This award supports theoretical research on the properties of disordered solids, a topic of wide applicability from rubbers to biology and one of deep and fundamental intellectual interest. In this project analytical methods will be applied to these materials to better understand their universal properties. Students will be trained in the methods of theoretical physics.
This project addresses the physical properties of a wide range of media that exhibit random solid states. Prototypical of these media is vulcanized rubber, a material whose constituents form a thermally fluctuating, macroscopic, random network. Systems such as this start out as fluids, but upon sufficient vulcanization (i.e. permanent chemical bonding between randomly selected constituents) they undergo a phase transition to a new state of matter: the random solid state. This state has a nonzero elastic shear modulus, and at least some of its constituent particles are localized in space. However, the randomness of the vulcanization process ensures that no crystallinity is exhibited, at least on sufficiently long length-scales.
Many forms of matter exhibit the random solid states. To give just a few examples, there are chemical gels made from randomly bonded small molecules, biologically vital structures (e.g. actin filaments), and colloidal particles attracting via depletion forces. In polymeric settings, the constituents may be flexible or rigid, linear or branched, neutral or charged, isotropic or liquid-crystalline, and they may be linked anywhere or solely via chain ends. One can also cross-link blends of polymers, in which case cross-linking can even be restricted to like constituents, in which case inter-penetrating random solid networks emerge.
Intellectual Merit: In view of the multiple levels of randomness that it must confront (e.g. thermal motion, quenched constraints, and emergent structure and response), the statistical physics of random solid media is among the most intellectually challenging frontiers in theoretical physics. Making progress continues to require the introduction, development and application of creatively crafted concepts and strategies for characterizing these media, as well as powerful analytical techniques for extracting pivotal physical information both static and dynamic especially about the heterogeneity of their structure and elastic response. Not only are random solids of broad interest due to their abundance in nature and technology, but also they serve as fascinating laboratories for exploring fundamentally disordered states of matter.
The project builds upon prior work done at UIUC and elsewhere on the theory of random-solid-forming systems. Among its aims is the widening of the range of media and phenomena encompassed, via extensions to novel systems with tendencies towards various forms of ordering, such as liquid-crystalline or phase-separational. The focus is on generic properties, i.e., ones reasonably expected to hold not just for specific brands of random-solid-forming matter but for wide classes it.
Regarding static properties, central questions include: What universal features can be understood? Can one devise useful alternatives to the replica method (e.g. cavities, or anti-commuting fields or coordinates)? Can methods specific to two-dimensions be invoked? Can quantal aspects be explored? How is the interplay between random solidification and other ordering (e.g. liquid-crystallization or phase separation) resolved? Emphasis is anticipated to glide from statics to dynamics, both near the random solidification transition and in the random solid state. Here, the central questions are: What are the optimal strategies and models for building a dynamical theory? What can one say about dynamic critical phenomena near the random solidification transition, or about hydrodynamics in the random solid state?
Media without permanent random constraints will be addressed, too. Thus, consideration will be given to systems in which associations are temporary rather than permanent, forming and disintegrating in equilibrium. This raises important questions, such as: To what extent does the physical gelling that such systems undergo resemble glass formation, and what can such systems teach us about glasses? Can concepts and techniques developed for vulcanized matter be useful for structural glasses and physically gelling systems? If so, how? To what extent can one bridge the gap between vulcanized matter (with its quenched disorder) and conventional glassy systems (which apparently have none)? Statistical mechanics will be the central tool. Statics issues will be tackled via replica field theory, the cavity method and anti-commuting variables methods. Issues of dynamics will be addressed via the MSR formalism and the cavity method, both elaborated to cope with the impact of quenched random constraints.
Broader Impact: The statistical physics of random systems has been outstanding for its reach. It has engendered seminal approaches and results in fields as diverse as neural networks and information processing, computer science and algorithmic complexity, aspects of biology (e.g. heteropolymer folding, analysis of treelike structures), and many aspects of mainstream condensed matter (such as disordered electronic, magnetic and superconducting media). This, coupled with the growing family of materials to which it directly applies, and its potential for shedding light on important areas such as glassy systems, makes it highly likely that the impact of research on the statistical physics of random solids will reach far beyond its intended domain.
