Obstructed swelling and fracture of hydrogels.

Obstructions influence the growth and expansion of bodies in a wide range of settings-but isolating and understanding their impact can be difficult in complex environments. Here, we study obstructed growth/expansion in a model system accessible to experiments, simulations, and theory: hydrogels swelling around fixed cylindrical obstacles with varying geometries. When the obstacles are large and widely-spaced, hydrogels swell around them and remain intact. In contrast, our experiments reveal that when the obstacles are narrow and closely-spaced, hydrogels fracture as they swell. We use finite element simulations to map the magnitude and spatial distribution of stresses that build up during swelling at equilibrium in a 2D model, providing a route toward predicting when this phenomenon of self-fracturing is likely to arise. Applying lessons from indentation theory, poroelasticity, and nonlinear continuum mechanics, we also develop a theoretical framework for understanding how the maximum principal tensile and compressive stresses that develop during swelling are controlled by obstacle geometry and material parameters. These results thus help to shed light on the mechanical principles underlying growth/expansion in environments with obstructions.

Geometric frustration of hard-disk packings on cones.

Conical surfaces pose an interesting challenge to crystal growth: A crystal growing on a cone can wrap around and meet itself at different radii. We use a disk-packing algorithm to investigate how this closure constraint can geometrically frustrate the growth of single crystals on cones with small opening angles. By varying the crystal seed orientation and cone angle, we find that-except at special commensurate cone angles-crystals typically form a seam that runs along the axial direction of the cone, while near the tip, a disordered particle packing forms. We show that the onset of disorder results from a finite-size effect that depends strongly on the circumference and not on the seed orientation or cone angle. This finite-size effect occurs also on cylinders, and we present evidence that on both cylinders and cones, the defect density increases exponentially as circumference decreases. We introduce a simple model for particle attachment at the seam that explains the dependence on the circumference. Our findings suggest that the growth of single crystals can become frustrated even very far from the tip when the cone has a small opening angle. These results may provide insights into the observed geometry of conical crystals in biological and materials applications.

Anomalous Thermal Expansion in Ising-like Puckered Sheets.

Motivated by efforts to create thin nanoscale metamaterials and understand atomically thin binary monolayers, we study the finite temperature statistical mechanics of arrays of bistable buckled dilations embedded in free-standing two-dimensional crystalline membranes that are allowed to fluctuate in three dimensions. The buckled nodes behave like discrete, but highly compressible, Ising spins, leading to a phase transition at T_{c} with singularities in the staggered "magnetization," susceptibility, and specific heat, studied via molecular dynamics simulations. Unlike conventional Ising models, we observe a striking divergence and sign change of the coefficient of thermal expansion near T_{c} caused by the coupling of flexural phonons to the buckled spin texture. We argue that a phenomenological model coupling Ising degrees of freedom to the flexural phonons in a thin elastic sheet can explain this unusual response.

Buckling and metastability in membranes with dilation arrays.

We study periodic arrays of impurities that create localized regions of expansion, embedded in two-dimensional crystalline membranes. These arrays provide a simple elastic model of shape memory. As the size of each dilational impurity increases (or the relative cost of bending to stretching decreases), it becomes energetically favorable for each of the M impurities to buckle up or down into the third dimension, thus allowing for of order 2^{M} metastable surface configurations corresponding to different impurity "spin" configurations. With both discrete simulations and the nonlinear continuum theory of elastic plates, we explore the buckling of both isolated dilations and dilation arrays at zero temperature, guided by analogies with Ising antiferromagnets. We conjecture ground states for systems with triangular and square impurity superlattices, and comment briefly on the possible behaviors at finite temperatures.

Discrete Eulerian model for population genetics and dynamics under flow.

Marine species reproduce and compete while being advected by turbulent flows. It is largely unknown, both theoretically and experimentally, how population dynamics and genetics are changed by the presence of fluid flows. Discrete agent-based simulations in continuous space allow for accurate treatment of advection and number fluctuations, but can be computationally expensive for even modest organism densities. In this report, we propose an algorithm to overcome some of these challenges. We first provide a thorough validation of the algorithm in one and two dimensions without flow. Next, we focus on the case of weakly compressible flows in two dimensions. This models organisms such as phytoplankton living at a specific depth in the three-dimensional, incompressible ocean experiencing upwelling and/or downwelling events. We show that organisms born at sources in a two-dimensional time-independent flow experience an increase in fixation probability.

Fixation probabilities in weakly compressible fluid flows.

Competition between biological species in marine environments is affected by the motion of the surrounding fluid. An effective 2D compressibility can arise, for example, from the convergence and divergence of water masses at the depth at which passively traveling photosynthetic organisms are restricted to live. In this report, we seek to quantitatively study genetics under flow. To this end, we couple an off-lattice agent-based simulation of two populations in 1D to a weakly compressible velocity field-first a sine wave and then a shell model of turbulence. We find for both cases that even in a regime where the overall population structure is approximately unaltered, the flow can significantly diminish the effect of a selective advantage on fixation probabilities. We understand this effect in terms of the enhanced survival of organisms born at sources in the flow and the influence of Fisher genetic waves.

Nanoscale hydrophilicity studies of Gulf parrotfish (Scarus persicus) scales.

The Gulf parrotfish (Scarus persicus) offers inspiration for a strategy to combat marine biofouling, a problem of great economic and environmental interest to the maritime community, through its use of a continually maintained, multifunctional, water-based mucus layer to cover its scales. In this study, to better understand the scale-mucus interface, we investigate the nanoscale hydrophilicity of the fish scales by comparing reconstructed force distance profiles obtained using an amplitude-modulation atomic force microscopy (AM-AFM) technique. We note significant differences between three morphologically distinct regions of each scale, as well as between scales from four spatially distinct regions of the fish. This study reveals a previously unreported property of fish scales and proves the value of a new AFM technique to the field of biomaterials.