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