Transcript New Materials from Mathematics – Real and Imagined

New Materials from
Mathematics –
Real and Imagined
on the occasion of the 10th anniversary of the Max Planck Institute for
Mathematics in the Sciences
Richard D. James
MIS-MPI and the University of Minnesota
October 2, 2006
Martensitic phase transformation
austenite
martensite
October 2, 2006
MPI-MISI
Free energy and energy wells
minimizers...
3 x 3 matrices
U2
RU2
U1
1
I
Cu69
Al27.5
Ni3.5
 = 1.0619
 = 0.9178
 = 1.0230
Ni30.5
Ti49.5
Cu20.0
 = 1.0000
 = 0.9579
 = 1.0583
October 2, 2006
MPI-MISI
Nonattainment
1
October 2, 2006
MPI-MISI
Austenite/Martensite Interface
Cu-14.0%Al-3.5%Ni
10 m
October 2, 2006
MPI-MISI
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
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L
Magnetism (domains)
L
Superconductivity
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
Hubert-Schäfer, Co
October 2, 2006
MPI-MISI
Ferromagnetic shape memory materials
+
(U1,m1)
(RU1,Rm1)
October 2, 2006
…etc.
MPI-MISI
Ferromagnetic shape memory
Ga
Ni
Mn
Ni2MnGa
N
S
H
October 2, 2006
MPI-MISI
Strain vs. field in Ni2MnGa
H
(010)
(100)
30 times the strain of giant magnetostrictive materials
October 2, 2006
MPI-MISI
Ferromagnetic shape memory materials
Ni2MnGa
Courtesy:
T. Shield
October 2, 2006
MPI-MISI
Main themes in science on hysteresis in
structural phase transformations
Pinning of interfaces by defects
October 2, 2006
System gets stuck in an energy
well on its potential energy
landscape
MPI-MISI
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
October 2, 2006
MPI-MISI
Hysteresis vs. λ2
Z. Zhang
October 2, 2006
MPI-MISI
Periodic Table of the Elements
1
1
2
3
4
5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
H
He
Hex
Hex
Li
Be
B
C
N
O
F
Ne
Cub
Hex
Rhom
Hex
Hex
Cub
Cub
Cub
Na
Mg
Al
Si
P
S
Cl
Ar
Cub
Hex
Cub
Cub
Mono
Ortho
Ortho
Cub
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Cub
Hex
Cub
Cub
Hex
Ortho
Cub
Rhom
Hex
Ortho
Cub
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
Cub
Cub
Hex
Hex
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Hex
Tet
Tet
Rhom
Hex
Ortho
Cub
Cs
Ba
*
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Cub
Cub
Hex
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Rhom
Hex
Cub
Rhom
Mono
?
Cub
6
October 2, 2006
MPI-MISI
Bravais lattice
FCC
October 2, 2006
MPI-MISI
Periodic Table: Bravais lattices
1
1
2
3
4
5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
H
He
Hex
Hex
Li
Be
Cub
= not a Bravais lattice
B
C
N
O
F
Ne
Hex
Rhom
Hex
Hex
Cub
Cub
Cub
Na
Mg
Al
Si
P
S
Cl
Ar
Cub
Hex
Cub
Cub
Mono
Ortho
Ortho
Cub
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Cub
Hex
Cub
Cub
Hex
Ortho
Cub
Rhom
Hex
Ortho
Cub
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
Cub
Cub
Hex
Hex
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Hex
Tet
Tet
Rhom
Hex
Ortho
Cub
Cs
Ba
*
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Cub
Cub
Hex
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Rhom
Hex
Cub
Rhom
Mono
?
Cub
6
October 2, 2006
MPI-MISI
Objective atomic structure (regular point
system)
October 2, 2006
MPI-MISI
Objective atomic structures
1
1
2
3
4
5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
?
18
H
He
Hex
Hex
Li
Be
B
C
N
O
F
Ne
Cub
Hex
Rhom
Hex
Hex
Cub
Cub
Cub
Na
Mg
Al
Si
P
S
Cl
Ar
Cub
Hex
Cub
Cub
Mono
Ortho
Ortho
Cub
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Cub
Hex
Cub
Cub
Hex
Ortho
Cub
Rhom
Hex
Ortho
Cub
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
Cub
Cub
Hex
Hex
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Hex
Tet
Tet
Rhom
Hex
Ortho
Cub
Cs
Ba
*
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Cub
Cub
Hex
Cub
Cub
Hex
Hex
Cub
Cub
Cub
Rhom
Hex
Cub
Rhom
Mono
?
Cub
?
6
October 2, 2006
MPI-MISI
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
October 2, 2006
MPI-MISI
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
October 2, 2006
MPI-MISI
Mechanism of infection
A 100nm bioactuator
October 2, 2006
MPI-MISI
Structure of T4 sheath
1) Approximation of molecules using electron density maps
Data from Leiman et al., 2005
October 2, 2006
Gives orientation
and position of
one molecule in
extended and
contracted
sheath
one molecule of
extended sheath
MPI-MISI
Bacteriophage T4 tail sheath
(extended to infinity)
center of mass
orientation
We assert a much stronger statement:
describes the molecule
October 2, 2006
MPI-MISI
Structure of this formula
The experimental values of
October 2, 2006
satisfy
MPI-MISI
Objective structures

is an objective
molecular structure if there are orthogonal
transformations
such that

M = 1: objective atomic structure
October 2, 2006
MPI-MISI
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
October 2, 2006
MPI-MISI
Examples

Bravais lattice

Multilattice (or, an arbitrary periodic structure)
October 2, 2006
MPI-MISI
C60 and most viral capsids
Icosahedral rotation group:
choose
October 2, 2006
MPI-MISI
Quantum mechanical significance of
objective molecular structures
where
October 2, 2006
MPI-MISI
Invariance
October 2, 2006
MPI-MISI
Equilibrium equations (atomic case)
If one atom is in equilibrium then all atoms are in equilibrium
October 2, 2006
MPI-MISI
3-term formula for objective molecular
structures
describes the molecule
Some structures generated by this formula
October 2, 2006
MPI-MISI
Four molecule arrays, eight molecule
arrays
October 2, 2006
MPI-MISI
Pairs of rings
unstaggered
October 2, 2006
staggered
MPI-MISI
Bilayers
October 2, 2006
MPI-MISI
Molecular fibers
s
u
staggered
October 2, 2006
unstaggered
MPI-MISI
Branden and Tooze, Introduction to
protein structure
October 2, 2006
MPI-MISI
First principles computations of the
energy of an objective structure
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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
October 2, 2006
MPI-MISI
Collective properties

Objective structures are the natural structures to exhibit
collective properties:
– Ferromagnetism
– Ferroelectricity
– Superconductivity
October 2, 2006
MPI-MISI