Miranda Holmes-Cerfon

A geometrical approach to computing free-energy landscapes from short-ranged potentials

Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free-energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system—equilibrium or nonequilibrium—can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker–Planck equation and show that it becomes restricted to the low-dimensional manifolds connected by “sticky” boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a unique approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.

Enumerating Rigid Sphere Packings

Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behavior at larger ones. In many systems there is no single, optimal packing that dominates, but rather one must understand the entire set of possible packings. As a step in this direction we enumerate rigid clusters of identical hard spheres for

Particle dispersion by random waves in the rotating Boussinesq system

n\leq 14

Low Rossby limiting dynamics for stably stratified flow with finite Froude number

and clusters with the maximum number of contacts for

Sticky-Sphere Clusters

n\leq 19

Particle dispersion by random waves in rotating shallow water

. A rigid cluster is one that cannot be continuously deformed while maintaining all contacts. This is a nonlinear notion that arises naturally because such clusters are the metastable states when the spheres interact with a short-range potential, as is the case in many nano- or microscale systems. We believe that our lists are nearly complete, except for a small number of highly singular clusters (linearly floppy but nonlinearly rigid). The data contains some major geometric...

Development of knife-edge ridges on ion-bombarded surfaces

We present a theoretical and numerical study of horizontal particle dispersion due to random waves in the three-dimensional rotating and stratified Boussinesq system, which serves as a simple model to study the dispersion of tracers in the ocean by the internal wave field. Specifically, the effective one-particle diffusivity in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) is computed for a smallamplitude internal gravity wave field modelled as a stationary homogeneous and horizontally isotropic Gaussian random field whose frequency spectrum is bounded away from zero. Dispersion in this system does not arise simply because of a Stokes drift effect, as in the case of surface gravity waves, but in addition it is driven by the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A formula for the one-particle diffusivity as a function of the spectrum of the random wave field is presented. It is shown that this diffusivity is much smaller than might be expected from heuristic arguments based on the magnitude of the Stokes drift or the pseudomomentum. This appears to stem from certain incompressibility constraints for the Stokes drift and the second-order velocity field. Finally, the theory is applied to oceanic conditions described by a typical model wave spectrum, the Garrett–Munk spectrum, and also by detailed field observations from the North Atlantic tracer release experiment.

Free energy of singular sticky-sphere clusters

In this paper we explore the fast rotation, nonhydrostatic limit of the rotating and stratified Boussinesq equations. We derive new reduced equations for the slow dynamics that describe Taylor-Proudman flows. One new aspect of the dynamics is a decoupling of the horizontal kinetic energy, described by 2D Navier-Stokes, from new dynamics that describe the coupling of vertical kinetic energy and buoyancy. We support the theory with high resolution numerical simulations of the full Boussinesq equations that, in this limit, reveal the spontaneous formation of Taylor-Proudman columns and their dynamics.

Designing steep, sharp patterns on uniformly ion-bombarded surfaces

Nano- and microscale particles, such as colloids, commonly interact over ranges much shorter than their diameters, so it is natural to treat them as “sticky,” interacting only when they touch exactly. The lowest-energy states, free energies, and dynamics of a collection of n particles can be calculated in the sticky limit of a deep, narrow interaction potential. This article surveys the theory of the sticky limit, explains the correspondence between theory and experiments on colloidal clusters, and outlines areas where the sticky limit may bring new insight.

Dynamics and unsteady morphologies at ice interfaces driven by D2O–H2O exchange

We present a theoretical and numerical study of wave-induced particle dispersion due to random waves in the rotating shallow-water system, as part of an ongoing study of particle dispersion in the ocean. Specifically, the e!ective particle di!usivities in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) are computed for a small-amplitude wave field modelled as a stationary homogeneous isotropic Gaussian random field whose frequency spectrum is bounded away from zero. In this case, the leading-order di!usivity depends crucially on the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave‐mean interaction theory. A closed-form analytic expression for the e!ective di!usivity is derived and carefully tested against numerical Monte Carlo simulations. The main conclusions are that Coriolis forces in shallow water invariably decrease the e!ective particle di!usivity and that there is a peculiar choking e!ect for the second-order particle flow in the limit of strong rotation.