Xiaoming Mao Physics, University of Michigan, Ann Arbor. IGERT Summer Institute 2017 Brandeis

Transcription

1 Xiaoming Mao Physics, University of Michigan, Ann Arbor IGERT Summer Institute 2017 Brandeis

2 Elastic Networks A family of discrete model networks involving masses connected by springs 1 2 kk( ll)2 Disordered Interactions Central force springs Angle bending stiffness 1 2 κκ( θθ)2 Ordered Simple, good for: Analytic calculations Numerical simulations In any dimension

3 Why Elastic Networks Why we want to use elastic networks models ( as opposed to continuum elasticity?) Because many soft matter systems are marginal solids (both solid and liquid characteristics). Elastic networks help understand the microscopic origin of different classes of marginal solids Elastic networks also provide ideas of making new metamaterials with novel properties Topological Mechanics

4 Experimental Systems as Elastic Networks Elastic networks capture the essential physics of many interesting systems Glasses (silica, TEM image, Huang et al, Science 2013) Granular matter Colloids and Emulsion (E. Weeks group, Emory) Biopolymer networks (collagen, Ovaska et al, 2017)

5 Experimental Systems as Elastic Networks Elastic networks capture the essential physics of many interesting systems Protein rigidity/function (Chubynsky et al, 2008) Mechanical metamaterials design (J. Greer s group, Caltech) Structural phase transitions in crystals (α-β transition of cristobalites)

6 Why Elastic Network Models Are Useful? With generalization, elastic networks can also be used to study: Foams (D. Durian s group, Penn) Biological tissues (Bi et al, Nat. Phys. 2015) Origami & Kirigami (Evans et al, PRE 2015) and many more!

7 How to Understand Elastic Networks? First Step: Mechanical Stability Floppy mode Vs. Floppy How to find out? Mechanically Stable

8 Maxwell s Counting # of zero modes (normal modes cost 0 elastic energy) NN 0 = NN dd.oo.ff NN CC Degrees of freedom Constraints Central force bonds between point-like particles NN dd NN bb # of particles * spatial dimension Bonds # of floppy modes (excluding trivial modes) NN mm = NN 0 dd(dd+1) 2 Trivial translations and rotations of the whole system NN 0 = = 4 NN 0 = = 3 NN mm = 4 3 = 1 Floppy mode Vs. NN mm = 3 3 = 0 Floppy Mechanically Stable J. C. Maxwell, Phil. Mag. 27, 598 (1864).

9 Maxwell s Counting on Infinitely Large Systems # of zero modes (normal modes cost 0 elastic energy) NN 0 = NN dd.oo.ff NN CC Degrees of freedom Constraints Central force bonds between point-like particles NN dd NN bb # of particles * spatial dimension Bonds # of floppy modes (excluding trivial modes) NN mm = NN 0 dd(dd+1) 2 of O(1) and can be dropped for large system Trivial translations and rotations of the whole system NN mm = NN dd NN bb = NN dd NN zz /2 In order to have mechanical stability zz : average coordination number e.g. zz = 5 particle zz 2222 for stability J. C. Maxwell, Phil. Mag. 27, 598 (1864).

10 Periodic Lattices Critical condition for mechanical stability: zz = 2222 (assuming only NN central force interactions) z = 3 < 2d z = 2d z = 6 > 2d Floppy Extensive # of floppy modes Maxwell Lattices How many floppy modes? Over-constrained No floppy modes

11 Mechanical Critical Systems With zz = 2dd Critical condition for mechanical stability: zz = 2222 Example 1: Rigidity percolation on a central-force lattice Randomly take off bonds zz = 6pp When does it lose rigidity? pp < 1 pp = 1 pp cc /3 pp Numerical value Using (meanfield) counting rule zz = 2222 Feng and Sen, PRL 52, 216 (1984). Feng and Thorpe, PRB 31, 276 (1985). Feng, Sen, Halperin, and Lobb, PRB 30, 5386 (1984).

12 Mechanical Critical Systems With zz = 2dd Critical condition for mechanical stability: zz = 2222 Example 2: Jamming of frictionless particles (contact repulsive interactions) Non-overlapped liquid Overlapped solid Increasing density J φ C φ Random close packing Isostaticity zz = zz CC = 2222 A. J. Liu and S. R. Nagel, Nature 396 N6706, 21 (1998). C. S. O Hern, et al., Phys. Rev. E 68, (2003).

13 Classification of Systems at zz = 2dd? Both rigidity percolation and jamming: zz = 2222 Different universality classes (correlation length scaling, elastic moduli scaling, etc ) Are there other universality classes of rigidity transitions at zz = 2222? Elastic network models can help us understand this Main Reference of my lectures: Lubensky, T. C., et al. "Phonons and elasticity in critically coordinated lattices." Reports on Progress in Physics 78.7 (2015):

14 Rich Physics of Critical Mechanical Structures Interesting fundamental questions: e.g.: Cut a finite piece of Maxwell lattice (zz = 2222) Deficit of constraints on the boundary # of floppy modes Size of boundary But where are these modes located? Control of floppy modes Program mechanical Response

15 Where Are The Floppy Modes? Case I. Floppy modes localized at where you cut Example: twisted kagome Floppy modes on all edges Described by conformal transformations Periodic BC in x K. Sun, A. Souslov, X. Mao, and T. C. Lubensky, PNAS 109, (2012).

16 Where Are The Floppy Modes? Case II. Floppy modes are plane waves Example: regular kagome regular square Periodic BC in x

17 Where Are The Floppy Modes? Case III. Floppy modes are topologically protected edge modes on one side Example: deformed kagome (arrows show a pair of modes) Soft edge Topological polarization RR TT Hard edge Periodic BC in x C. L. Kane and T. C. Lubensky, Nat. Phys. 10, 39 (2014).

18 Where Are The Floppy Modes? Case IV. Floppy modes are topologically protected Weyl modes in the bulk Example: deformed square (4 site per cell) (arrows show a Weyl mode) Rocklin, Chen, Falk, Vitelli & Lubensky, PRL, 116, (2016)

19 Nonlinear Floppy Modes: Mechanisms What if an elastic network has NN 0 > 0? Finite systems: A more interesting example: Floppy mode Following the floppy mode to nonlinear order: Mechanism Theo Jansen s walking mechanism and Strandbeest (wind powered robot using this mechanism) Critical mechanical structures : structures close to mechanical instability and exhibit a small number of floppy modes

20 Second Lecture Slides

21 Topological 1D Chain 1D Rotor Chain C. L. Kane and T. C. Lubensky, Nat. Phys. 10, 39 (2014). Chen, Upadhyaya and Vitelli, PNAS 111, (2014).

22 Topological 1D Chain: Soliton 1D Rotor Chain: floppy mode beyond linear order topological soliton C. L. Kane and T. C. Lubensky, Nat. Phys. 10, 39 (2014). Chen, Upadhyaya and Vitelli, PNAS 111, (2014).

23 2D Lattices: Topological Kagome Lattice Soft edge Topological polarization RR TT Hard edge Periodic BC in x C. L. Kane and T. C. Lubensky, Nat. Phys. 10, 39 (2014).

24 Reconfigure Topological Kagome Using Soft Strain All Maxwell lattices have at least one macroscopic soft deformation: θθ Does it change the topological structures of the phonons? Non-polarized RR TT RR TT Non-polarized θθ 1 θθ 2 θθ 3 θθ aa 2 aa 1 θθ What happens at the topological transitions θθ 1, θθ 2, θθ 3? Rocklin, Zhou, Sun, and Mao, Nat Comm, 8, (2017)

25 What happens at the transition? Evolution of a pair of floppy edge modes (determined by topological polarization RR TT ) a b Metallic state c Soft edge Insulating states with different topologies θθ < θθ 1 d 1 Hard edge θθ = θθ 1 RR TT Hard edge θθ > θθ 1 Edge Stiffness Soft edge θθ Transformable mechanical metamaterial Rocklin, Zhou, Sun, and Mao, Nat Comm, 8, (2017)

26 Rigid connector Hinge Webpage of video: /research/ttmm/ttmm.html

27 Brandeis Page 1 Tuesday, May 30, :03 AM

28 Brandeis Page 2

29 Brandeis Page 3

30 Brandeis Page 4

31 Brandeis Page 5

32 Brandeis Page 6

33 Brandeis Page 7