Leopoldo R. Gómez

Shocks near Jamming

Nonlinear sound is an extreme phenomenon typically observed in solids after violent explosions. But granular media are different. Right when they jam, these fragile and disordered solids exhibit a vanishing rigidity and sound speed, so that even tiny mechanical perturbations form supersonic shocks. Here, we perform simulations in which two-dimensional jammed granular packings are dynamically compressed and demonstrate that the elementary excitations are strongly nonlinear shocks, rather than ordinary phonons. We capture the full dependence of the shock speed on pressure and impact intensity by a surprisingly simple analytical model.

Uniform shock waves in disordered granular matter

The confining pressure P is perhaps the most important parameter controlling the properties of granular matter. Strongly compressed granular media are, in many respects, simple solids in which elastic perturbations travel as ordinary phonons. However, the speed of sound in granular aggregates continuously decreases as the confining pressure decreases, completely vanishing at the jamming-unjamming transition. This anomalous behavior suggests that the transport of energy at low pressures should not be dominated by phonons. In this work we use simulations and theory to show how the response of granular systems becomes increasingly nonlinear as pressure decreases. In the low-pressure regime the elastic energy is found to be mainly transported through nonlinear waves and shocks. We numerically characterize the propagation speed, shape, and stability of these shocks and model the dependence of the shock speed on pressure and impact intensity by a simple analytical approach.

Phase nucleation in curved space

Nucleation and growth is the dominant relaxation mechanism driving first-order phase transitions. In two-dimensional flat systems, nucleation has been applied to a wide range of problems in physics, chemistry and biology. Here we study nucleation and growth of two-dimensional phases lying on curved surfaces and show that curvature modifies both critical sizes of nuclei and paths towards the equilibrium phase. In curved space, nucleation and growth becomes inherently inhomogeneous and critical nuclei form faster on regions of positive Gaussian curvature. Substrates of varying shape display complex energy landscapes with several geometry-induced local minima, where initially propagating nuclei become stabilized and trapped by the underlying curvature.

Coupling between mean curvature and textures in block copolymer thin films deposited on curved substrates

We study the dynamics of coarsening in a cylinder-forming polystyrene-block-poly(ethylene-alt-propylene) block copolymer thin film deposited on a topographically patterned substrate. Thermal annealing leads to highly ordered arrays of polystyrene cylinders embedded in the poly(ethylene-alt-propylene) matrix, lying in-plane and oriented perpendicular to the trenches of the substrate. We show that this configuration corresponds to an equilibrium state. The coupling between the block copolymer morphology and the mean curvature of the substrate is dictated by the out-of-plane deformations of the block copolymer structure. Thus, with appropriate control over the substrate features, it should be possible to obtain novel structures with controlled orientation.

Crystallization dynamics on curved surfaces

We study the evolution from a liquid to a crystal phase in two-dimensional curved space. At early times, while crystal seeds grow preferentially in regions of low curvature, the lattice frustration produced in regions with high curvature is rapidly relaxed through isolated defects. Further relaxation involves a mechanism of crystal growth and defect annihilation where regions with high curvature act as sinks for the diffusion of domain walls. The pinning of grain boundaries at regions of low curvature leads to the formation of a metastable structure of defects, characterized by asymptotically slow dynamics of ordering and activation energies dictated by the largest curvatures of the system. These glassylike ordering dynamics may completely inhibit the appearance of the ground-state structures.

Amorphous precursors of crystallization during spinodal decomposition.

A general Landaus free energy functional is used to study the dynamics of crystallization during liquid-solid spinodal decomposition (SD). The strong length scale selectivity imposed during the early stage of SD induces the appearance of small precursors for crystallization with icosahedral order. These precursors grow in densely packed clusters of tetrahedra through the addition of new particles. As the average size of the amorphous nuclei becomes large enough to reduce geometric frustration, crystalline particles with a body-centered cubic symmetry heterogeneously nucleate on the growing clusters. The volume fraction of the crystalline phase is strongly dependent on the depth of quench. At deep quenches, the SD mechanism produces amorphous structures arranged in dense polytetrahedral aggregates.

Curvature as a Guiding Field for Patterns in Thin Block Copolymer Films

Experimental data on thin films of cylinder-forming block copolymers (BC)-free-standing BC membranes as well as supported BC films-strongly suggest that the local orientation of the BC patterns is coupled to the geometry in which the patterns are embedded. We analyze this phenomenon using general symmetry considerations and numerical self-consistent field studies of curved BC films in cylindrical geometry. The stability of the films against curvature-induced dewetting is also analyzed. In good agreement with experiments, we find that the BC cylinders tend to align along the direction of curvature at high curvatures. At low curvatures, we identify a transition from perpendicular to parallel alignment in supported films, which is absent in free-standing membranes. Hence both experiments and theory show that curvature can be used to manipulate and align BC patterns.

Thermal properties of vortices on curved surfaces

We use Monte Carlo simulations to study the finite temperature behavior of vortices in the XY model for tangent vector order on curved backgrounds. Contrary to naive expectations, we show that the underlying geometry does not affect the proliferation of vortices with temperature respect to what is observed on a flat surface. Long-range order in these systems is analyzed by using two-point correlation functions. As expected, in the case of slightly curved substrates these correlations behave similarly to the plane. However, for high curvatures, the presence of geometry-induced unbounded vortices at low temperatures produces the rapid decay of correlations and an apparent lack of long-range order. Our results shed light on the finite-temperature physics of soft-matter systems and anisotropic magnets deposited on curved substrates.