A mathematical perspective on the structure of matter

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1 A mathematical perspective on the structure of matter Richard James University of Minnesota Tutorial for the Workshop on Physics and Mathematics of Multiscale Modeling for Chemistry and Materials Science

2 Plan of the lecture Objective structures Objective MD

3 Structure of viruses Animal viruses Plant viruses Bacteriophages TMV CCMV

4 Examples HIV

5 Examples Flu Hepatitis B

6 Examples TMV Ebola

7 Examples Bacteriophage T4

8 Mechanism of infection Thomasson and Raaij We focus on the tail sheath A 100nm bioactuator (joint work with Wayne Falk)

position of one molecule in extended and contracted

9 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

molecules on the main helix For contracted sheath there

10 Structure of T4 sheath 2) Helices I: the 8/3 rule 3 consecutive molecules on the lowest annulus 8 consecutive molecules on the main helix For contracted sheath there is a similar 12/1 rule 3) Helices II: formulas for the helices Let

11 Structure of T4 sheath where, Parameters:

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

14 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

15 Bacteriophage T4 tail sheath (extended to infinity) describes the molecule

16 Periodic Table of the Elements H Hex He 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

17 Bravais lattice e 3 e2 FCC e 1

18 Periodic Table: Bravais lattices H He Hex = not a Bravais lattice 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 Hex

19 Objective atomic structure

Objective atomic structures 1 2 3 4 5 6 7 8 9 10

Li Be B C N O F Ne Cub Hex Rhom Hex Hex Cub Cub

Ortho Ortho Cub K Ca Sc Ti V Cr Mn Fe Co Ni Cu

Cub Hex Cub Cub Hex Ortho Cub Rhom Hex Ortho Cub

Xe Cub Cub Hex Hex Cub Cub Hex Hex Cub Cub Cub

Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Cub Cub Hex

20 Objective atomic structures H Hex He 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??

21 Before trying to understand why?, some further clarification Every monatomic two-lattice is an objective atomic structure (e.g., HCP, diamond lattice, graphene sheet) Single walled carbon nanotubes of all chiralities are objective atomic structures

22 Proof. Iterate g 1 : More generally,

23 Two lemmas

24 Proof of Lemma 1.2 We have to show, using the definition that?

25 ? Explicit forms

26 Return to two-lattices and carbon nanotubes

27 Bacteriophage T4 tail sheath, revisited describes the molecule parameters Experimental values of the parameters satisfy, for both short or tall forms, (structural parameters: )

28 A formula for every objective structure Dayal, Elliott, James

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

30 Torsion-tension-bending of a beam

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

32 Self-assembly and objective structures Consider structures assembled from one kind of molecule. This molecule could be a collection of several actual molecules a a a a Crane, 1950 Caspar and Klug, 1962

33 Quantum mechanical significance of objective molecular structures (notation!) where

34 Invariance The atomic case:

35 Equilibrium equations (objective atomic structure) If one atom is in equilibrium then all atoms are in equilibrium Objective structures have free parameters: atomic case structural equilibrium structural equilibrium implies atomic equilibrium if

36 Part II. Objective MD Joint work with Traian Dumritica, Kaushik Dayal and Stefan Müller

37 Fluid Mechanics viscometric flows Simple shearing flow, for example normal stress (non-newtonian fluids)

38 Viscometric flows Constitutive equation for the Cauchy stress relative deformation gradient Ordinary Lagrangian description of motion Formula for the relative deformation gradient cone and plate flow Definition of a viscometric flow

39 Solid Mechanics the bending and twisting of beams B. de St. Venant, Memoire sur la torsion des prismes, Mem. Des Savants Etrangers 14 (1855),

40 Bending and twisting of beams J. L. Ericksen, Adv. Appl. Mech. (ed. C-S Yih) 17 (1977), : or

41 Claim There is a universal atomic level interpretation of these motions (universal = independent of material) From this viewpoint the bending and twisting of beams and viscometric flows of fluids are the same subject

42 Examples Translation group Theorem: If a discrete group of isometries does not contain a translation and does not consist entirely of rotations, it is expressible in one of the forms where Full list: Dayal, Elliott, James

43 A time-dependent invariant manifold of the equations of molecular dynamics simulated atoms a discrete group of isometries (N can be infinite) all of the atoms The elements can depend on t>0, but this time dependence must be consistent with

44 Atomic forces The force is smooth and satisfies Frame-indifference (These conditions satisfied, e.g., by the Hellmann-Feynman force based on Born- Oppenheimer quantum mechanics) Permutation invariance where Π is a permutation that preserves species.

45 Potential energy These conditions can be found by formally differentiating the frame-indifference and permutation invariance of the potential energy, ( but of course this calculation would not make sense when N = )

46 Theorem Assume the restrictions on the potential energy above and let be a time-dependent discrete group of isometries satisfying the restriction on the timedependence given above. If satisfy the equations of molecular dynamics, i.e., then also satisfy the equations of molecular dynamics:

47 Proof

48 Allowed time dependence of the group elements The permitted time-dependence, that is, This is satisfied (in the absence of excessive assumptions on the solution) if and only if

49 Simplest case - translation group affine in t discrete translation group all of the atoms simulated atoms permitted time-dependence

50 Naïve passage to continuum level affine motion Recall, for simple materials because

51 Evidently Some viscometric flows, i.e., cone and plate flow, seem to have no analog at atomic level. My personal view: failure of full timedependent frame-indifference in molecular dynamics The definition of viscometric flows misses some affine motions having all the good properties of viscometric flows

52 Key question: how representative are these solutions? Attracting in some sense? If one takes more and more simulated atoms (and a fixed group), with certain averages of initial conditions prescribed, the behavior is statistically like the large body/large time limit of a suitable boundary value problem (?). Another way to explore this issue: look at other theories of physics intermediate between MD and continuum Maxwell-Boltzmann Equation molecular density function Maxwell-Boltzmann equation UMD

53 OMD solutions have their own statistics Use translation group (i.e., gases fill volumes) y 0

54 This yields an exact reduction of the Maxwell-Boltzmann equation g(t,w) satisfies Apparently includes all known exact solutions of the equations for the moments for special molecular force laws

55 H theorem Prototypical molecular density corresponding to these flows g(t,w) w Müller, James

56 Principle of Material Frame Indifference change of frame objectivity transformations Cauchy stress objective tensor PMFI:

57 Suggested modification of PMFI Add: (homogeneous materials only) Satisfied by every successful constitutive equation of which I am aware Interesting to examine the implications for turbulence models, Langevin equation

58 Other groups besides the translation group (joint work with Traian Dumitrica, Kaushik Dayal)

59 Objective MD study of a carbon nanotube under torsion Three-body Tersoff potential for carbon Twist was controlled by controlling the group parameters (interesting open question: what generalized forces answer to variations of group parameters?) The groups chosen were various subgroups of the following group (with no translational subgroup) listed earlier: No time-dependence of the group elements

60 Objective MD: study of buckling of C nanotube under torsion (12, 12) CNT a t 1 b b a b t 2 ~3 deg/nm twist

61 Effect of different choices of the fundamental domain bifurcation diagram

62 Objective MD simulation of bending of a carbon nanotube Is there a St. Venant s principle at atomic level, with these solutions playing the role of the St. Venant solutions?

63 A time dependent group: viscometry of nanostructures Tersoff potential for carbon, again Same group as in the static simulations of the carbon nanotubes, but introduce time-dependence consistent with the main theorem Replace

64 Lots of interesting things happen, depending on initial conditions (mainly initial temperature), strain rate t = 4.6 ps t = 5.7 ps initial state, simulated atoms in red, initial temperature = 320 K, strain rate = 0.1 1/ps t = 4 ps T = 336 K T = 6143 K T = 6452 K

65 A different fundamental domain, lower strain rate initial state, simulated atoms in red, strain rate = /ps t = 412 ps t = 432 ps initial temperature = 315K T = 358 K T = 4452 K

66 Higher initial temperature, different fundamental domain initial state, simulated atoms in red, strain rate = 0.001, initial temperature = 1500 K t = 310 ps t = 311 ps t = 325 ps T= 1390 K T = 1831 K T = 4991 K

67 Higher initial temperature, lower strain rate t = 300 ps t = ps t = ps t = 1815 ps simulated atoms in red, strain rate = , initial temperature = 1800 K T = 1873 K T = 3179 K T = 3405 K T = 4510 K

68 Notes The bending and twisting of beams in solid mechanics and viscometry in fluid mechanics are the same subject, just different isometry groups Solutions work in all materials Suggest that these solutions be the basis of determination of material properties of nanostructures Needs: many other groups to be explored, formulas for forces and moments, better understanding of dependence of solution on fundamental domain, nonequilibrium statistical mechanics of these solutions

Objective Molecular Dynamics

Objective Molecular Dynamics Richard D. James University of Minnesota Joint work with Kaushik Dayal, Traian Dumitrica, Stefan Müller Fluid Mechanics viscometric flows Simple shearing flow, for example