March 28, 2008 New Materials from Mathematics – Real and Imagined Richard James University of Minnesota Thanks: John Ball, Kaushik Bhattacharya, Jun Cui, Traian Dumitrica, Stefan Müller, Ichiro Takeuchi, Rob Tickle, Manfred Wuttig, Jerry Zhang

March 28, 2008UMD Martensitic phase transformation austenite martensite

March 28, 2008UMD Free energy and energy wells Ni 30.5 Ti 49.5 Cu 20.0  =  =  = Cu 69 Al 27.5 Ni 3.5  =  =  = minimizers... 1 U 1 U 2 RU 2 I 3 x 3 matrices 2 2 1

March 28, 2008UMD Nonattainment 1

March 28, 2008UMD A minimizing sequence min n There are four normals m to such austenite-martensite interfaces. n There are two volume fractions λ of the twins. From analysis of this sequence (= the crystallographic theory of martensite), : m

March 28, 2008UMD 10  m Austenite/Martensite Interface Cu-14.0%Al-3.5%Ni

March 28, 2008UMD + Ferromagnetic shape memory materials (U1,m1)(U1,m1) (RU 1,Rm 1 ) …etc.

March 28, 2008UMD Ferromagnetic shape memory N S Ga Mn Ni Ni 2 MnGa H

March 28, 2008UMD Strain vs. field in Ni 2 MnGa H (010) (100) 30 times the strain of giant magnetostrictive materials

March 28, 2008UMD Ferromagnetic shape memory materials Ni 2 MnGa Courtesy: T. Shield

March 28, 2008UMD Low hysteresis materials Hysteresis

March 28, 2008UMD 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

March 28, 2008UMD austenite two variants of martensite, finely twinned A rather different hypothesis on the origins of hysteresis What if we tune the composition of the material to make

March 28, 2008UMD Data on one graph. Hysteresis = A s + A f – M s – M f Jerry Zhang

March 28, 2008UMD Hysteresis vs. λ 2 Z. Zhang Triangles (NiTiCu) from combinatorial measurements of Cui, Chu, Famodu, Furuya, Hattrick- Simpers, James, Ludwig,Theinhaus, Wuttig, Zhang, Takeuchi

March 28, 2008UMD Local minimizers? A = I B φ There is no existing framework within the calculus of variations for discussing the concept of metastability relevant to the above.

March 28, 2008UMD Periodic Table of the Elements HHe Hex 2 LiBeBCNOFNe CubHexRhomHex Cub 3 NaMgAlSiPSClAr CubHexCub MonoOrtho Cub 4 KCaScTiVCrMnFeCoNiCuZnGaGeAsSeBrKr Cub Hex Cub HexCub HexOrthoCubRhomHexOrthoCub 5 RbSrYZrNbMoTcRuRhPdAgCdInSnSbTeIXe Cub Hex Cub Hex Cub HexTet RhomHexOrthoCub 6 CsBa*HfTaWReOsIrPtAuHgTlPbBiPoAtRn Cub HexCub Hex Cub RhomHexCubRhomMono?Cub

March 28, 2008UMD Bravais lattice FCC e1e1 e3e3 e2e2

March 28, 2008UMD Periodic Table: Bravais lattices HHe Hex 2 LiBeBCNOFNe CubHexRhomHex Cub 3 NaMgAlSiPSClAr CubHexCub MonoOrtho Cub 4 KCaScTiVCrMnFeCoNiCuZnGaGeAsSeBrKr Cub Hex Cub HexCub HexOrthoCubRhomHexOrthoCub 5 RbSrYZrNbMoTcRuRhPdAgCdInSnSbTeIXe Cub Hex Cub Hex Cub HexTet RhomHexOrthoCub 6 CsBa*HfTaWReOsIrPtAuHgTlPbBiPoAtRn Cub HexCub Hex Cub RhomHexCubRhomMono?Cub = not a Bravais lattice

March 28, 2008UMD Objective atomic structure (regular point system)

March 28, 2008UMD Objective atomic structures HHe Hex 2 LiBeBCNOFNe CubHexRhomHex Cub 3 NaMgAlSiPSClAr CubHexCub MonoOrtho Cub 4 KCaScTiVCrMnFeCoNiCuZnGaGeAsSeBrKr Cub Hex Cub HexCub HexOrthoCubRhomHexOrthoCub 5 RbSrYZrNbMoTcRuRhPdAgCdInSnSbTeIXe Cub Hex Cub Hex Cub HexTet RhomHexOrthoCub 6 CsBa*HfTaWReOsIrPtAuHgTlPbBiPoAtRn Cub HexCub Hex Cub RhomHexCubRhomMono?Cub ??

March 28, 2008UMD 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

March 28, 2008UMD Mechanism of infection A 100nm bioactuator

March 28, 2008UMD Structure of T4 sheath 1) Approximation of molecules using electron density maps Data from Leiman et al., 2005

March 28, 2008UMD Bacteriophage T4 tail sheath (extended to infinity) describes the molecule We assert a much stronger statement: center of mass orientation

March 28, 2008UMD Objective structures n M = 1: objective atomic structure n is an objective molecular structure if there are orthogonal transformations such that Can write the definition using a permutation: where is a permutation.

March 28, 2008UMD Theorem Dayal, Elliott, James

March 28, 2008UMD Quantum mechanical significance of objective molecular structures where

March 28, 2008UMD Invariance

March 28, 2008UMD Equilibrium equations (objective atomic structure) If one atom is in equilibrium then all atoms are in equilibrium

March 28, 2008UMD First principles computations of the energy of an objective structure n For full quantum mechanics we do not know how to write a cell problem n 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 n The same simplifications are possible for objective structures – Use density functional theory – Replace periodic boundary conditions by objective boundary conditions

March 28, 2008UMD Objective structures should exhibit collective properties n Objective structures are the natural structures to exhibit collective properties: – Ferromagnetism – Ferroelectricity – Superconductivity Suggestion: search systematically among objective structures for those with collective properties, using DFT and the formulas for OS based on isometry groups

March 28, 2008UMD The end