Lecture 17. Insights into disorder

Transcription

1 Lecture 17 Insights into disorder

2 C 4 C 6 Order is convenient Structures can be classified in a compact way A certain amount of defects can be introduced and classified Order is efficient The closest sphere packs are the Barlow packings! = ! = Disorder is everywhere There are local motifs which repeat (a-periodically) in space. They are organized hierarchically. They results in efficient configurations both geometrically and topologically. disorder is NOT randomness

3 DISORDERED PACKING ARE INTRINSICALLY COMPLEX COMPLEX SYSTEM a a system that is long to describe (G. Parisi)

4 phy.duke..duke.edu/research/ltb/ltbgroup.html

5 Why disorder is different from randomness?

6 Organization (Local) Hierarchy (Global)

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8 Can we identify typical properties of disordered systems? -packing grains-

9 Give, and it shall be given to you. Good measure, pressed down, shaken together, running over, will be put into your lap. For with the measure you use it will be measured back to you. Luke 6:38 the method of filling the measure has to be very particularly stipulated Different preparation methods give results which differ by up to 15% thought among themselves the difference is less than 0.5%. Knight et al. PRE 51 (1995) 3957; E. R. Nowak et al. PRE 57 (1998) Ribiere et al. EPJE 22 (2007) 249

10 Packings of equal spheres Science is measurement, but what is a good measure? r = % less dense than the best packing 15% more dense than the worst packing J. D. Bernal, A geometrical approach to the structure of liquids, Nature 183 (1959) J. D. Bernal and J. Mason, Co-ordination of Randomly Packed Spheres, Nature 188 (1960) G. D. Scott, Radial Distribution of Random Close Packing of Equal Spheres, Nature 194 (1962) Random loose packing RLP gas 0.5 fluid 0.55 ~0.62 ~0.58 ~0.6 glass Random close packing RCP 0.64 crystal 0.74 Density (Packing Fraction)! Lennard Jones MD, hard spheres, colloids, sands, powders, grains

11 At this stage the problem resolves into two. what is the structure? why has it got this structure? Statistical Geometry Statistical Mechanics

12 LOOKING INSIDE COMPLEX MATTER (ANU XCT facility) THE EXPERIMENTS acrylic Spheres with diameter 1 ± 0.05 mm packing fractions (A) acrylic Spheres with diameter 1.59 ± 0.05 mm packing fractions (B) (D) (E) (F) glass Spheres (C) XCT Tomogram ( voxels) XCT Tomogram ( voxels) 12 samples, packing fractions XCT Tomogram ( voxels) Dry packings The whole range of packing fractions between 0.56 to 0.64 (T Senden, M Saadatfar) Fluidized beds (+ M Schroter)

13 THE SIMULATIONS Packings of ,000 identical spheres Modified Jodrey-Tory algorithm (200 samples) [W. S. Jodrey and E. M. Tory, Phys. Rev. A. 32, 2347 (1985). A. Bezrukov, M. Bargiel, and D. Stoyan, Part. Past. Syst. Char. 19, 111 (2002). K. Lochmann, A. Anikeenko, A. Elsner, N. Medvedev, and D. Stoyan, Eur. Phys. J. B 53, 67 (2006). ] Modified Lubachevsky-Stillinger algorithm (~50,000 samples) [M. Rintoul and S. Torquato, Phys. Rev. E. 58, 532 (1998). M. Skoge, A. D. F. H. Stillinger, and S. Torquato, Phys. Rev. E. 74, (2006). A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Phys. Rev. E. 71, (2005). ] Discrete elements simulations and virtual experiments (~100 samples) [Gary W. Delaney, Shio Inagaki and Tomaso Aste Fine tuning DEM simulations to perform virtual experiments with three-dimensional granular packings, Lecture Notes in Complex Systems Vol.8 (World Scientific, Singapore 2007) p ]

14 What the structure? Divide the system into parts: simplicial decomposition Voronoï Delaunay

15 quasi-regular tetrahedra Edge differences, T -measure 1 T = #(l 15 l 2 i " l j ) 2 i< j Procrustean distance, d-measure $ 4 1 ' d 2 = min R,t,P % # y i " (Rx i + t) 2 ( & 4 ) i=1 Maximal edge length,!-measure T = 0 T = T < d = 0 d t < d q d = " = min( l i ) #1 " = 0 " < " =

16 What happens at the Bernal limit? 1/3 quasi -perfect tetrahedra Experiments 1/3 = fraction of regular tetrahedra in close packed structures

17 Polytetrahedral aggregate

18 Fraction of polytetrahedral aggregate RCP A.V. Anikeenko, N.N. Medvedev, T. Di Matteo, G. Delaney and T. Aste, Delaunay simplex analysis of the structure of equal sized sphere packings, Lecture Notes in Complex Systems Vol.8 (World Scientific, Singapore 2007) p

19 WHY HAS IT GOT THIS STRUCTURE? A STATISTICAL MECHANICS APPROACH The goal is to achieve the analogous of the Maxwell - Boltzmann distribution Given the temperature calculate the Probability Distribution for the velocity of each molecule. Given the packing fraction calculate the Probability Distribution for the Voronoi volume of each grain.

20 A DEDUCTIVE STATISTICAL MECHANICS APPROACH MECHANICS APPROACH (Lecture 5) Complete (microscopic) description of the SYSTEM STATIC STATE A mechanism (trial) to drive the system from a (static) state to another Transition probability that during t trials a system prepared in the state " can jump to the state # Assume that the transition probability depends only on the two states " and # P t (" # $) P t (" # $ 0 ) = & P 1 (" # $ t%1 )P 1 ($ t%1 # $ t%2 )LP 1 ($ 1 # $ 0 ) {$ 1,$ 2...$ t } & $ P t (") = w(" # $)P t%1 ($) P t (" # $ 0 ) P 1 (" # $) P t"1 (# $ # 0 ) " Chapman-Kolmogorov

21 " 0 # V 0 " 1 # V 1 w(" 1 # " 0 ) > 0 w(" 0 # " 1 ) = 0 Transient Ergodic Set: Statistical Equilibrium p " (V ) = $ w(v # V ') p " (V ') V ' Persistent <V> Z(V) V V

22 CALCULATE THE PROBABILITY BY MAXIMIZING ENTROPY S(Z(V )) = " % p # (V )logp # (V ) + % p # (V )S(V ) V $Z S(V ) = log"(v ) V $Z #(V )exp($ V % ) p " (V ) = &#(V ')exp($v ' % ) V ' V = V p " (V ) # V

23 THE DEGREES OF SPACE PARTITION 1) The microscopic state of the system can be encoded in terms of the properties of k local cells " = { c,c,...,c } 1 2 k 2) The properties of these cells are either fully described by their volumes v i or they are independent form v i 3) Any space partitions with cells of volumes v i! v min satisfying the condition v 1 +v v k = V corresponds to mechanically stable packings "(V ) = (V #V min )k#1 (k #1)!$ dk p " (V ) = k k (V #V min ) k#1 $ exp #k V #V min & (k #1)! (V #V min ) k % V #V min ' ) ( granular temperature " = V #V min k

24 P(V) WHY HAS IT GOT THIS STRUCTURE? p " (V ) = VORONOI CELLULAR PARTITION k k (V #V min ) k#1 $ exp #k V #V min & (k #1)! (V #V min ) k % V #V min ' ) ( k!12 k=5.6 k!20 3/4 ( ) V min = /2 " % $ ' # 2 & The analogous of the MAXWELL-BOLTZMANN DISTRIBUTION F. W. Starr et al., Phys. Rev. Lett.89 (2002) Chain of 20 monomers with Lennard-Jones potential at fixed density but different temperatures. V "V min V "V min V P(v) P(v) T. Aste, T. Di Matteo, M. Saadatfar, T. J. Senden, Matthias Schröter and Harry L. Swinney, An invariant distribution in static granular media, Euro Phys. Lett. 79 (2007) T. Aste, T. Di Matteo A deductive statistical mechanics approach for granular matter, arxiv: , J.Stat. Phys. (2007) submitted. T. Aste, T. Di Matteo Emergence of kgamma distributions in granular materials and packing models, Phys. Rev. E (2007) submitted. v Coagulated colloidal particles v I. Schenker, F. T. Filser, T. Aste and L. J. Gauckler, Microstructures and Mechanical Properties of Dense Particle Gels: Microstructural Characterization, Journal of the European Ceramic Society (2007).

25 SPECIFIC HEAT RCP k = V "V min ( ) 2 V 2 " V 2 (over 50,000 simulations) k N = 1 N "V "# Amount of volume per grain which must be given to the system in order to increase by one degree the granular temperature

26 ENTROPY * $ S(V ) = k 1+ log V "V min, & + % k# d '- )/ (. numerical simulation: hard spheres Lubachevsky-Stillinger algorithm Newtonian dynamics S RLP RCP Kauzmann Density 0.66 numerical simulation: Jodrey-Tory algorithm repulsive overlapping spheres Experiments: dry acrylic beads in air random disordered mix crystal Experiments: fluidized beads glass spheres in water!

27 Sixty Four -Per-Cent Solution What is the structure? at the fraction q-r tetrahedra reach saturation the polytetrahedral network percolates through all spheres between and 0.66 the network disassembles a (poly) crystalline phase is formed Why has it got this structure? a statistical mechanics approach can predict the structural properties the analogous of the Maxwell-Boltzmann distribution is derived at the analogous of the specific heat (k) has a sharp peak at the entropy jumps at 0.66 the (extrapolated) entropy of the disordered phase goes to zero