Understanding how to make materials with features on the micro- or nano-scale is increasingly important, with applications ranging from drug delivery, energy storage, and meta-materials, to understanding biological processes such as protein folding and viral capsid assembly. Because directly manufacturing small structures is expensive and inefficient, there is a need for techniques stemming from self-organizing principles, whereby macroscale or disordered systems dynamically evolve to complex structures on the small scale. Inventing such techniques requires input from applied mathematics in several ways: to understand the physics at these scales, characterize the qualitative dynamics that it gives rise to, and formulate a quantitative theory that can be used to predict the complex structures arising from a given set of initial conditions, and then to answer the inverse problem of which ingredients to start with so a system evolves to a given target structure. Below I describe a collection of projects with these aims.
Colloidal energy landscape and kinetics
Colloids are particles with diameters of nano- to micro-metres that typically live in a solution where they diffuse as Brownian particles and interact in various ways. They are a promising system to study for self-assembly, because experimentalists have recently gained a wealth of control over features such as shape, size, charge, specificity, and valency of the particles, so an enormous variety of structures can potentially be assembled. For example, by coating different particles with complementary strands of DNA, one can in principle program exactly the set of interactions that one wants. Finding ways to explore the parameter space of possibilities is an exciting arena where applied mathematics and computation can help to guide experiments and to predict bounds on what can possibly be assembled.
One major feature of many colloids (and other mesoscale particles) is that their interactions are very short-ranged compared to their diameters. This means that traditional concepts such as local minima and transition states are no longer adequate to characterize their dynamics. We proposed a new framework for investigating such questions, based on the limit of short-ranged, strong interactions. In this framework, the free energy landscape is given by a set of geometrical manifolds (shapes of varying dimensions) plus a single control parameter, while the dynamics on the manifolds are given by a hierarchy of Fokker-Planck equations coupled by "sticky" boundary conditions. This allows the landscape to be completely enumerated and the dynamics to be exactly calculated, and we have verified that it quantitatively describes experiments with colloids. This framework has the potential to be a powerful approach to analyzing mesoscale particle dynamics and my group is developing a set of computational tools to implement it.
Monte Carlo on manifolds
Relatedly, my group and collaborators are developing Monte Carlo methods to sample and integrate functions on manifolds. Current work can do this on a single manifold, defined implicitly by a set of constraints. Our method is simple to implement, requiring only first derivatives of the constraint functions. Future work will consider how to sample a collection of manifolds of different dimensions (a particular type of stratification.) And maybe it will eventually be possible to sample kinetic, path-dependent quantities, using metropolis-type ideas.
Friction and noise
Friction (the kind you feel when you push a box across a floor) is at the heart of much of our lives: it is the reason why we can walk, drive, skate, build sand castles, etc. So is noise: random jiggling of particles at small scales provides the energy needed to overcome energy barriers, so our proteins can fold, our blood can mix, our lungs can expel waste, our salt can form nice crystals, etc. Both friction and noise have been studied for centuries separately, but what happens when a system (say a collection of particles) is both frictional, and noisy? Such systems are rarely been examined, though they are increasingly important, especially in materials science.
My group is interested in the mathematics of how to model systems that are both frictional and noisy. What are the right equations to write down for a collection of particles that interact with friction? And how do you analyze them? These turn out to be very challenging, mathematically subtle questions that bring together some disparate ideas in stochastic analysis, differential geometry, and PDE theory; there is much work still to do.
We are also interested in physical questions involving the collective effects of noisy frictional particles. For example, try this experiment: take a container, put in a few balls (say ball bearings), and swirl the container (translate it around in a circle.) The balls will move around the container in the same direction as your swirling. Now add more balls, so the density is quite high, and repeat. The balls will now move in the opposite direction as the swirling. Why?
The answer will be forthcoming, and shows that this is a richer problem than one might think....
How many ways are there to arrange N hard spheres in space, so they form a rigid cluster? This is a mathematical question, but the solution could bring insight to a range of disciplines, including what structures colloids can assemble into and how they might do it. It is also a challenging computational problem, that brings up interesting issues in rigidity theory and numerical algebraic geometry.
Current work includes extending our current algorithm to handle more general packings of objects, and developing robust numerical tools to test for geometrical rigidity.
Please see the sphere packing data page for more information and the most current datasets.
When the interactions between colloidal particles are irreversible, equilibrium theories do not apply because the particles cannot explore their configuration space. However, it is still possible to control such systems: we have shown that with only two types of particles it is possible to create tetramers with 100% yield. This surprising result came by analysing a random parking model (to be distinguished from random packing), and was verified experimentally, theoretically, and in simulations. Recently, with Beth Chen we have considered random parking on other curved spaces: the sphere, plane, hyperbolic plane, and projective plane, and shown some counterintuitive properties.
One promising method of creating structures on surfaces is to bombard them with ions. The ions transmit energy to the surface, causing the atoms to locally redistribute, or if they have enough energy, to sputter away, creating a small crater near the impact location of each ion. The average effect over many ion impacts is to cause the surface height to erode. Because it erodes with different rates depending on the slope of the surface, it can develop intricate patterns depending on the initial shape of the surface and the parameters involved in ion bombardment. These patterns can have features at a very small scale, so if we understand how they arise we can learn how to exploit them to create desired target patterns. Uniform ion bombardment is cheap compared to other methods of nanofabrication, so this is a promising avenue to pursue.
My aim is to further a theoretical understanding of the structures that can be formed, in particular by the nonlinear dynamics. Along with experimental and numerical collaborators, we have shown have the striking property that steep, sharp features can arise spontaneously on a surface provided it is initially gently patterned on the macroscale. These features have length scales on the order of nanometers, much smaller than anything realizable by linear instabilities. They can be fully understood by a discrete set of traveling-wave solutions to the PDE governing the macroscale dynamics, that are special in that they are undercompressive (they violate the Lax entropy condition). Because of this they act as attractors for the dynamics, and allow for low-dimensional parameterizations of the nonlinear dynamics. This parameterization allows one to characterize the nonlinear dynamics using only 3 material-dependent parameters, that are much easier to measure experimentally than the full erosion function and smoothing physics. Additionally, we have used it to develop a computationally efficient method for solving the inverse problem of finding the initial condition whose evolution leads to a desired target pattern.
The ocean is filled with dynamics at a multitude of scales that are intertwined with no sharp separations, so that understanding the dynamics at one scale requires understanding it at many. Because it is not possible to conceptualize nor to numerically resolve all of the scales simultaneously, there is a need to understand the processes and mechanisms acting at each scale, so they can be modelled or parameterized at others. My interest is in small-scale processes: in how these affect and can be parameterized at the larger scales, in mechanisms for energy transfer to and from the smaller scales, and in processes that depend on the small scales themselves. Because these scales are so complex and ubiquitous, it is natural to model them stochastically; I outline below two of areas of research with this aim.
As the tides slosh back and forth in the ocean's irregular basins, they generate energetic internal motions that cascade to smaller scales and eventually dissipate. The mixing that results is critical to ocean circulation patterns, providing roughly half of the amount required to maintain the global abyssal density distribution, so understanding the mechanisms and geography of the different steps in the tidal energy cascade is fundamental to our overall understanding of ocean dynamics.
One question is in the role that topography can play in transferring energy to small scales, and in particular in the small-scale topography the baroclinic waves encounter away from the large features where they were generated. We have looked at tidal scattering by small-scale random topography, and found two things of note: (i) a formula for the scattering rate that can be used to parameterize the small-scale bottom topography as a form of friction; and (ii) random topography focuses the internal wave energy at localized regions in space — a surprising qualitative feature that suggests an alternative explanation for observations of localized internal wave activity. This study was limited to one-dimensional subcritical topography, however, and it remains an interesting question what happens when one allows for two-dimensional and/or supercritical topography — in the latter case the equations then allow for the presence of wave attractors that may dramatically change the calculations.
- Buhler, O., Holmes-Cerfon, M. Decay of an internal tide due to random topography in the ocean. Journal of Fluid Mechanics, 678 (2011): 271-293.
Mixing by internal waves
The ocean is filled with an energetic background of small-scale motions called internal waves that are too small to be resolved in basin-scale ocean models, so their effects on large-scale circulation patterns must be parameterized. It is natural to take a stochastic approach, by modelling the internal waves as a random wave field, and to consider how nonlinear wave-wave or wave-vortical interactions can contribute to mean dynamics at larger scales. One possibility is in mixing at horizontal scales of 1--10km, over time scales of days to weeks — measurements show that the mixing rates are much higher than any explainable by known mechanisms.
We considered the role that internal waves might play in mixing at these scales, by calculating the contribution from the nonlinear wave-wave interactions in both the shallow-water and Boussinesq equations. This yielded an expression that we can apply to any Gaussian random wave field in these systems. However, when we applied this formula to oceanic conditions described by a typical model wave spectrum, it gave values well below those observed in measurements. Subsequent numerical experiments and theoretical analyses suggested that the nonlinear effects are actually negligible as soon as a small amount of viscosity is added to the system; this singular perturbation causes the linear wave field to do most of the mixing. More work is needed to determine the appropriate dynamical and statistical model to properly evaluate the mixing due to internal waves in the ocean.
- Buhler, O., Grisouard, N., Holmes-Cerfon, M. Strong particle dispersion by weakly dissipative random internal waves. Journal of Fluid Mechanics, 719 (2013) doi: 10.1017/jfm.2013.71
- Holmes-Cerfon, M., Buhler, O., Ferrari, R. Particle dispersion by random waves in the rotating Boussinesq system. Journal of Fluid Mechanics, 670 (2011): 150-175.
- Buhler, O., Holmes-Cerfon, M. Particle dispersion by random waves in rotating shallow water. Journal of Fluid Mechanics, 638 (2009): 5-26.
MORE FLUID DYNAMICS
Length of a lava tube
Lava tubes are tunnels that form during a volcanic eruption to carry hot liquid lava away from the source. They can be extremely long -- the Mauna Loa flow tube in Hawaii is over 50 km long, while the Toomba and Undara flows in Queensland, Australia are 123km and 160km long respectively. The longest known tubes are found on Mars and are over 200km. What sets the length of a lava tube? Could they be arbitrarily long, if there were no oceans or other geometrical obstructions? Or are there physical constraints that govern their lengths? We analyzed an idealized model of a lava tube, and found that the length is very sensitive to conditions at the entrance: if the source has constant flux, the tube can be arbitrarily long, but if the source is at constant pressure, then there is a maximum length that we can calculate exactly. Experiments with wax have verified our model.
- Holmes-Cerfon, M., Whitehead, J. Instability and freezing in a solidifying melt conduit. Physica D: Nonlinear Phenomena, 240 (2011), 131-139.