Richard D. James Research Group

Mathematics, materials and mechanics


James' group uses mathematical ideas to discover new materials and structures. We seek the fundamental causes of the unique behavior of functional materials, with a special focus on phase transformation, magnetism, ferroelectricity, shape memory and light-matter interaction. To construct predictive theory, we use a variety of tools from nonlinear analysis, the calculus of variations, partial differential equations and group theory, and we invent simulation tools for the theory we create. We make materials in our laboratory and study them using both local and major national characterization facilities. Our ultimate measure of success is to put a spectacular new theory-designed material on the table.

Highly Reversible Phase-transforming Materials

New phase transforming materials, even with strong first-order transformations, can be designed with low hysteresis and multimillion cycle reversibility.

Video of successive heating/cooling cycles in an alloy which has been tuned, by systematically changing composition, so as to satisfy the strongest known conditions of compatibility between phases.

Phase transforming materials have an abrupt change of crystal structure at the transformation temperature. This change can be quite drastic. So it can happen, for example, that one phase is a strong magnet and the other is non-magnetic. Such materials also are usually ferroelastic. We call this "multiferroism by reversible phase transformation". Over the past 10 years we have been seeking the fundamental explanation of what causes some phase transforming materials to fail after a few transformation cycles, while others can go many millions of cycles. The Zn45Au30Cu45 alloy at left was discovered using a systematic theory-designed procedure. It is a big first-order phase transformation, exceedingly low hysteresis, and excellent reversibility. During heating and cooling it exhibits the delightful (and very unusual) non repeating microstructures shown. See [ 115] and [ 116] for more details. We are collaborating closely with researchers from Kiel, Germany and Hong Kong in this work.

Origami structures

World population is growing approximately linearly at about 80 million per year. As time goes by, there is necessarily less space per person. Perhaps this is why the engineering community seems to be obsessed with folding things. Here is our approach.

Origami structures

We are developing a mathematical approach to "rigid folding" based on the use of piecewise rigid isometric mappings that have a group structure. Our ideas are inspired by the way atomistic structures form naturally. There are also fascinating mathematical analogs to the way microstructures form, as in the video above. Our methods imply simple schemes for actuation and control. The resulting structures can also interact with solutions of Maxwell's equations exhibiting constructive or destructive interference. Currently in this work, we are collaborating with researchers at Caltech, Carnegie Mellon and Princeton.

Objective structures

Graphene, carbon nanotubes, phosphorene and buckyballs have a peculiar structural feature that can be used in many ways for the discovery and analysis of new nanostructures.

Objective structures

In the simplest case "Objective structures" are atomic structures with the following property. Imagine a structure consisting of identical atoms. Now imagine sitting on an atom and looking around at your environment. Nearby, you see nearest neighbors, and, as you look further, you see more and more atoms of the structure. Now go to another atom of the structure. Objective structures have the defining property that, if you reorient yourself in just the right way, you see exactly the same environment. Briefly, atoms "see identical environments". There is a natural generalization of this idea to molecular structures. What is so surprising is that many of the most intensely studied nanostructures today - graphene, single-walled carbon nanotubes of any chirality, phosphorene, buckyballs, many viral structures - have this property. We are using this property in many ways, from the design of new fast methods of doing molecular dynamics to new methods for accurate first principles calculations. In this work, we are collaborating with a broad collection of researchers from Berkeley, Caltech, Carnegie Mellon, Cornell, Georgia Tech, Lawrence Berkeley Laboratory, TU Munich, UCSB, UMN and Rome.

Energy conversion

Phase transforming materials can be made to have one phase that is a strong magnet and the other phase nonmagnetic, or, one phase is a strong ferroelectric and the other non ferroelectric. We call this "multiferroism by reversible phase transformation". Such materials can be used for the direct conversion of heat to electricity.

Energy conversion

The concept of "multferroism by reversible phase transformation" can be used in many ways, and we are particularly exploring its possible application to direct energy conversion. In this context "direct" means that the material itself generates the electricity, without a separate electrical generator. We started by using a NiCoMnSn alloy (discovered in our lab) for which the high temperature phase is strongly magnetic an the low temperature phase is nonmagnetic. Passing through the phase transformation by heating, one can convert the abruptly changing magnetization into electricity, by induction. We studied this theoretically and realized it has some disadvantages: good properties for induction conflict with good heat transfer. Guided again by theory, and with Prof. Bharat Jalan, we are now working on the case of an abrupt ferroelectric transformation, which is proving to be extremely promising. Generally, these methods are good for the "small temperature difference regime", e.g., solar-thermal sources, waste heat from air conditioners, data centers, exhaust systems, or even hand-held electronic devices, for which there currently exists no reasonable method of energy conversion.

Interaction of radiation and objective structures

Almost everything we know about the structure of matter comes from X-ray methods. To implement them, one needs a crystal. But what if the material is impossible to crystallize, or, say, prefers a helical structure rather than a periodic one?

Interaction of radiation and objective structures

Almost everything we know about the structure of matter, whether materials science or biology, comes from X-ray methods. (We are trying to change this situation - see above - but the theoretical methods are not yet accurate enough to compete with X-ray methods.) How do they work? One shines plane waves at X-ray wavelength on a crystal, and by changing the relative orientation of plane wave and crystal, one gets constructive interference, that is, strong spots on the detector. Tune the wavelength slightly, and suddenly one gets strong destructive interference. The mathematical reasons for this behavior comprise a fascinating chapter in Fourier analysis, with a digression into number theory and the invariance of differential equations. Could one do that same thing for other "regular" structures such as objective structures. With Prof. Gero Friesecke and his group at TU Munich, we are investigating this possibility. Some of the new solutions of Maxwell's equations that interact constructively with objective structures also have other unexpected properties such as orbital angular momentum, and interact with materials in unusual ways.

Magnetic hysteresis

Hysteresis is synonomous with loss, and thus is usually unwanted in a vast array of applications. We learned a lot about hysteresis in phase transformations, which is primarily related to the fitting together of the phases. Magnetic hysteresis is different, but can we borrow some of the modes of thought to understand the origins of magnetic hysteresis in a predictive way?

Magnetic hysteresis

Hysteresis in phase transformations refers to the property of transforming at a higher temperature on heating than on cooling. Similarly, in stress-induced phase transformation, beginning in one phase, the transformation stress with increasing stress is higher than that of the reverse transformation. Although there remain significant questions in the case of stress-induced transformation, we learned a lot about the fundamental origins of hysteresis in phase transformations. Using these ideas we now routinely make alloys with big first order phase transformations and thermal hysteresis of 2-3 C. Mathematically, hysteresis in phase transformations concerns primarily the fitting together of phases, and the energy barrier that results when they do not fit together so well. Recently, with groups at Colorado School of Mines and Case, we decided to tackle magnetic hysteresis, particularly in soft magnetic materials. We believe there may be a similar energy barrier, but not due to fitting and decidedly more subtle in this case. The fundamental mathematical challenge here is that this barrier (also in the case of phase transformations) is not discovered by doing a linear stability analysis of the state that is just about to transform.

International Projects

1. University of Kiel, Germany, Prof. Dr. Eckhard Quandt, "Reinhart Koselleck Project on Crystallographically Compatible Ceramic Shape Memory Materials". Also, joint UMN/Kiel "Materials for Brain" project (DFG)

2. Hong Kong University of Science and Technology, Prof. Xian Sherry Chen, "Highly reversible shape memory alloys"

3. University of Oslo and SINTEF, Prof. Ole Martin Løvvik, "Conversion between Magnetic, Electric, and Thermal energies in phase change materials (COMET)"

4. Institute of Thermomechanics, Prague, Czech Repbulic, Prof. Hanus Seiner, currently visiting AEM as a Fulbright Fellow, "High mobility and micromechanics of macro-twin interfaces in modulated martensites"

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Aerospace Engineering and Mechanics
University of Minnesota
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