-
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
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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)
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MotivationMotivation
• Nature:- sand, gravel, etc.
• Science:- colloids- granular media
• Technology:- catalyst carriers - food technology- reinforced
composites
Packings in
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Ordered sphere packingOrdered sphere packing
Kepler’s
conjecture : you can’t pack spheres denser than to asolid volume
fraction of = 0.7405./ 18
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Disordered, Disordered, ‘‘randomrandom’’ sphere packingsphere
packing
Disorded
spheres pack at a lower density of about 0.64 (the Bernal sphere
packing).
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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
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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
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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
Colloidal silica ellipsoids Granular matter
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?
Colloidal silica ellipsoids Granular matter
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
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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
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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
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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
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Long thin rodsLong thin rods
Simulations Experiments
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Long thin rodsLong thin rods
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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
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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
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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
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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:
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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.
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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
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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
-
Packing (rodPacking (rod--sphere mixture)sphere mixture)
composition: x = 0.1L/D = 10
L/D = 0 L/D = 1
L/D = 5
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composition: x = 0.5
L/D = 0.1 L/D = 1
L/D = 10 L/D = 100
Packing (rodPacking (rod--sphere mixture)sphere mixture)
-
composition: x = 0.5
L/D = 0.1 L/D = 1
L/D = 10 L/D = 100
Packing (rodPacking (rod--sphere mixture)sphere mixture)
-
composition: x = 0.9
L/D = 0.5 L/D = 2
L/D = 100L/D = 10
Packing (rodPacking (rod--sphere mixture)sphere mixture)
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Packing (rodPacking (rod--sphere mixture)sphere mixture)
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
-
Packing (rodPacking (rod--sphere mixture)sphere mixture)
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
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Packing (rodPacking (rod--sphere mixture)sphere mixture)
composition: x = 0.5
L/D = 1
L/D = 10
L
D
-
Packing (Packing (bidispersebidisperse rod mixture)rod
mixture)
composition: x = 0.5
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
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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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