Xiaoming Mao. Department of Physics and Astronomy, University of Pennsylvania. Collaborators: Tom Lubensky, Ning Xu, Anton Souslov, Andrea Liu

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1 Xiaoing Mao Departent of Physics and Astronoy, University of Pennsylvania Collaborators: To Lubensky, Ning Xu, Anton Souslov, Andrea Liu Feb., 009

2 What is isostaticity? Isostatic systes are at the onset of echanical rigidity Central force syste of particles in diensions N total d.o.f. dn # of constraints N C # of soft odes dn N Large lattice, ean coordination # of constraints N C zn / Isostaticity: z d at bulk # of soft odes per particle O( N ) Soft odes are associated with boundary C d d( d ) / z N 6, N 7, d, dn C N C translations rotation soft odes 5 soft odes J. C. Maxwell, Philosophical Magazine 7, 598 (864). S. Alexander, Physics reports 96, 65 (998). C. S. O Hern, et al., Phys. Rev. E 68, 0306 (003). M. Wyart, et al., Phys. Rev. E 7, (005).

3 What is interesting about isostaticity? Zero-odes are present because of insufficient coordination, rather than broken syetry Jaing Granular packings Glasses and the Boson peak Eulsions, foas, colloids Rigidity percolation Courtesy of S. R. Nagel Courtesy of D. J. Durian Networks of sei-flexible polyers Applications in engineering Courtesy of D. A. Weitz Courtesy of D. A. Weitz

4 General uestion: How rigidity eerges in isostatic systes? Add ore bonds Negative pressure: stretch Add angle-dependent force Theral fluctuations (+ excluded volue repulsion) rubber cheical gels

5 Current work: randoness and isostaticity Interplay of randoness and extra coordination Each NNN bond is present with a given probability For large syste, threshold for rigidity P ~ N 0 How does rigidity scale with? What is new fro randoness P P Nonaffine deforations Scattering of phonons: dissipation

6 Granular aterials and point J Motivation: jaing Frictionless soft spheres One-sided repulsion At T=0, conjugate-gradient energy-iniization T unjaed G=0 G>0 J jaed shear stress C Packing fraction / A. J. Liu and S. R. Nagel, Nature 396 N6706, (998). C. S. O Hern, et al., Phys. Rev. E 68, 0306 (003).

7 Point J is isostatic Jaed solids at point J are isostatic G>0 J C Packing fraction Z C d A. J. Liu and S. R. Nagel, Nature 396 N6706, (998). C. S. O Hern, et al., Phys. Rev. E 68, 0306 (003).

8 Jaing: scalings G=0 G>0 Coordination nuber: Shear odulus: Bulk odulus: Pressure: Characteristic freuency: z ~ C ( ) / G C 3/ ~ ( ) ( z B C ~ ( ) ( z p C ~ ( ) ( z Haronic α= ( )/ ~ ( C ) ( z ) ) 0 ) ) c L ~ B c T ~ G l L ~ ( z) l T ~ ( z) / C. S. O Hern, et al., Phys. Rev. E 68, 0306 (003). L. E. Silbert, et al., Phys. Rev. Lett. 95, (005). M. Wyart, et al., Phys. Rev. E 7, (005).

9 The characteristic freuency scale DOS for jaed solids Heuristic arguents by M. Wyart l Cut a region of size l # of bonds severed: ~ l d # of extra bonds in the region: ~ l d z The region has soft odes if l l ~ ( z) Isostaticity length scale l Isostaticity freuency scale ~ / l ~ ( z) M. Wyart, et al., Phys. Rev. E 7, (005).

10 Diensional crossover at DOS for jaed solids DOS for continuous elastic edia D or right at isostaticity D() ~ D D( ) ~ c c D D/isostaticity Below the syste behaves D, and above, like D/isostaticity. Mao, Xu and Lubensky, anuscript in preparation (009). Souslov, Liu and Lubensky, anuscript in preparation (009).

11 Our goal Can we systeatically understand isostaticity using the tools in condensed atter physics? Lattice odels Naturally sall paraeter for perturbation What is the role of disorder? Nonaffine deforations Daping of phonons P ~ z Transport properties --- theral conductivity of glasses

12 Our odel Suare lattice and kagoe lattice Spring constant of NN bonds k Spring constant of NNN bonds Probability for each NNN bond to be present P Map to effective ediu periodic lattice with NNN bonds all present with Mao, Xu and Lubensky, anuscript in preparation (009).

13 Elastic expansion for lattice odels Displaceent field and potential energy Reference state r Displaceent r R Potential energy of a bond r u V b'( rb ) 3 u u O Vb ( R R ' ) Vb"( rb ) b b u r spring constant b force u R r u b r b u R b u ' r' R ' u'

14 Elasticity of the effective ediu elastic free energy for suare lattice F u D Dynaical atrix D D D xx xy Eigenvalues: u D D xy yy odes x u y u NN bonds k NNN bonds Landau theory: continuous elasticity strain tensor u ij u i j j u i u i l j u l C ~ k C ~ C44 ~ soft odes u xy

15 The characteristic freuency scale Dispersion relation F u D u D( ) x x x y Phonon density of states D D/isostatic D x k k l Van Hove singularity D/isostaticity NN bonds k NNN bonds

16 Longitudinal wave & characteristic length scale l Ising odel Isostaticity (longitudinal waves) Free energy Correlation function Characteristic length above assive D below assless/critical isostatic/d x y x x x u k u k F / sin 4 F ~ / sin 4 ~ x y x x k u u / k l

17 How to relate to the extra coordination z? Scalings suare lattice ~ l / ~ / ~ / Scalings in jaing NN bonds k NNN bonds l L ~ z ~ / ~ ( z) How to relate the? Siple guess ~ z ~ ( z) inconsistent /

18 Nearly isostatic rando lattices Suare lattice with rando additional NNN bonds k Spring constant of NN bonds k Spring constant of NNN bonds Probability for each NNN bond to be present P

19 Nearly isostatic rando lattices Suare lattice with rando additional NNN bonds k Spring constant of NN bonds k Spring constant of NNN bonds Probability for each NNN bond to be present P Length scales in the nearly isostatic suare lattice Copare with the case of jaing: l ~ / P ~ ( z) l ~ P c L ~ B c T ~ G l L ~ ( z) l T ~ ( z) /

20 Method: Coherent Potential Approxiation (CPA) Mapping to effective ediu ap P ' i" Self-consistency euation P P 0 ' " i P. Soven, Phys. Rev. 78, 36 (969). S. Feng, et al., PRB, 3, 76 (985).

21 CPA: Green s function and perturbation The effective ediu Elastic free energy u u F D The perturbation ' The self-consistency euation ', ', ', V D D ) ( V G Tr V V G V G V V G V V T G T G G G Mao, Xu and Lubensky, anuscript in preparation (009). Phonon Green s function

22 Static solution 0 The self-consistency euation At 0, for P asyptotic euation: nonaffine affine / k nonaffine affine P

23 Coparison with siulation The self-consistency euation At 0, for P asyptotic euation: / k nonaffine P affine C44 / k 0 / k Siulation with 00*00 lattice Nuerical solution fro the CPA / k 0 Asyptotic for (nonaffine) Asyptotic for (affine)

24 Coparison with jaing The self-consistency euation: / k nonaffine affine P Coparison with scaling at point J It is nonaffine near point J / k ~ P ~ z ~ P CPA agrees with the scalings of jaing! ~ l / ~ / ~ P ~ / ~ P P ~ z l L ~ z ~ / ~ ( z)

25 Discussion of 0 solution How do we understand the scaling ~ P? Length scale in the effective ediu Length scale in the rando lattice l ~ / P F u x x k u k x x 4 sin k l x y k / u Length scale for waves propagating in direction l ~ P x x x l ~ P ~ P

26 CPA at finite freuency 0 The self-consistency euation At 0 Results, the solution gives a coplex effective ediu spring constant ' " i Rescaled " ~ viscous elastic edia

27 Density of states D() Phonon DOS derived fro Green s function DOS of jaing DOS for various P P 0 3 P 0 P 0

28 Density of states: siulation Coparison with siulation: P=0. siulation with 00*00 lattice DOS fro CPA (infinite vol) DOS fro CPA (0000 particles)

29 Response functions (, ) Phonon response function ab, ' ( t, t') u f a b ' ( t) ( t') xx (, ) k 4 sin ( x y / ) ) y 0 no daping ) strong daping y I xx x

30 The kagoe lattice k 0; p0 0 k ; p0 " K M K K M K 0. k0.0; p0 0 0 '

31 Suary and future work Conclusions We did a systeatic study on rando nearly isostatic lattices, in particular, suare lattices with rando additional NNN bonds. Crossover between affine and nonaffine regie ~ P ~ P jaing Scaling of characteristic length and freuency scale in jaing can be recovered by CPA calculation of rando nearly isostatic lattices. Strong daping in phonon propagation sooth out Van Hove singularity, siilar DOS with jaing. Rando nearly isostatic lattice odels are able to capture soe essential physics in jaed solids Future work Transport properties Isostaticity in other systes