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
Martensitic phase transformation austenite martensite
Free energy and energy wells minimizers... 3 x 3 matrices U 1 2 1 1 I U 2RU2 2 Cu 69 Al 27.5 Ni 3.5 α = 1.0619 β = 0.9178 γ = 1.0230 Ni 30.5 Ti 49.5 Cu 20.0 α = 1.0000 β = 0.9579 γ = 1.0583
Nonattainment 1
A minimizing sequence m min From analysis of this sequence (= the crystallographic theory of martensite), : There are four normals m to such austenite-martensite interfaces. There are two volume fractions λ of the twins.
Austenite/Martensite Interface Cu-14.0%Al-3.5%Ni 10 µm
Ferromagnetic shape memory materials + (U 1,m 1 ) (RU 1,Rm 1 ) etc.
Ferromagnetic shape memory H Ni Ga Mn Ni 2 MnGa N S
Strain vs. field in Ni 2 MnGa H (010) (100) 30 times the strain of giant magnetostrictive materials
Ferromagnetic shape memory materials Ni 2 MnGa Courtesy: T. Shield
Low hysteresis materials Hysteresis
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
A rather different hypothesis on the origins of hysteresis austenite two variants of martensite, finely twinned What if we tune the composition of the material to make
Data on one graph. Hysteresis = A s + A f M s M f Jerry Zhang
Hysteresis vs. λ 2 Triangles (NiTiCu) from combinatorial measurements of Cui, Chu, Famodu, Furuya, Hattrick- Simpers, James, Ludwig,Theinhaus, Wuttig, Zhang, Takeuchi Z. Zhang
Local minimizers? φ A = I B There is no existing framework within the calculus of variations for discussing the concept of metastability relevant to the above.
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 1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Bravais lattice e 3 e2 FCC e 1
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 1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 = not a Bravais lattice
Objective atomic structure (regular point system)
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 1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1??
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
Mechanism of infection A 100nm bioactuator
Structure of T4 sheath 1) Approximation of molecules using electron density maps Data from Leiman et al., 2005
Bacteriophage T4 tail sheath (extended to infinity) center of mass orientation We assert a much stronger statement: describes the molecule
Objective structures is an objective molecular structure if there are orthogonal transformations such that M = 1: objective atomic structure Can write the definition using a permutation: where is a permutation.
Theorem Dayal, Elliott, James
Quantum mechanical significance of objective molecular structures where
Invariance
Equilibrium equations (objective atomic structure) If one atom is in equilibrium then all atoms are in equilibrium
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
Objective structures should exhibit collective properties 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
The end