University of Minnesota
University of Minnesota

IMA Workshop:

Atomic Motion to Macroscopic Models: The Problem of Disparate Temporal and Spatial Scales in Matter

April 11-15, 2005

Photo Gallery     Materials from Talks

Ken A. Dill (Department of Pharmaceutical Chemistry, University of California)

Protein folding: A problem in global optimization

Joint work with Banu Ozkan, John Chodera, and Thomas Weikl.

To compute the folded structure of a protein using a physical model and the monomer sequence has been regarded as a challenge because it involves multiple time and space scales. Our approach has been to understand how proteins physically fold and to try to apply that strategy to protein structure prediction. We believe that proteins break their large global optimization problem into smaller local optimizations. We have been exploring various methods for exploiting that idea for protein structure prediction.

Antonio DiCarlo (Department of Studies on Structures, Università degli Studi "Roma Tre")

Microtwists & nanodefects
Poster:  pdf   Paper:  pdf
Note: This paper is a (slightly corrected) reprint of the paper: A. Di Carlo, M. Monteferrante, P. Podio-Guidugli, V. Sansalone, L. Teresi: How (and why) twisting cycles make individual MWCNTs stiffer. In: Electronic Properties of Synthetic Nanostructures, Proceedings of the XVIII International Winterschool/Euroconference on Electronic Properties of Novel Materials held in Kirchberg, Tirol, Austria (March 6-13, 2004), pp. 355-358 (H. Kuzmany et al., eds.), AIP Conf. Proc. 723, American Institute of Physics, Melville (2004).

Joint work with Luciano Teresi.

Prototype nanoelectromechanical devices incorporating individual multiwall carbon nanotubes as torsion bars or rotary bearings have been fabricated and tested by various groups. Typical length scales are: 1 micron for the overall span, 25 nm for the diameter, 0.5 nm for the interwall gap. The experimental evidence collected so far is puzzling, pointing out a need for a better understanding of the interwall mechanical coupling mechanisms. We speculate that the basic mechanism behind progressive interwall coupling is the formation of bridging defects, that is, covalent links between adjacent walls, triggered by inward migration of chromium atoms (which are evaporated onto the outer wall of the nanotube when fabricating the device).

Weinan E (Department of Mathematics & Program in Applied Computational Mathematics, Princeton University)

Cauchy-Born rule, generalized Cauchy-Born rule and applications to nanomechanics

The focus of this talk is to develop continuum mechanics type of models for studying the deformation of nano-scale objects such as nano-tubes, nano-rods and DNA. The basic tool we will use is various generalizations of the classical Cauchy-Born rule. For this purpose, we will first review the classical Cauchy-Born rule for bulk crystals. We will discuss the validity of the Cauchy-Born rule and give a precise characterization of its boundary of invalidity. We then discuss the generalization of the Cauchy-Born rule to curved low dimensional objects, including the local Cauchy-Born rule and the exponential Cauchy-Born rule. Finally we turn our attention to nano-tubes and nano-rods, and examine whether their deformation can be described by these continuum theories.

Ron Elber (Department of Computer Science, Cornell University)

Calculations of activated trajectories with boundary value formulation

Activated trajectories are paths that pass over a significant energy barrier between two (or more) stable states. A typical activated trajectory "incubates" in one of the wells for a significant time period, but once the process is initiated the transition time can be extremely short. A boundary value formulation is proposed that computes efficiently rare and activated trajectories. A large integration step is used in two examples: the Mueller potential, and a conformational transition in alanine dipeptide, a model for protein backbone.

Ron Elber (Department of Computer Science, Cornell University)

Computing timescales by milestoning

An algorithm is presented to compute time scales of complex processes following predetermined milestones along a reaction coordinate. A non-Markovian hopping mechanism is assumed and constructed from underlying microscopic dynamics. General analytical analysis, a pedagogical example, and numerical solutions of the non- Markovian model are presented. No assumption is made in the theoretical derivation on the type of microscopic dynamics along the reaction coordinate. However, the detailed calculations are for Brownian dynamics in which the velocities are uncorrelated in time (but spatial memory remains).

Ryan S. Elliott (Aerospace Engineering & Mechanics, University of Minnesota )

Stability and bifurcation of multilattice crystals undergoing martensitic transformation

Understanding thermoelastic martensitic transformations is a fundamental component in the study of shape memory alloys. These transformations involve a hysteretic change in stability of the crystal lattice between an austenite (high symmetry) phase and a martensite (low symmetry) phase within a small temperature range. To study these transformations, a set of phenomenological temperature-dependent atomic pair-potentials is used to derive the crystal's energy density W(F,S1, S2 , ... ;T) as a function of a uniform deformation F and a set of internal atomic shift degrees of freedom S1 Special attention is paid to the evaluation of crystal structure stability. Using a specific set of temperature-dependent pair-potentials a stress-free bifurcation diagram is generated for the B2 binary crystal structure (with temperature serving as the loading parameter). A hysteretic transformation is suggested by the existence of certain stable equilibrium branches corresponding to B2 (CsCl) and B19 (orthorhombic) crystal structures. These results indicate the ability of temperature-dependent atomic potential models to provide valuable insight into the behavior of shape memory alloys such as NiTi, AuCd, and CuAlNi.

Jim Evans (Department of Mathematics, Iowa State University)

From atomic scale ordering to mesoscale spatial patterns in surface reactions: Heterogeneous coupled Lattice-Gas (HCLG) simulation approach
Slides:  pdf

A challenge for the modeling of surface reaction-diffusion systems is to connect-the-length-scales from a realistic atomistic treatment of local ordering of reactants and reaction kinetics to an "exact'' description of mesoscale spatial pattern formation. We discuss a heterogeneous coupled lattice-gas (HCLG) approach which utilizes parallel kinetic Monte Carlo simulations of a lattice-gas reaction model to simultaneously determine the local reaction kinetics and diffusive transport properties at various macroscopic "points'' distributed across the surface. These simulations are coupled to reflect macroscopic mass transport via surface diffusion.

This method is demonstrated for bistable CO-oxidation reactions on surfaces. We first discuss a simple model where surface transport of CO is reduced from a many-particle to a single-particle problem (although one with even more complexity than an "ant-in-the-labyrinth" type percolative diffusion problem). However, we have also applied HCLG to analyze reaction front propagation in a realistic model for CO-oxidation on Pd(100).

Ref. Da-Jiang Liu and JWE, PRB 70 (04) 193408; SIAM MMS 3 (05).

Jim Evans (Department of Mathematics, Iowa State University)

Atomistic and continuum modeling strategies for homoepitaxial thin film growth
Slides:   pdf

Homoepitaxial thin film growth produces a rich variety of far-from-equilibrium morphologies. Atomistic lattice-gas models analyzed by KMC simulation have been most successful to date in predicting behavior observed in specific experiments.

However, 2D continuum formulations (level-set, phase-field, geometry-based-simulation = GBS) retaining discrete layers have been explored as alternatives, especially for the regime of highly reversible island formation where KMC becomes inefficient. Exploiting GBS, we present the first precise results for the submonolayer island size distribution in this regime [1].

3D continuum formulations have been applied to describe multilayer kinetic roughening where step edge barriers inhibit downward transport and produce unstable growth (mound formation). We analyze this phenomonon using realistic atomistic modeling to show that Ag/Ag(100) [regarded as the prototype for smooth growth] actually grows very rough [2]. Furthermore, mound dynamics is seen to be more complex than predicted by standard 3D continuum models.

[1] PRB 68 (03) 121401; SIAM MMS 3 (05); [2] PRB 65 (02) 193407.

Gero Friesecke (University of Warwick and Technical University Munich)

Interatomic 'van der Waals' forces and the Schroedinger equation

Long range interatomic 'Van der Waals' forces play an important role for equilibrium structure and nonequilibrium behaviour of complex molecular systems (carbon nanotubes, DNA, proteins, ...), but must at present be modelled empirically: ab initio computation remains out of reach. The latter would be particularly desirable because of the huge chemical specificity (i.e., atom dependence) of the VdW force, e.g. for a pair of sodium atoms it is bigger by a factor 1000 than for two Helium atoms.

The difficulty is that 'order N' computational quantum models like Hartree-Fock theory and density functional theory (and their variants integrated into molecular dynamics in the spirit of Car-Parinello) do not resolve VdW forces, only short range covalent bonding. No model short of full all-N-body Schroedionger is known which captures VdW - but that is order eN (prohibitive).

Our main result is that the presence, magnitude and underlying mechanism (quantum electron-electron correlations) of this attraction is in fact a rigorous theorem about the many-body Schroedinger equation. This leads, in particular, to an explicit expression at long range with greatly reduced computational complexity: roughly speaking, enumber of electrons in a single atom.

In the talk, we will start by reviewing the interesting history (starting from van der Waals 1873) and the well developed chemistry 'lore' (starting with Eisenschitz and London 1927) of van der Waals forces.

Matthias Kurzke (IMA, University of Minnesota)

Boundary vortices in thin magnetic films
Poster:  pdf

We analyze a model for thin ferromagnetic films that leads to the formation of vortices at the boundary. The energy asymptotically splits into a singular part depending only on the number of vortices and a finite part depending on their position. This finite part, the renormalized energy, is shown to also control the gradient flow motion associated to the boundary vortex functional. The results and proofs are similar to the theory for Ginzburg-Landau vortices by Bethuel-Brezis-Helein for the static and Sandier-Serfaty for the dynamic case.

Claude Le Bris (Ecole Nationale des Ponts et Chaussées (ENPC), CERMICS)

Discrete to continuum limit for various microscopic lattices: a static picture

We overview some recent work, joint with xavier blanc and pierre-louis lions, on the derivation of macroscopic densities of mechanical energy from energies of microscopic lattices. A particular emphasis is laid on the definition of the energy of some microscopic stochastic lattices, with possible applications to the modelling of materials.

Frederic Legoll (Institute for Mathematics and its Applications, University of Minnesota)

Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed, that aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We present here a theoretical analysis for such a coupling in a one-dimensional setting. We study both the general case of a convex energy and a specific example of a nonconvex energy, the Lennard-Jones case.

In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We also consider the effect of the finite element discretization of the continuum model on the behaviour of the coupled model.

This work is joint with Xavier Blanc (Paris 6) and Claude Le Bris (CERMICS, ENPC).

Igor Mezic (Department of Mechanical Engineering, University of California - Santa Barbara)

Dynamics and control of large-scale molecular motion

Finding rules for design of complex systems that have sufficient flexibility to execute varied tasks while at the same time performing robustly against a large class of perturbations is of considerable interest. Biological networks are of particular interest in this context, given that they have those desired features. We study a network of nearest-neighbor coupled oscillators starting with a simple coarse-grained model of a molecule with a backbone and side-chains. We show that this system is particularly good at reacting responsively to localized disturbances by amplifying them. We present analysis of an interesting transition phenomenon between global energy minima of the system and relate this to recent results on controllability of Hamiltonian systems. More generally, oscillator networks that exhibit such phenomena are characterized by nearest neighbor interactions of a node that are, in most of the phase space, much stronger than the nonlinear oscillations of the local dynamics at the node.

Alexander Mielke (Institut für Mathematik, Humboldt-Universität zu Berlin)

Macroscopic equations for microscopic dynamics in periodic crystals
Poster:   pdf    ps

In infinite periodic lattices the solutions can be studied by Fourier analysis on the associated dual torus. However, in doing a limit procedure with vanishing atomic distance, one observes new phenoma which are usually studied by WKB mehtods. We show that similar results can be obtained under much weaker assumptions by using weak convergence methods.

First we show that linearized elastodynamics can be obtained by a gamma limit procedure which automatically produces the effective elastic tensor. Second we study the transport of energy in the lattice which occurs on quite different wave speeds as the macroscopic elastic waves. It is possible to derive a energy transport equation for a Wigner measure which depends on time, space and the wave vector on the dual torus.

Alexander Mielke (Institut für Mathematik, Humboldt-Universität zu Berlin)

Macroscopic dynamics in discrete lattices
Slides:   pdf

We study effective macroscopic models for oscillations in oscillator chains. We consider small-amplitude waves which are modulations of a basic periodic pattern. If certain nonresonance copndtions hold and if the envelope function satisfeis an associated nonlinear Schroedinger equation then the dynamics of the oscillator chain remains a modulated wave which can be described by the modulational theory on a suitably long time scale.

Similar results are observed for slow modulations of large-amplitude traveling waves, where the macroscopic dynamics is now described by Witham's modulation equation. However, rigorous proofs only exist in simplified cases.

Julie Mitchell (Mathematics and Biochemistry, University of Wisconsin-Madison)

Computer prediction of protein docking and analysis of binding interfaces
Slides:   pdf

Recent work on the development of methods for protein docking and analysis of binding interfaces will be discussed. One of the methods presented is the Docking Mesh Evaluator that uses an implicit solvent model for electrostatics. The Docking Mesh Evaluator is capable of exhaustive search as well as of local and global optimization of binding energies, all of which can be performed using parallel computation.

The Fast Atomic Density Evaluator is a method for analyzing protein shape, and shape complementarity within binding interfaces. The Fast Atomic Density Evaluator's complementarity "hot spots" correlate with residues in which mutation is known to impact binding. This has recently been used in the development of engineered ribonucleases able to kill cancer cells. Shape complementarity analysis can also aid docking prediction, either as a post-filter for exhaustive search results or as a means of dynamic parameterization for flexible docking calculations.

Mirko Hessel-von Molo (Department of Mathematics, University of Paderborn)

Identification of macroscopic dynamics

Several (combined) approaches for the identification of macroscopic (or essential) dynamical features of certain systems are presented. In particular the use of topological entropy with respect to special partitions to identify almost invariant sets and "most expanding sets" of a system has been investigated. Possible future work includes constructing a Freidlin-Wentzell-like quasipotential to construct simple models for high-dimensional systems.

Jonathan A. Othmer (Applied & Computational Mathematics, California Institute of Technology)

Coarse-grained phenomenological rate laws for nucleic acid hybridization kinetics

Joint work with Justin S. Bois and Niles A. Pierce (California Institute of Technology).

Given a nucleic acid energy landscape defined in terms of secondary structure microstates, we describe a coarse-graining approach for performing kinetic simulations that accurately captures the temporal evolution of physically meaningful macrostates. The method is based on the solution of local eigenvalue problems that identify the dominant local relaxations between interacting macrostates. The objective is the quantitative mapping of important landscape features for functional nucleic acid devices.

Anja Riegert (Max-Planck Institute for the Physics of Complex Systems)

Modeling fast Hamiltonian chaos by suitable stochastic processes

Projection operator techniques known from nonequilibrium statistical mechanics are applied to eliminate fast chaotic degrees of freedom in a low-dimensional Hamiltonian system. A perturbative approach, involving a Markov approximation, yields a Fokker-Planck equation in the slow subspace which respects the conservation of energy. A numerical and analytical analysis of suitable model systems demonstrates the feasibility of obtaining the system specific drift and diffusion terms and the accuracy of the stochastic approximation on all time scales. Non-Markovian and non-Gaussian features of the fast variables are negligible.

Christof Schuette (Mathematics and Computer Science, Freie Universität Berlin )

Automated model reduction for complex molecular systems

A novel method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information will be presented. The novel approach represents the effective dynamics of the full system by a Markov jump process between its metastable states / conformations, and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of Hidden Markov Models (HMMs) with output behavior given by SDEs. The numerical effort of the method is linear in the length of the given time series, and quadratic in terms of the number of metastable states. The performance of the resulting method is illustrated by numerical tests and by application to molecular dynamics time series of a DNA oligomer.

Florian Theil (Mathematics Institute, University of Warwick)

Crystallization in two dimensions
Slides:  pdf

Why do so many materials have a crystalline phase at low temperatures? The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type E(y)= sum0 < x < x' < N+1 V(|y(x)-y(x')|) where y(x) in R2 is the position of particle x and V(r) in R is the pair-interaction energy of two particles which are placed at distance r. We show rigorously that under natural assumptions on the potential V the ground state energy per particle converges to an explicit constant E*:

limN imply.gifoo 1/N miny E(y) = E*,

where E* is the minimum of a simple function on [0, oo). Furthermore, if suitable Dirichlet- or periodic boundary conditions are used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry group consisting of rotations and translations.

Giovanni Zanzotto (Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate (DMMMSA), University of Padua)

Stressed microstructures in M9R-M18R martensites

Joint work with Xavier Balandraud (Laboratoire de Mécanique et Ingénieries (LaMI), Institut Français de Mécanique Avancée (IFMA), Université Blaise Pascal (UBP).

We revisit the phase transformation that produces monoclinic 'long-period stacking' M9R or M18R martensites in Cu-based shape-memory alloys, and analyze some associated microstructures, in particular a typical wedge-shaped configuration (Fig.). The basic premise is that the cubic-to-monoclinic martensitic phase change in such alloys is, geometrically, a slight modification of the well-known bcc-to-9R transformation occurring for instance in Li and Na, whose basic strain, at the micro level, is the same Bain strain as for the bcc-to-fcc transition. One then determines the 'near-Bain' microstrain variants pertaining to these elements and alloys, and analyze the long-period stacking martensite as a mesoscale 'adaptive phase.' Twins, habit planes, and also more complex microstructures, such as the CuZnAl wedge, can be analyzed in this way. Earlier conclusions that this microstructure is not kinematically compatible at zero stress are confirmed. However, one can check the wedge is `close enough' to compatibility and compute the corresponding stresses, which turn out to be low, causing only minimal plastification and damage in the crystal. This microstructure is therefore rationalized as a viable path for the transformation also in these alloys. One can moreover verify this to be true for all the known lattice parameters reported for materials exhibiting long-period M9R-M18R martensites. The general conclusion is that the observed martensitic microstructures can be stressed to various degrees also in good memory alloys; and that there seem to be no need for material tuning in order tgat such stresses be low. Indeed, the lattice-parameter relations, guaranteeing the zero-stress compatibility of special configurations, favoring the transformation and its reversibility, do not need to be strictly enforced because microstructural stresses are not very sensitive to lattice parameter values.