture)Packing (binary sphere mixture)

lD

)6.2/( sl DD

A.V. Kyrylyuk, A. Wouterse and A.P. Philipse, Prog Colloid Polym Sci, 2010

A.B.

Yu

and

N. S

tand

ish.

, Pow

der

Tech

.,19

93

sD

Packing (binary sphere mixture)Packing (binary sphere mixture)

A.V. Kyrylyuk, A. Wouterse and A.P. Philipse, Prog Colloid Polym Sci, 2010

I. B

iazz

oet

al.,

Phy

s Re

v Le

tt,2

009

M. C

luse

let

al.,

Nat

ure,

2009


composition: x = 0.1L/D = 10

L/D = 0 L/D = 1

L/D = 5

composition: x = 0.5

L/D = 0.1 L/D = 1

L/D = 10 L/D = 100



L/D = 0.5 L/D = 2

L/D = 100L/D = 10



Universality + Ideality: the value of the density maximum depends linearly on the mixture composition

A.V. Kyrylyuk, A. Wouterse and A.P. Philipse, AIP Conf. Proc., 2009


A.V.

Kyr

ylyu

ket

al.,

in p

repa

ratio

n

Linearity for aspect ratios up to 1.7

+ = =

Φ

= Φs

xs

+ Φr

(1-xs

)

(law of mixtures)

Equality of mixed and demixed packings

Mixing Entropy = 0 !

0.72



L/D = 1

L/D = 10

L

D

Packing (Packing (bidispersebidisperse rod mixture)rod mixture)


L1 / D1 = 3

L2

D2

)( 21 DD )1( 2 L

L1

D1

A.V.

Kyr

ylyu

k an

d A.

P. P

hilip

se, i

n pr

epar

atio

n

Packing (Packing (polydispersepolydisperse rods)rods)

Uniform length distributionminmax

1)(LL

Lf

LmaxLmin

Glass transition of nearGlass transition of near--spheresspheres

F. Sciortino and P. Tartaglia, Adv. Phys., 2005

S.H

. Cho

ng a

nd W

. Got

ze, P

RE,2

002

M. L

etz,

R. S

chill

ing

and

A. L

atz,

PRE

,200

0

Ideal MCT glass transition for symmetric hard dumbbell systems

Ideal MCT glass transition for hard ellipsoids

Ideal glass transition for rod-like particlesG. Y

atse

nko

and

K.S.

Sch

wei

zer,

PRE

,200

7

ConclusionsConclusions• Bernal packing of spheres: no ideality

• Long thin rods: an ideal packing of uncorrelated mechanical contacts

• Non-monitonic packing behavior: deviation from spheres to near- spheres produces a density maximum

• Random packing of a rod-sphere mixture also has a density maximum for near-spheres:

- Universality: Positions of the density maximum and intersection point depend only on the rod aspect ratio and not on the composition

- Ideality: the height of the maximum depends linearly on the rod-sphere mixture composition

• The density maximum is also present in bidisperse and polydisperse rod mixtures

Universality: Position of the density maximum holds for one unique

rod aspect ratio and does not depend on the rod aspect ratio of the second component

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Ideality in Granular Mixtures: Random Packing of Non ...€¦ · Ideality in Granular Mixtures: Random Packing of Non-Spherical Particles. ... hard dumbbell systems. ... PRE, 2007