Ideality in Granular Mixtures: Random Packing of Non-Spherical Particles
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1 Ideality in Granular Mixtures: Random Packing of Non-Spherical Particles Andriy Kyrylyuk Van t Hoff Lab for Physical and Colloid Chemistry, Utrecht University, The Netherlands
2 Outline 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)
3 Motivation Packings in Nature: - sand, gravel, etc. Science: - colloids - granular media Technology: - catalyst carriers - food technology - reinforced composites
4 Ordered sphere packing Kepler s conjecture : you can t pack spheres denser than to a solid volume fraction of = / 18
5 Disordered, random sphere packing Disorded spheres pack at a lower density of about 0.64 (the Bernal sphere packing).
6 Hard sphere phase diagram a) Colloids 0 Shape anisotropy fluid fluid + crystal crystal 0.50 fluid glass (RCP) b) Granular matter?? Shape anisotropy J jamming 0.74 Volume fraction A.J. Liu and S.R. Nagel, Nature, 1998
7 Bernal random sphere packing Classical reference system for amorphous matter, colloidal glasses, etc.
8 Bernal random sphere packing S.R. Wiliams and A.P. Philipse, Phys. Rev. E, 2003 A. Wouterse et al., J. Chem. Phys., 2006 J.D. Bernal, Nature, 1960 radial distribution function volume fraction =
9 Spheres are exceptional Colloidal silica ellipsoids Granular matter S. Sacanna et al., J. Phys.: Condens. Matter, 2007 Failure to analyse these packings in terms of effective spheres
10 Generalize Bernal to particles of any shape Colloidal silica ellipsoids Granular matter S. Sacanna et al., J. Phys.: Condens. Matter, 2007 Conjecture: any particle shape has a unique, size-invariant maximum random packing density
11 Where and how to start? Colloidal silica ellipsoids Granular matter S. Sacanna et al., J. Phys.: Condens. Matter, 2007 Is any of these (or other) random packings truly random, in the sense that all spatial and orientational correlations are absent?
12 The ideal packing 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)
13 Particle contacts Counting uncorrelated contacts: f( r) 1 inside V ex T T r f( r) 0 outside V ex V ex Orientationally averaged exclude volume: V ex V f( r) d r
14 Ideal packing law Contact number c f( r) ( r) d r; ( r) T V local nr.density ~ f ( r ) dr; V average nr. density V ex Ideal packing law for uncorrelated contacts: c V ex
15 Ideal packing law Ideal packing law for uncorrelated contacts: c V ex c = average contact number on a particle Particle volume fraction : V p V p = particle volume c V V p ex V V p ex = fixed by particle shape. But do uncorrelated contacts exists in dense granular packings?
16 Long thin rods Simulations Experiments
17 Long thin rods
18 Non-ideal packings 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.
19 Packing (spherocylinders) L Random contact equation: S.R. Williams and A.P. Philipse, Phys. Rev. E, 2003 Volume fraction D spherocylinder L 2 c D for L/D >> 1; < c > 10 Aspect ratio
20 Packing (spherocylinders) L Random contact equation: S.R. Williams and A.P. Philipse, Phys. Rev. E, 2003 Volume fraction D spherocylinder density maximum L 2 c D for L/D >> 1; < c > 10 Aspect ratio
21 Packing (ellipsoids) Packing (ellipsoids) (triangles) Ellipsoids (circles) Spherocylinders A. Donev et al., Science, 2004 A. Wouterse et al., J. Phys.: Condens. Matter, 2007 Is there universality in the density maximum?
22 Colloidal rods (spheroids) Colloidal rods (spheroids) S. Sacanna et al., J. Phys.: Condens. Matter, 2007
23 Packing (rod-sphere mixture) rod/sphere mixture: sphere + spherocylinder
24 Mechanical contraction method (MCM) System: L D (a) spheres (b) spherocylinders Procedure: V V V V s V 1 V 1/ 3 Dilute system is mechanically contracted until overlaps cannot be removed anymore. Result is a reproducible random packing density.
25 Approach rate of overlap changing: t ij vi i rcij nˆ ij A. Wouterse et al., J. Phys.: Condens. Matter, 2007 overlap removal speed: constraint: v s i i v i C j1 ij t ij I 1 Lagrange multiplier method direction of overlap removal: C v i nˆ i j1 ij i ij ( ) i 1 I C j1 ij ( ) ( ) ( ) ( ) n r n r,,, x, y, z ij cij ij cij
26 Packing (binary sphere mixture) D l D s ( D l / D s 2.6) A.B. Yu and N. Standish., Powder Tech., 1993 A.V. Kyrylyuk, A. Wouterse and A.P. Philipse, Prog Colloid Polym Sci, 2010
27 Packing (binary sphere mixture) I. Biazzo et al., Phys Rev Lett, 2009 M. Clusel et al., Nature, 2009 A.V. Kyrylyuk, A. Wouterse and A.P. Philipse, Prog Colloid Polym Sci, 2010
28 Packing (rod-sphere mixture) L/D = 0 L/D = 1 L/D = 5 composition: x = 0.1 L/D = 10
29 Packing (rod-sphere mixture) L/D = 0.1 L/D = 1 L/D = 10 L/D = 100 composition: x = 0.5
30 Packing (rod-sphere mixture) L/D = 0.1 L/D = 1 L/D = 10 L/D = 100 composition: x = 0.5
31 Packing (rod-sphere mixture) L/D = 0.5 L/D = 2 L/D = 10 composition: x = 0.9 L/D = 100
32 Packing (rod-sphere mixture) A.V. Kyrylyuk, A. Wouterse and A.P. Philipse, AIP Conf. Proc., 2009 Universality + Ideality: the value of the density maximum depends linearly on the mixture composition
33 Packing (rod-sphere mixture) Linearity for aspect ratios up to = = Φ = Φ s x s + Φ r (1-x s ) (law of mixtures) Equality of mixed and demixed packings 0.72 A.V. Kyrylyuk et al., in preparation Mixing Entropy = 0!
34 Packing (rod-sphere mixture) L D L/D = 1 L/D = 10 composition: x = 0.5
35 Packing (bidisperse( rod mixture) L 1 L 2 D 1 D 2 ( D1 D2) ( L 2 1) A.V. Kyrylyuk and A.P. Philipse, in preparation L 1 / D 1 = 3 composition: x = 0.5
36 Packing (polydisperse( rods) Uniform length distribution f ( L) L max 1 L min L min L max
37 Glass transition of near-spheres M. Letz, R. Schilling and A. Latz, PRE, 2000 S.H. Chong and W. Gotze, PRE, 2002 Ideal MCT glass transition for hard ellipsoids G. Yatsenko and K.S. Schweizer, PRE, 2007 Ideal glass transition for rod-like particles Ideal MCT glass transition for symmetric hard dumbbell systems F. Sciortino and P. Tartaglia, Adv. Phys., 2005
38 Conclusions Bernal packing of spheres: no ideality Long thin rods: an ideal packing of uncorrelated mechanical contacts Non-monitonic packing behavior: deviation from spheres to nearspheres 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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