Phase transition creates the geometry of the continuum from discrete space.

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may help tame the infinities arising from quantizing gravity, and remove the need for the machinery of the real numbers; a construct with no direct observational support. However, despite many attempts to build discrete space, researchers have failed to produce even the simplest geometries. Here we investigate graphs as the most elementary discrete models of two-dimensional space. We show that if space is discrete, it must be disordered, by proving that all planar lattice graphs exhibit a taxicab metric similar to square grids. We then give an explicit recipe for growing disordered discrete space by sampling a Boltzmann distribution of graphs at low temperature. Finally, we propose three conditions which any discrete model of Euclid's plane must meet: have a Hausdorff dimension of 2, support unique straight lines, and obey Pythagoras' theorem. Our model satisfies all three, resulting in a discrete model in which continuum-like behavior emerges at large lengths.

Flow and evolution of ice-sucrose crystal mushes.

We study the rheology of suspensions of ice crystals at moderate to high volume fractions in a sucrose solution in which they are partially soluble, a model system for a wide class of crystal mushes or slurries. Under step changes in shear rate, the viscosity changes to a relaxed value over several minutes, in a manner well fitted by a single exponential. The behavior of the relaxed viscosity is power-law shear thinning with shear rate, with an exponent of -1.76±0.25, so that shear stress falls with increasing shear rate. On longer time scales, the crystals ripen (leading to a falling viscosity) so that the mean radius increases with time to the power 0.14±0.07. We speculate that this unusually small exponent is due to the interaction of classical ripening dynamics with abrasion or breakup under flow. We compare the rheological behavior to mechanistic models based on flow-induced aggregation and breakup of crystal clusters, finding that the exponents can be predicted from liquid phase sintering and breakup by brittle fracture.

Optimal counter-current exchange networks.

We present a general analysis of exchange devices linking their efficiency to the geometry of the exchange surface and supply network. For certain parameter ranges, we show that the optimal exchanger consists of densely packed pipes which can span a thin sheet of large area (an "active layer"), which may be crumpled into a fractal surface and supplied with a fractal network of pipes. We derive the efficiencies of such exchangers, showing the potential for significant gains compared to regular exchangers (where the active layer is flat), using parameters relevant to biological systems.

Simple heuristic for the viscosity of polydisperse hard spheres.

We build on the work of Mooney [Colloids Sci. 6, 162 (1951)] to obtain an heuristic analytic approximation to the viscosity of a suspension any size distribution of hard spheres in a Newtonian solvent. The result agrees reasonably well with rheological data on monodispserse and bidisperse hard spheres, and also provides an approximation to the random close packing fraction of polydisperse spheres. The implied packing fraction is less accurate than that obtained by Farr and Groot [J. Chem. Phys. 131(24), 244104 (2009)], but has the advantage of being quick and simple to evaluate.

Easily repairable networks: reconnecting nodes after damage.

We introduce a simple class of distribution networks that withstand damage by being repairable instead of redundant. Instead of asking how hard it is to disconnect nodes through damage, we ask how easy it is to reconnect nodes after damage. We prove that optimal networks on regular lattices have an expected cost of reconnection proportional to the lattice length, and that such networks have exactly three levels of structural hierarchy. We extend our results to networks subject to repeated attacks, in which the repairs themselves must be repairable. We find that, in exchange for a modest increase in repair cost, such networks are able to withstand any number of attacks.

Close packing density of polydisperse hard spheres.

The most efficient way to pack equally sized spheres isotropically in three dimensions is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution based upon a mapping onto a one-dimensional problem. To test this theory we performed extensive simulations for mixtures of elastic spheres with hydrodynamic friction. The simulations show a general (but weak) dependence of the final (essentially hard sphere) packing density on fluid viscosity and on particle size but this can be eliminated by choosing a specific relation between mass and particle size, making the random close packed volume fraction well defined. Our theory agrees well with the simulations for bidisperse, tridisperse, and log-normal distributions and correctly reproduces the exact limits for large size ratios.

Sintering behavior of two roughened crystals just after contact.

We consider two spherical, roughened crystals with approximately isotropic surface free energy, which are brought into contact and begin to sinter. We argue that the geometry immediately postcontact is two dimensional and Cartesian and can be approximated by the evolution of a slot-shaped cavity. On this basis, we construct traveling wave solutions for the crystal shape in the limits of bulk- and surface-diffusion-limited kinetics. These solutions are then used to calculate scalings for the neck size as a function of time t after contact: We predict that neck size is proportional to t 1/4 for the bulk-diffusion-limited case and (following a single pinch-off event) approximately proportional to t 1/3 for the surface-diffusion-limited case.

Fractal design for efficient brittle plates under gentle pressure loading.

We consider a plate made from an isotropic but brittle elastic material, which is used to span a rigid aperture, across which a small pressure difference is applied. The problem we address is to find the structure which uses the least amount of material without breaking. Under a simple set of physical approximations and for a certain region of the pressure-brittleness parameter space, we find that a fractal structure in which the plate consists of thicker spars supporting thinner spars in an hierarchical arrangement gives a design of high mechanical efficiency.

Fractal design for an efficient shell strut under gentle compressive loading.

Because of Euler buckling, a simple strut of length L and Young modulus Y requires a volume of material proportional to L3f12} in order to support a compressive force F, where f=F/YL2 and f<<1. By taking into account both Euler and local buckling, we provide a hierarchical design for such a strut consisting of intersecting curved shells, which requires a volume of material proportional to the much smaller quantity L3f exp[2 square root(ln 3)(ln f-1)].