Collaborative Research: Unified Field Theory of Soft Amorphous Solids

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Non-technical summaryFrom collections of grains to aggregates of proteins, colloids or polymers, soft solids have a variety of structures and exhibit a broad range of physical response. They often exist at the margin of mechanical stability, which leads to the adaptability exploited in their applications. Examples include glass, cement, compacted sand, and even yogurt or chocolate mousse. This award supports theoretical and computational research and education focused on the physics of soft solids, with the objective to develop and test a new theoretical framework for their dynamics. Distinct from crystals, the structures of soft solids, generically, do not exhibit any order. Therefore, their mechanical response cannot be described by the conventional paradigm of broken symmetry and long-range order that define the behavior of crystals. Conventional elasticity theories are built on principles of momentum (mechanical equilibrium) and energy conservation, from which symmetries, order parameters, geometry and topology of patterns emerge. The absence of energy conservation in marginal soft solids, where dissipative or active processes can be at play, invalidates these theories. In the new framework proposed here, conservation principles emerge from just the constraints of mechanical equilibrium. This approach provides a natural way of incorporating the coupling between stress and structural rearrangements inherent in soft solids, which is missing in existing theories, to construct an effective field theory for amorphous materials with heterogeneities in stress and deformation fields. The PIs will engage as role models to inspire a more diverse population of students to theoretical condensed matter physics by promoting outreach activities that communicate and discuss how theories are built, how they connect to phenomena and experiments, and what specific skills theorists develop. Outreach activities will also disseminate the excitement of condensed matter physics to K-12 students and the general public.Technical SummaryThis award supports research and education aimed at understanding the structure-function relationship in amorphous soft solids such as jammed grains, gels, and even biological tissues. It has become increasingly clear that localized, sub-dimensional, stress patterns emerging from the constraints of mechanical equilibrium, determine the non-equilibrium mechanical response of a wide range of soft matter. Sub-dimensional excitations have also emerged in tensor gauge theories, a class of field theories recently developed for quantum spin liquids. A recently discovered rigorous mapping of such a tensor gauge theory to mechanics of amorphous solids forms the basis of the proposed research. This mapping has the potential to solve the problem of how stresses get transmitted and why they localize in soft, amorphous solids. An outstanding challenge in amorphous systems is identifying an order parameter that distinguishes between different stress-carrying states. Remarkably, the absence of an order parameter is also a feature of quantum spin liquids, where topological indices such as winding numbers can distinguish between the states. For mechanical structures, topological mechanics provides an index that can do precisely that. Interestingly enough, the elasticity theories of soft matter and tensor gauge theories for quantum spin liquids are field theories that emerge at some level of coarse-graining, whereas topological mechanics explicitly takes into account the network architecture in which the mechanical constraints operate, suggesting that topological mechanics may be the right tool to establish the connection between the microscopic mechanical constraints at play in soft amorphous materials and the appropriate tensor gauge theory framework. A combination of theory and numerical simulations will be used to explore the implications of a new paradigm emerging from tensor gauge theory for soft amorphous materials, and establish connections between this continuum theory and topological mechanics, which provides a network-specific description of the ability of amorphous solids to sustain and evolve under external stresses. Given the ubiquitous presence of soft amorphous solids, a unified field theory of their mechanical response will be transformative for soft condensed matter physics, and the associated applied disciplines of materials science, chemical and structural engineering. The collaboration will develop new computational tools to complement and inform theory and identify new experimental tests. The feedback between theory and simulations will be reflected in the research training of postdocs, graduate and undergraduate students. The bridge created between soft and hard condensed matter physics through the shared framework of tensor gauge theories offers new opportunities for training at the interface between soft and hard condensed matter physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.