Geometry, topology and soft matter

Geometry, disorder and topology have a central role in condensed matter. Their interplay and synergy lead to some of the most intriguing phenomena in soft-matter physics. I have studied crystalline lattices whose disordered topology is crafted to model the elastic properties of jamming (a ubiquitous transition describing many systems from molecular glasses and granular media to dislocation tangles and biological tissues). I have also used the mathematics of martensites (iron, shape-memory alloys) to study focal conic domains — the most iconic defect of smectic liquid crystals displaying Lorentz invariance and neat arrangements of ellipses and hyperbolas. Much of the current frontier in soft-matter science involves a deep understanding of the interplay between geometry, disorder and topology.

Plasticity, elasticity and jamming

Confluent cell tissues, amorphous solids, and ice single crystals are prototypical examples of systems with peculiar elastic properties and complex scaling behavior. Plastic flow on small-scale crystals starts at history-dependent yielding points, and manifests as intermittent strain bursts (avalanches) with scale-free size distributions that are similar (but not equal) to depinning. Randomly-diluted networks lose rigidity at the isostatic point of mechanical stability, which was described by J. C. Maxwell in the nineteenth century, and that plays a seminal role at the celebrated jamming transition, rigidity transitions of actin networks, topological mechanical metamaterials and several systems of soft and biological matter.

I have participated in the analysis of a vast set of experiments performed by Julia Greer's group at Caltech, in which micro and nano Cu pillars (Figure 1) are subject to uniaxial compression and cyclic loading. We find that the amplitude and decay time of precursor avalanches diverge as the peak stress approaches the failure stress for each pillar, with a power-law scaling virtually equivalent to reversible-to-irreversible transitions in other systems [read more].

Figure 1: Image of a 300nm Cu pillar fabricated by the Greer group at Caltech: Under uniaxial compression and cyclic loading, these crystals exhibit nontrivial stress-strain response with signatures of a ‘reversible-to-irreversible’ transition.

With T. C. Lubensky and colleagues at Penn, I have developed a new theory of the jamming transition [read more] that is both analytically tractable, and that clarifies the relation between jamming and rigidity percolation for disordered elastic systems. Note that the jamming critical point is very unusual; it has properties of both a first-order transition (with a discontinuous jump in the bulk modulus) and a second-order transition (with a continuous growth of the shear modulus), in contrast with rigidity percolation, where both moduli continuously grow from zero. This peculiar behavior prompted us to consider compression-resistant honeycomb and diamond lattices in two and three dimensions, respectively. In a previous publication, we had used these lattices as prototypes to describe the role played by bond-bending interactions on phonon and elastic properties as well as rigidity transitions of under-coordinated networks [read more]. To describe jamming, we randomly populate these lattices with harmonic springs connecting nearest and next-nearest-neighbor pairs of sites (see Figure 2). In the space of bond occupation probabilities, jamming emerges as a multicritical line terminating a surface of rigidity-percolation transitions. Our effective-medium theory yields a faithful description of the jamming transition, featuring explicit forms for the scaling variables and universal scaling functions near both jamming and rigidity percolation, including their spatial and temporal dependencies in the elastic and the fluid phases.

Figure 2: Effective-medium theory for jamming: Bond occupation probabilities of randomly-diluted honeycomb/triangular lattice (blue/red solid lines) can be tuned to yield jamming or rigidity-percolation transitions. Our Effective-Medium Theory (illustrated by faint lines) provides analytical formulas for universal scaling variables and functions, including frequency and spatial behavior, as well as the crossover between jamming and rigidity percolation.

More recently, we have used similar ideas to design metamaterials with rich criticality and flexible mechanical properties [read more]. We have developed efficient protocols to tune the Poisson ratio v of disordered elastic networks by leveraging special features of three two-dimensional lattices: the honeycomb lattice with a finite bulk modulus B > 0 and vanishing shear modulus G = 0 and thus v =(B - G)/(B + G) = 1; the untwisted kagome lattice with B > 0, G > 0 and v = 1/3; and the twisted-kagome lattice with B = 0, G > 0 and thus auxetic behavior with v = -1. Randomly-diluted versions of suitable combinations of these lattices (see Figure 3) give rise to model mechanical metamaterials that are experimentally accessible, have controllable Poisson ratio easily tuned from -1 to +1 (the entire range for v in 2D), and rich phase diagrams showing jamming, shear-jamming, rigidity-percolation and auxetic transitions.

Figure 3: Multifunctional elastic network: A combination of the honeycomb (red) and twisted kagome (gray) lattices gives rise to a mechanical metamaterial that has a Poisson ratio which can be tuned from -1 to +1, and rich phase diagrams featuring jamming, shear-jamming, rigidity percolation and auxetic transitions.

Constrained geometry, martensites and the smectic spacetime

For long the tools of geometry have been applied to understand and control several properties of complex materials. Current applications permeate state-of-the-art topics in materials science and technology, from self-folding origami to the unfolding of solar sails. Under constrained geometries, martensitic crystals self-accommodate into finely-arranged patterns of low-energy variants described by a sophisticated mathematical theory. In turn, our profound understanding of the smectic state involves a deep geometrical analysis (from the inference of its layered structure to the discovery of Poincaré symmetries).

With the Sethna group at Cornell, I have used modern GPUs to develop what to our knowledge is the first smectic simulation to spontaneously generate focal conic domains [read more]. We developed new tools for visualizing the smectic microstructure [see Figure 4(a), (b), (c) and (d) corresponding to visualizers for energy, centers of curvature, layers and microscopy], and studied rheology and coarsening.

Figure 4: Visualizing the smectic geometry and defects: Our GPU simulations provide powerful visualization tools that lead to deep insights into the three-dimensional smectic microstructure. (a) Energy density. (b) Centers of curvature of the smectic layers [shown in (c)]. (d) 2D section mimicking microscopy images.

Ultimately, we were able to bridge the geometry of these liquid crystals to the mathematical analysis used to describe crystal martensites [read more]. Smectics are the weirdest martensites: instead of rotations of crystalline variants (top-left of Figure 5), smectics are Weyl-Poincaré transformations ('special relativity' plus dilatations) of domains of concentric tori (Figure 5). Crystalline variants (dark and light red in the top left) merge via twin boundaries and form fine laminate structures. Smectic variants (different colors Figure 5) merge via conic sections (intersections of light cones) and form a complex mosaic of focal conic domains.

Figure 5: Smectics and martensites: Nearly perfect ellipses and hyperbolas emerge in the smectic microstructure as a consequence of local minimization over constrained geometries, and are interpreted using a generalization of the theory of martensites. For comparison, an example of crystal martensitic microstructure is displayed at the top corner (courtesy of C. Chu and R. D. James).

Statistical mechanics of liquid crystals

Unsurprisingly, liquid crystals occupy a central position in soft-matter research, as they provide rich laboratory models to study spontaneous symmetry breaking and topological defects, and have applications ranging from well-known LCDs to holographic laser tweezers and photoactuators. Despite past and ongoing advances in the theory of liquid crystals, the statistical mechanics of even the simplest kind (the nematic) remains largely unexplored. In this context, elementary statistical models have the important purpose of providing a bridge between microscopic and heuristic approaches, global phase diagrams and qualitative insights into liquid-crystalline behavior.

With Yokoi and the Salinas group at the University of São Paulo, I have used elementary models to clarify the interplay between quenched and annealed disorder and the stability of the elusive biaxial nematic state [read more]. I have also obtained global phase diagrams for a coupled system of two nematic subsystems to investigate the enhancement of the nematic phase in suspensions of ferroelectric nanoparticles embedded in a nematic environment [read more]. Finally, I have extended our understanding of the phase behavior of liquid crystal elastomers --- hybrids of nematics and rubber --- to describe their peculiar critical properties [read more] and soft behavior [read more]. Figure 6 shows a phase diagram for a model of nematic elastomers subject to uniaxial stress [read more] that captures the existence of a biaxial nematic state, clarifies the mapping between strain and magnetic fields, and displays nontrivial critical behavior with critical, triple and tricritical points.

Figure 6: Elementary models of liquid-crystal elastomers display rich criticality: Phase diagram in terms of temperature and engineering stress showing two paranematic phases and one nematic biaxial phase, first and second-order transition lines and a triple point, a tricritical point and a critical endpoint.

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