Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities

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Focus Groups

• Monday Focus Group: Multiscale Modeling and Computing
• Wednesday Focus Group: Singularities
• Thursday Focus Group: Molecular Dynamics and Sampling
• Friday Focus Group: Soft Matter

Introduction

During the academic year 2004-2005, the IMA will host a program aimed at a synthesis of the problems at the interface between mathematics, materials science, condensed matter physics, and biology. We believe that a program on Mathematics and Matter is highly timely as the traditional barriers between the fields mentioned above are slowly disintegrating, providing rich interdisciplinary opportunities for the interplay between mathematics and the more traditional disciplines which have involved the study of matter. We will focus on phenomena that require modeling that integrates the atomic to the continuum scales. Though we acknowledge that much can change in the years between now and the beginning of this program, we have targeted several representative topics which appear to provide particularly exciting opportunities.

A remarkably broad spectrum of modern mathematics is being utilized to understand matter. Several of the most active research efforts in nonlinear partial differential equations have been motivated by the need to model the structure and dynamics of defects and microstructure. Current research in stochastic differential equations is being driven by the need to model microscale and nanoscale devices and phenomena. Topological and geometric concepts are being developed to understand defects in crystals, the structure of DNA, protein folding, and the topology of cellular organisms. Symmetry and representation theory are being utilized to identify order parameters in complex materials and to study phase transitions.

Scientific computation is playing a major role in the development of materials theory and its validation by experiment. Mathematicians and materials scientists are beginning to confront the computational challenges of multiscale modeling, singularities, and disorder. However, contemporary computational algorithms for the study of matter, such as "hyperdynamics," are often developed in the context and language of physical theories that are not part of the traditional education of computational mathematicians. This IMA program will strive to enable interactions and research between mathematicians and materials scientists on such computational problems that are important in the study of matter, but have been given little attention by the mathematics community.

Organization

Our goal in the IMA Program is to facilitate the development of multi-disciplinary research efforts for which mathematics can play a role at the cutting edge of research in matter. We believe that the major themes of multiscale modeling, singularities, and disorder must be studied in the context of specific physical and biological systems. We think that our goals for this program can be best achieved by the organization of several focused research groups composed of both senior and junior members that will investigate a physical or biological system starting from elementary aspects and progressing to research projects. During the first weeks of the Fall semester, each IMA post-doc and long-term visitor will join one of more focus groups. We expect that each focus group will meet regularly and will invite short-term visitors to discuss current developments.

Each focus group will organize a workshop in which leading experts will be invited to the IMA for several days to participate in a forum to discuss current theories, identify research opportunities, and make connections to related physical and mathematical theory. We will also welcome the participation of mathematicians and scientists who are not expert in the focus area. The format for the workshops will be structured to faciliate active discussion and exploration. There will be a few general lectures, and then participants will take turns leading the discussion by describing their own work or other ideas that intrigue them. Several members of this organizing committee have participated in very successful workshops with this format.

During the Spring semester, the focus groups will continue to meet, but in addition periods of concentration will be organized on the general themes of multiscale modeling, singularities, and disorder. We will organize new focus groups to investigate these general themes and each focus group will organize a workshop on its theme with the participatory format described above for the Fall semester.

In addition to the focus areas descibed below, we will invite one or more scientific leaders to organize other focus groups during the next year as research opportunities are identified. To maintain the vitality of the research groups and the direction of the post-docs, we expect the focus group organizers to be resident at the IMA for long-term visits.

The publication of the Human Genome is perhaps the most visible outcome of the many remarkable advances that have recently been made in the experimental techniques of molecular and cell biology. In addition to the detailed sequence information now available for Nucleic Acids and Proteins, it is currently possible to manipulate individual macro-molecules and molecular assemblies through Atomic Force Microscopy and other techniques to measure their mechanical material properties, to observe 3D crystal structures of biological macro-molecules through ever more accurate X-ray diffraction data, and to begin to observe accurate 3D solution structures through NMR techniques and Cryo-Electron Microscopy. The recent progress in infrared spectroscopy even makes it possible to observe biomolecular dynamics in full temporal resolution at extremely short time scales.

The area is quite unusual within the history of science because of the increasing wealth of (rather accurate) experimental data that is available as compared to the paucity of mathematical and computational models that are adequate to understand and simulate the data in a quantitative way.

There are many efforts throughout the world that are trying to exploit this unusual imbalance between experiment and theory through the establishment of programs in mathematical and computational molecular biology. Consequently the molecular and cell biology sciences are currently entering a key developmental stage in which the quantitative tools and outlook of mathematics, physics and mechanics are being brought to bear on biological problems at the length scale of the cell and downward (microns to nanometers). These problems range from the role of the sequence-dependent elastic properties of DNA in its biological function, to the operation of molecular motors such as kinesin, to the equilibrium shapes of biological membranes, and the active transport of molecules and ions through membranes, to the mechanics of flagellar motion, to the mechanics of micro-tubules, and the mechanisms by which cells divide.

All these problems have the common feature they need mathematical models for mesoscopic quantities such as elastic properties and coefficients, membrane shapes, relations between protein distribution and activity, or mean flows and diffusion coefficients. On the one hand, all of this mesoscopic information needs to be consistently incorporated into longer-scale macroscopic models that allow simulation, for example, on the cellular level. On the other hand, the same mesoscopic information needs to be reliably derived from shorter-scale microscopic models at, for example, atomistic resolution, or appropriate computations are required to extract information from microscopic experiments. Mathematical research in both of these passages of scale is still in its infancy; it is surprising how many fundamental questions are still essentially unaddressed after decades of development in molecular dynamics and theoretical bio-physics and bio-chemistry. Three examples serve to illustrate this observation:

(a) In molecular dynamics a kind of ergodicity principle is used to compute certain (mesoscopically relevant) averages by means of long-term trajectory sampling. But the construction of reliable and algorithmically feasible, error estimates is still an open problem.

(b) Mesoscopic models for the dynamics of larger biomolecules require knowledge about the molecule's effective dynamics, at best in terms of some essential variables, and its sensitivity to external parameters. However, useful mathematical principles for the definition and identification of essential variables are still largely unknown. Even in cases where appropriate essential variables can be identified, the necessary computations of statistical averages, which are unavoidable in order to determine the effective forces in terms of the essential variables, brings us back to the difficulties addressed in point (a).

(c) The simulation of many cell processes will require the combination of macroscopic and microscopic models; for example the description of a membrane as an elastic surface coupled to some model for its transport channels at atomic resolution. The mathematical and computational treatment of such combined, multi-scale models is still in its very early stage.

Questions such as these will be addressed in periodic, intensive discussion groups, centered around researchers who have already personally persevered through sufficient ad hoc attacks on specific problems of this type as to be deeply motivated to develop and benefit from, more general, concept-oriented approaches. A more traditional workshop will also compile some of the substantially different perspectives on these topics.

Activities will be coordinated to benefit from the available synergies with the other focus groups of the special year, particularly the programs in condensed soft matter and in multi-scale modeling.

Soft Condensed Matter Physics

Organizers: Maria Carme T. Calderer (School of Mathematics, University of Minnesota), Chun Liu (Department of Mathematics, Penn State University) and Eugene Terentjev (Cavendish Laboratory, University of Cambridge, England).

The proposed research work will address physical systems such as liquid crystals, polymers, membranes, gels and micelles. Molecular chirality and its ubiquitous macroscopic manifestation will be an underlying property linking many of such systems. Electromagnetic and mechanical interactions as well as related flow phenomena are fundamental issues concerning such materials, and their applications.

Liquid crystal systems and models, in addition to their intrinsic theoretical and application values, often serve as tools to investigate fundamental issues in soft matter. One major theoretical issue in condensed matter systems is to understand the phenomenon of self-organization such as order-disorder phase transitions (that may eventually lead to crystallization), the self-assembly of biological membranes, the occurrence of liquid crystal phases in several families of organic materials and polymers, the formation of self-assembled monolayers on solid substrates, and, in general, processes that involve cell or molecular packing. Currently, these areas are being intensely investigated and many new results and data will likely be available by 2004-05, and in need of theoretical models and mathematical tools to help sorting them out. The successful combination of optical and electron microscopy drives the research endeavors. (The former provides information on the geometric structure of the assembly, whereas the latter gives detailed views of individual molecules).

The leading work of Onsager in the 1940's can help illustrate our approach to using liquid crystal phases as models of self-assembly. Indeed, Onsager's work explains the parallel ordering of non-chiral, rod-like molecules as driven by entropy principles. Macromolecules in suspension at high densities, such as those found in the cytoplasm or nucleus, undergo packing constraints that lead to self-assembly, with a tendency to form orientationally or positionally ordered structures. If in addition such molecules are chiral, then frustration of the order may occur, resulting in topological defects. From a different point of view, as the molecular arrangements achieve increased order, spontaneous polarization often emerges (less ordered phases are mostly dielectric). Therefore, ferroelectricity will also provide a unifying framework to the proposed research. In particular, one approach will explore interconnections between mechanical and biological systems through studies of ferroelectric models, and their application to sensor and activator devices. Ferroelectric models are at the core of important research themes such as artificial muscles and robotics, video display, organic semiconductors, optical switching and telecommunications. Prototype ferroelectric models can be found in solid mechanics as well as in liquid crystals, polymeric systems, membranes, liquid crystal elastomers, gels and micelles. Another research aspect to address deals with the applications of liquid crystals as pattern templates for nonomaterials.

Flow phenomena in soft matter arise in material processing as well as a result of the high nonlinear response to electromagnetic fields (in crystalline solids it would be prevented by the lattice structure). Consequently, Rheology becomes an important research field in soft condensed matter, and it is linked as well to nanomaterial processing. Mathematical modeling in Rheology will be another main research theme in the program; it also arises as companion as well counterpart to the ordering arrangements previously discussed. Indeed, self-assembling structures of atoms and molecules measured in nanometers occur naturally in living organisms, human attempts at nanoscale manufacturing are still at the incipient stages. Modeling and mathematical studies of polymer systems lack behind synthesis and experimental work. Current systems of interest for many different applications include block copolymers, polyelectrolytes, ionomers, liquid crystalline polymers, miscible polymer blends, branched polymers, networks of both charged and uncharged polymers, surfactants, and colloidal suspensions. Multiscale modeling is at the core of mathematical studies of such systems.

The program will be structured in two research themes:

1. Self-assembly, chirality and ferroelectricity in soft condensed matter systems, and

2. Nanomaterials and Rheology.

Leading senior scientists in these areas have agreed to participate for extended periods of time. The selection and organization of topics will progress following special input of senior participants. We intend to structure participants into continuing research themes. Lectures on the scientific background of the selected topics will initiate the research work.

Molecular Dynamics and Sampling

Organizers: Benedict Leimkuhler (Department of Mathematics, University of Leicester) and Frederic Legoll (IMA, University of Minnesota)

The group will investigate problems associated to the sampling of the phase space of biological or chemical systems, and will address both theoretical and numerical issues, as well as challenges encountered in the chemistry and biology communities. Fundamental issues include efficient sampling of corrugated landscapes, computation of free energy along reaction paths, accelerated dynamics, and stochastic vs. dynamical models and methods. The aim is to identify challenges and make rapid progress through the collaboration of people with diverse research backgrounds. Discussion will be led by current visitors at the IMA, U of Minnesota faculty and recognized experts in these fields. Participants from mathematics as well as chemistry, physics and biology are all welcome.

Multiscale Modeling, Singularities, and Disorder Focus Areas

Multiscale Modeling and Computing: the Problem of Disparate Time Scales

Organizer: Richard James (Aerospace Engineering and Mechanics, Minnesota)
Co-organizer: Mitchell Luskin (Mathematics, Minnesota)

It is probably fair to say that the single most important theme in science today is the problem of relating phenomena on different scales of length and time. Partly this problem arises from the significant advances on the calculation of properties on the atomic scale at zero temperature, as embodied in methods like density functional theory. The great hope in areas like materials science and biochemistry is to relate the functionality of the material or organism to its fundamental constituents, their chemical nature and geometric arrangement.

A great deal more progress is being made on the length scale problem than on the analogous time scale problem. This perhaps can be attributed to the commonness of the situation of having gradual variation of relevant quantities across a lattice, or the prevalence of macroscopic homogeneity in disordered systems, which gives a basis for the approximation of atomic arrangements by smooth fields. Operationally, it may also be attributed partly to the great advances in the techniques of microscopy - atomic probe, electron and optical microscopy - that have given an accurate picture of many systems over a large range of spatial scales - but always averaged in time.

This is one of the most active areas in applied mathematics and physics. A central theme is the determination of what information on the finer scale is needed to formulate an equation for the "effective" behavior on the coarser scale. Mathematical and computational techniques such as homogenization, multi-grid, relaxation, Young and H and Wigner-measures, quasiconvexity, Gamma convergence and more generally weak convergence methods have provided new insights and improved both the analytical understanding and the design of numerical algorithms. A broad set of ideas in physics and engineering, including the renormalization group, the quasicontinuum method, and a host of new methods that blend ideas from statistical mechanics, transition state theory, many body physics and continuum mechanics are being formulated, tested and refined for the change-of-scale.

Given this present scenario, and the significant lead time associated with the formulation of IMA programs, we propose the following plan. We propose two levels of activity. One level recognizes the significant advances that have been made, and possibly more significant advances that will come in the interim, on multiscale methods. This part of the program will involve a series of extended lectures (1-2hrs.) on recent multiscale advances. It will serve as a educational forum for postdocs (and participants and organizers!) to survey many of the concepts under development. This activity may include relatively mature work on homogenization, the quasicontinuum method, Gamma-convergence in magnetism and superconductivity, and the theory of effective Hamiltonians and cluster expansions.

The second activity will be focused exclusively on the time-scales. As explained above, the progress on time scales has been significantly slower than the analogous problem for the length scales, and this is not likely to be remedied in the interim. For the passage from atomic to macroscopic scales the time scale at the fine scale is determined by the frequency of atomic vibrations. The oscillations at finite temperature have amplitude of order 1 on atomic scale, but they are of extremely short duration compared to the time scale of macroscopic kinetic events of interest, such as the growth of grains or phases, diffusion in polymers and glasses, the growth of a precipitate, the folding of a protein, or the motion of a dislocation line or a magnetic domain wall. These phenomena involve a still large collection of rare events each of which occurs only after many vibrational periods. There are all the issues of large Hamiltonian systems, ergodicity, and the rigorous derivation of statistical mechanics that have led to important mathematical theory, but have ultimately resisted a completely satisfying resolution. There is the possibility of new ideas in this area springing from many different fields - atomic physics, nonlinear dynamics, pde, turbulence, numerical analysis, materials science, wave propagation, statistical physics. - such as the temperature-accellerated dynamics methods of Voter and Sorensen, kinetic Monte Carlo methods, and combined quantum and classical methods. It is hoped that the simplifications afforded by separation of scales, together with the special features of atomistic problems, can in some way lead to new ways of understanding macroscopic kinetics.

For this activity on the difficult problem of time-scales, we believe a traditional workshop format will not be the most productive and will instead host periodic intensive discussion groups, centered around researchers who seem to have the germ of an idea that might work.

There is substantial overlap between this program and the others in the proposed year.

Fall Semester: Atomic force and interatomic potential energy, coarse grain Monte Carlo methods, effective Hamiltonians. Applications to crystalline materials and polymers.

Spring Semester: Nonequilibrium statistical mechanics, accelerated molecular dynamics, kinetic Monte Carlo, transition rates, metastability. Applications to biology and phase transformations.

Singularities

Organizer: Peter Sternberg (Mathematics, Indiana)
Co-organizers: Fanghua Lin (Mathematics, NYU) and J. Rubinstein (Mathematics, Indiana)

The focus group on singularities will work on topics from superconductivity, optics, and continuum mechanics; depending on senior participants as well as scientific developments over the next three years. We outline below the general landscape in each of these areas. We should also mention that the prevalence of singularities in other scientific areas such as in biology, neural networks, ferroelectric materials and liquid crystals will likely lead to fruitful interactions between the focus group on singularities and the other focus groups within this proposed year of study.

Towards the end of the year, we propose to hold a workshop on the general subject of singularities in materials.

Superconductivity

We propose a focus on the behavior of superconductors in the presence of applied magnetic fields. Magnetic fields tend to impede the ability of such materials to carry resistance-free currents by forcing the appearance of "defects" inside the material known as vortices. Many crucial questions remain regarding the location and dynamics of vortices-questions that inevitably lead mathematicians to exciting intersections of analysis, geometry and topology. Certainly one area of special focus would likely involve the modeling of high-temperature superconductors. At present, there are many competing models for these complicated, layered materials. Given the complexity of the models so far introduced and the lack of agreement surrounding them, there has been relatively little contribution made by mathematicians to date. We would hope to initiate (or further advance) progress in this area through such a focus group, comprised of physicists, mathematical analysts and computational experts. A second subject of interest would likely involve the 3-d Ginzburg-Landau model for superconductivity. While the vortex behavior of samples in equilibrium as modeled by 2-d Ginzburg-Landau is relatively well-understood, the 3-d picture is far from clear. Experiments indicate the possibility of quite complicated vortex configurations and it remains a challenge to capture these co-dimension two singularities in a rigorous mathematical way. Even less well developed is the dynamical picture, where questions still remain on the level of modeling as regards the validity of the time-dependent Ginzburg-Landau system (even in 2d). This should again provide a fruitful context for collaboration between analysts, computational experts and physicists.

Optics

The guiding and manipulation of light revolves around the physics of solitons in one, two or three dimensions. Significant recent experimental progress revealed a rich family of optical singularities. For example we mention the observations of short pulse propagation in birefringent optical fibers and spatial and temporal vector solitons in nonlinear media. How are they formed? How do they propagate? Can they be controlled? How to use them in the design of optical fibers and in photonics? All these questions require the modeling nonlinear optics and the analysis of singularities formation and dynamics.

Continuum Mechanics

Many types of singularities are encountered in continuum mechanics. For example, let us mention dislocations (appearing in crystals, sand piles, etc.), folding patterns in thin film blisters and domain walls in micromagnetics. Each of these problems involves many length scales; each combination of them leads to an entirely different type of solution. Many of these solutions involve singularities (domain walls, line singularities, etc.). A thorough understanding of the relative importance of the different terms in the energy functionals is crucial to the modeling of technologies based on such materials. The entire discipline is closely related to liquid crystal theory, thus the activity proposed within the current framework can naturally merge into the proposed activity on liquid crystals and ferroelectricity.

Long Term Visitors:

The following scientists are confirmed or highly likely as long-term visitors during the program. Other long-term visitors are currently being arranged.