New Materials from Mathematics Real and Imagined

Transcription

1 New Materials from Mathematics Real and Imagined on the occasion of the 10 th anniversary of the Max Planck Institute for Mathematics in the Sciences Richard D. James MIS-MPI and the University of Minnesota

2 Martensitic phase transformation austenite martensite

3 Free energy and energy wells minimizers... 3 x 3 matrices U I U 2RU2 2 Cu 69 Al 27.5 Ni 3.5 α = β = γ = Ni 30.5 Ti 49.5 Cu 20.0 α = β = γ =

4 Nonattainment 1

5 Austenite/Martensite Interface Cu-14.0%Al-3.5%Ni 10 µm

6 Many studies at MPI-MIS Ben-Belgacem, Carstensen, Choksi, Conti, DeSimone, Dolzmann, Kohn, Kruzik, Kühn, Kurzke, Melcher, Moser, Müller, Ortiz, Otto, Plechac, Schäfer, etc. Later, many examples Magnetism (domains) Superconductivity θ L Soft matter (nematic elastomers) Thin films (wrinkling) Plasticity (dislocation patterns) and a fundamental new understanding of pde Chaudhuri, Conti, DeLillis, Dolzmann, Faraco, Kirchheim, Kristensen, Müller, Rieger, Sverak, Szekelyhidi L Hubert-Schäfer, Co

7 Ferromagnetic shape memory materials + (U 1,m 1 ) (RU 1,Rm 1 ) etc.

8 Ferromagnetic shape memory Ni Ga Mn Ni 2 MnGa N S H

9 Strain vs. field in Ni 2 MnGa H (010) (100) 30 times the strain of giant magnetostrictive materials

10 Ferromagnetic shape memory materials Ni 2 MnGa Courtesy: T. Shield

11 Main themes in science on hysteresis in structural phase transformations Pinning of interfaces by defects System gets stuck in an energy well on its potential energy landscape

12 A radically different hypothesis on the origins of hysteresis austenite What if we tuned the composition of the material to make two variants of martensite, finely twinned

13 Hysteresis vs. λ 2 Z. Zhang

14 Periodic Table of the Elements? Mono Rhom Rhom Rn At Po Bi Pb Tl Hg Au Pt Ir Os Re W Ta Hf * Ba Cs 6 Ortho Rhom Tet Tet Xe I Te Sb Sn In Cd Ag Pd Rh Ru Tc Mo Nb Zr Y Sr Rb 5 Ortho Rhom Ortho Kr Br Se As Ge Ga Zn Cu Ni Co Fe Mn Cr V Ti Sc Ca K 4 Ortho Ortho Mono Ar Cl S P Si Al Mg Na 3 Rhom Ne F O N C B Be Li 2 He H

15 Bravais lattice FCC

16 Periodic Table: Bravais lattices? Mono Rhom Rhom Rn At Po Bi Pb Tl Hg Au Pt Ir Os Re W Ta Hf * Ba Cs 6 Ortho Rhom Tet Tet Xe I Te Sb Sn In Cd Ag Pd Rh Ru Tc Mo Nb Zr Y Sr Rb 5 Ortho Rhom Ortho Kr Br Se As Ge Ga Zn Cu Ni Co Fe Mn Cr V Ti Sc Ca K 4 Ortho Ortho Mono Ar Cl S P Si Al Mg Na 3 Rhom Ne F O N C B Be Li 2 He H = not a Bravais lattice

17 Objective atomic structure (regular point system)

18 Objective atomic structures? Mono Rhom Rhom Rn At Po Bi Pb Tl Hg Au Pt Ir Os Re W Ta Hf * Ba Cs 6 Ortho Rhom Tet Tet Xe I Te Sb Sn In Cd Ag Pd Rh Ru Tc Mo Nb Zr Y Sr Rb 5 Ortho Rhom Ortho Kr Br Se As Ge Ga Zn Cu Ni Co Fe Mn Cr V Ti Sc Ca K 4 Ortho Ortho Mono Ar Cl S P Si Al Mg Na 3 Rhom Ne F O N C B Be Li 2 He H ??

19 D. L. D. Caspar and A. Klug, Physical principles in the construction of regular viruses, Cold Spring Harbor Symp. Quant. Biol. 27 (1962), 1-24 Page 11: Buckyballs Fullerenes Buckminster Fuller s geodesic dome

20 Bacteriophage T4: a virus that attacks bacteria Bacteriophage T-4 attacking a bacterium: phage at the right is injecting its DNA Wakefield, Julie (2000) The return of the phage. Smithsonian 31:42-6

21 Mechanism of infection A 100nm bioactuator

22 Structure of T4 sheath 1) Approximation of molecules using electron density maps Gives orientation and position of one molecule in extended and contracted sheath Data from Leiman et al., 2005 one molecule of extended sheath

23 Bacteriophage T4 tail sheath (extended to infinity) center of mass orientation We assert a much stronger statement: describes the molecule

24 Structure of this formula The experimental values of satisfy

25 Objective structures is an objective molecular structure if there are orthogonal transformations such that M = 1: objective atomic structure

26 Preservation of species Can write the definition using a permutation: where is a permutation. is the species of atom j (any molecule) An objective molecular structure preserves species if

27 Examples Bravais lattice Multilattice (or, an arbitrary periodic structure)

28 C 60 and most viral capsids Icosahedral rotation group: choose

29 Quantum mechanical significance of objective molecular structures where

30 Invariance

31 Equilibrium equations (atomic case) If one atom is in equilibrium then all atoms are in equilibrium

32 3-term formula for objective molecular structures Some structures generated by this formula describes the molecule

33 Four molecule arrays, eight molecule arrays

34 Pairs of rings unstaggered staggered

35 Bilayers

36 s u Molecular fibers staggered unstaggered

37 Branden and Tooze, Introduction to protein structure

38 First principles computations of the energy of an objective structure For full quantum mechanics we do not know how to write a cell problem For simpler atomic models, e.g., Density Functional Theory (DFT), we do, and this is what underlies the success of DFT: periodic BC for the density The same simplifications are possible for objective structures Use density functional theory Replace periodic boundary conditions by objective boundary conditions

39 Collective properties Objective structures are the natural structures to exhibit collective properties: Ferromagnetism Ferroelectricity Superconductivity