IMA Tutorial/Workshop:

Composites: Where Mathematics Meets Industry

February 7-9, 2005

Organizer:

Composites play a vital role in industry, from carbon-fibre materials, to polycrystalline alloys with the crystal microstructure tailored to achieve desired design parameters, to rocket fuels of metallic particles in an oxidizing matrix, to porous materials for filtering and storage, to electro and magneto rheological fluids, to photonic and phononic band gap structures, and to novel nanostructured materials. Composites are also a source of fascinating mathematics in the quest to understand how features of the microstructure determine the overall macroscopic properties of a material. This tutorial/workshop will consist of three parts. On the first day there will be a series of tutorials, as listed below, on problems of direct interest to industry. On the second day 5 or 6 industrial scientists will make presentations on their work which involves multiscale modelling. These will mix research reports with the posing of problems. On the third day we will break out into groups where we will discuss the problems posed and techniques that could be applied to solve them. The day will finish with each discussion group giving a report.

Tutorial Lectures:

Modeling the pipeline of high performance, nano-composite materials and effective properties

M. Gregory Forest
Professor of Mathematics
Applied Mathematics Program Leader Co-Director, Institute for Advanced Materials, Nanoscience & Technology University of University of North Carolina at Chapel Hill
forest@amath.unc.edu
http://www.amath.unc.edu/Faculty/forest/

Composite properties and microstructure

Robert P. Lipton
Professor of Mathematics
Louisiana State University
lipton@math.lsu.edu
http://www.math.lsu.edu/~lipton/

Nanoparticle suspensions with giant electrorheological response

Ping Sheng
Professor of Physics and Department Head
Director, Institute of Nano Science and Technology
Hong Kong University of Science and Technology
sheng@ust.hk
http://physics.ust.hk/department/staff_detail.php?action=1

Speaker Biographies and Lecture Abstracts

M. Gregory Forest (Mathematics, University of North Carolina at Chapel Hill) http://www.amath.unc.edu/Faculty/forest/

Biography: M. Gregory Forest is Grant Dahlstrom Distinguished Professor in the Department of Mathematics at University of North Carolina at Chapel Hill, where he also serves as Co-Director of the Institute for Advanced Materials, NanoScience and Technology, and the founding leader of the Program in Applied Mathematics. He holds a PhD in Applied Mathematics from the University of Arizona, and has served on the faculty of Ohio State University and extensively consulted to industrial and government research laboratories. His current research efforts are in complex fluids and soft matter applied to high performance materials and biological systems. He is on the editorial board of SIAM Journal on Applied Mathematics and Continuum Mechanics and Thermodynamics.

Lecture Title: Modeling the pipeline of high performance, nano-composite materials and effective properties
Part I Slides:  html    pdf    ps    ppt
Part II Slides:   html    pdf    ps    ppt

Abstract. We focus these lectures on the class of nano-composites comprised of nematic polymers, either rod-like or platelet-like macromolecules, together with a matrix or solvent. These materials are designed for high performance, multifunctional properties, including mechanical, thermal, electric, piezoelectric, aging, and permeability. The ultimate goal is to prescribe performance features of materials under conditions they are likely to be exposed, and then to reverse engineer the pipeline by picking the composition and processing conditions which generate properties with those performance characteristics. These lectures will address two critical phases of this nano-composite materials pipeline. First, we model flow processing of nematic polymer films, providing information about anisotropy, dynamics, and heterogeneity of the molecular orientational distributions and associated stored elastic stresses. Second, we determine various effective property tensors of these materials based on the processing-induced orientational distribution data. Underlying these technological applications is a remarkable sensitivity of nematic polymer liquids to shear-dominated flow, which must be understood from rigorous multiscale, multiphysics theory, modeling and simulation in order to approach the ultimate goal stated above.

This research is based on multiple collaborations and supported by various federal sponsors, to be highlighted during the lectures.

Robert P. Lipton (Department of Mathematics, Louisiana State University)

Biography: Robert Lipton is Professor of Mathematics and founding member of the Mathematical Materials Science Group at Louisiana State University. Currently a visiting scholar at the Division of Engineering and Applied Sciences at Harvard University, he obtained his Ph.D. from the Courant Institute of Mathematical Sciences in 1986 and, after a postdoc at the Mathematical Sciences Institute at Cornell University, became Charles B. Morrey Assistant Professor at the University of California at Berkeley in 1988. He served on the Mathematical Sciences Faculty at WPI from 1990-2001. He collaborates and consulted with with scientists at Wright Patterson Air Force Base. His current research is in the area of failure initiation in composite materials.

Lecture Title: Composite properties and microstructure

Abstract. We begin with an overview of composite materials and their effective properties. Most often only a statistical description of the microstructure is available and one must assess the effective behavior in terms of this limited information. To this end approximation schemes such as effective medium schemes and differential schemes are discussed. Variational methods for obtaining tight bounds on effective properties for statistically defined microgeometries are reviewed. Formulas for the effective properties of extremal microgeometries are presented. Such microgeometries include layered materials and sphere and ellipsoid assemblages.

Next we focus on physical situations where the interface between component materials play an important role in determining effective transport properties. This is relevant to the study of nanostructured materials in which the interface or interphase between materials can have a profound effect on overall transport properties. Variational methods and bounds are presented that illuminate the effect of particle size and shape distribution inside random composites with coupled heat and mass transport on the interface.

We conclude by introducing methods for quantifying load transfer between length scales. This is motivated by the fact that many composite structures are hierarchical in nature and are made up of substructures distributed across several length scales. Examples include aircraft wings made from fiber reinforced laminates and naturally occurring structures like bone. From the perspective of failure initiation it is crucial to quantify load transfer between length scales. The presence of geometrically induced stress or strain singularities at either the structural or substructural scale can have influence across length scales and initiate nonlinear phenomena that result in overall structural failure. We examine load transfer for statistically defined microstructures. New mathematical objects beyond the well known effective elastic tensor are presented that facilitate a quantitative description of the load transfer in hierarchical structures. Several physical examples are provided illustrating how these quantities can be used to quantify the stress and strain distribution inside multi-scale composite structures.

Ping Sheng (Department of Physics, Hong Kong University of Science and Technology)

Biography: Ping Sheng is head of the Physics Department and director of the Institute of Nano Science and Technology at the Hong Kong University of Science and Technology. He obtained his PhD in physics from Princeton University in 1971, and later worked at the Institute for Advanced Study, the RCA David Sarnoff Research Center, and the Exxon Corporate Research Lab, where he headed the theory group from 1982-86. Professor Sheng's research interests include many areas of composites and materials science. He is a fellow of the American Physical Society, a member of the Asia Pacific Academy of Materials, and was elected the 2001 Technology Leader of the Year by the Sing Tao Group of Hong Kong.

Lecture Title: Nanoparticle suspensions with giant electrorheological response
Slides:   html    pdf    ps    ppt

Abstract. In this talk I wish to tell the story of a 10-year effort in search of a better electrorheological (ER) fluid material, leading to the discovery of the giant ER effect, and the crucial role that mathematics and simulations has played in the whole process.

Electrorheology denotes the control of a material's flow properties (rheology) through the application of an electric field. ER fluid was discovered sixty years ago. In the early days the ER fluids, generally consisting of solid particles suspended in an electrically insulating oil, exhibited only a limited range of viscosity change under an electric field, typically in the range of 1-3 kV/mm. The study of ER fluid was revived in the 1980's, propelled by the envisioned potential applications, as well as the successful fabrication of new ER solid particles that, when suspended in a suitable fluid, can "solidify" under an electric field, with the strength of the high-field solid state characterized by a yield stress (breaking stress under shear). However, further progress was hindered by the barrier of low yield stress (typically in the range of a few kPa).

Starting in 1994, we have adapted the mathematics of composites, in particular the Bergman-Milton representation of effective dielectric constant, to the study of ER mechanism(s) [1-4]. The questions we aim to answer are: (1) the role of conductivity in the ER effect, (2) the role multipole interaction, (3) the ground state microstructure of the high-field state and most importantly (4) the upper bounds in the yield stress and shear modulus of the high field solid state. Finding the answer to (4) led to the suggestion of the coating geometry for the ER solid particles which can optimize the ER effect, but at the same time also pointed out the limitation of the ER mechanism based on induced polarization. The subsequent study of adding controlled amount of water to the ER fluid pointed to the intriguing possibility of using molecular dipoles as the new "agent" for enhancing the ER effect [5]. Working along this direction, the experimentalist W.J. Wen was able to synthesize urea-coated nanoparticles of barium titanyl oxalate which exhibited yield stress in excess of 100 kPa, breaking the yield stress upper bound and pointing to a new paradigm in ER effect in which the molecular dipoles can be harnessed to advantage in controllable, reversible liquid-solid transitions with a time constant on the order of 1 msec. We propose the model of aligned surface dipole layers in the contact area of the coated nanoparticles to explain the observed giant ER effect [6], with the electric-field induced dissociation (the Poole-Frenkel effect) of the molecular dipoles accounting for the observed ionic conductivity. Quantitative agreement between theory and experiment was obtained. The talk concludes with an outline of the intriguing questions yet to be answered, and the problems to be solved before ER fluids can become a commercial reality.

[1]   H.R. Ma, W.J. Wen, W.Y. Tam, and P. Sheng, Phys. Rev. Lett. 77, 2499 (1996).
[2]   W.Y. Tam, G.H. Yi, W.J. Wen, H.R. Ma, M.M. T. Loy, and P. Sheng, Phys. Rev. Lett. 78, 2987 (1997).
[3]   W.J. Wen, N. Wang, H.R. Ma, Z.F. Lin, W.Y. Tam, C.T. Chan, and P. Sheng, Phys. Rev. Lett. 82, 4248 (1999).
[4]   H.R. Ma, W.J. Wen, W.Y. Tam and P. Sheng, Adv. Phys. 52, 343 (2003).
[5]   W.J. Wen, H.R. Ma, W.Y. Tam and P. Sheng, Phys. Rev. E55, R1294 (1997).
[6]   W.J. Wen, X.X. Huang, S.H. Yang, K.Q. Lu and P. Sheng, Nature Materials 2, 727 (2003).

LIST OF CONFIRMED PARTICIPANTS