Person:

Brenner, Michael

Numerical modeling of global atmospheric chemical dynamics presents an enormous challenge, associated with simulating hundreds of chemical species with time scales varying from milliseconds to years. Here we present an algorithm that provides a significant reduction in computational cost. Because most of the fast reactants and their quickly decomposing reaction products are localized near emission sources, we use a series of reduced chemical models of decreasing complexity with increasing distance from the source. The algorithm diagnoses the chemical dynamics on-the-run, locally and separately for every species according to its characteristic reaction time. Unlike conventional time-scale separation methods, the spatial reduction algorithm speeds up not only the chemical solver but also advection–diffusion integration. Through several examples we demonstrate that the algorithm can reduce computational cost by at least an order of magnitude for typical atmospheric chemical kinetic mechanisms.

We calculate the ground states of hard-sphere clusters, in which n identical hard spherical particles bind by isotropic short-ranged attraction. Combining graph theoretic enumeration with basic geometry, we analytically solve for clusters of n<or=10 particles satisfying minimal rigidity constraints. For n<or=9 the ground state degeneracy increases exponentially with n, but for n>9 the degeneracy decreases due to the formation of structures with >3n-6 contacts. Interestingly, for n=10 and possibly at n=11 and n=12, the ground states of this system are subsets of hexagonal close-packed crystals. The ground states are not icosahedra at n=12 and n=13. We relate our results to the structure and thermodynamics of suspensions of colloidal particles with short-ranged attractions.

The linear dynamics of ion sputtered solids is essential to understanding the evolution of ordered and disordered surface patterns. We review the existing models of linear dynamics and point out qualitative discrepancies between theory and experimental observations that characterize the linear regime. In particular, we emphasize the importance of experimental and theoretical analysis of bifurcation points: certain values of control parameters such as ion beam angle or energy, where flat surfaces undergo a transition from stability to instability.

A recent paper of B. Naundorf et al. described an intriguing negative correlation between variability of the onset potential at which an action potential occurs (the onset span) and the rapidity of action potential initiation (the onset rapidity). This correlation was demonstrated in numerical simulations of the Hodgkin-Huxley model. Due to this antagonism, it is argued that Hodgkin-Huxley-type models are unable to explain action potential initiation observed in cortical neurons in vivo or in vitro. Here we apply a method from theoretical physics to derive an analytical characterization of this problem. We analytically compute the probability distribution of onset potentials and analytically derive the inverse relationship between onset span and onset rapidity. We find that the relationship between onset span and onset rapidity depends on the level of synaptic background activity. Hence we are able to elucidate the regions of parameter space for which the Hodgkin-Huxley model is able to accurately describe the behavior of this system.

The classical theory of ion beam sputtering predicts the instability of a flat surface to uniform ion irradiation at any incidence angle. We relax the assumption of the classical theory that the average surface erosion rate is determined by a Gaussian response function representing the effect of the collision cascade, and consider the surface dynamics for other physically motivated response functions. We show that although instability of flat surfaces at any beam angle results from all Gaussian and a wide class of non-Gaussian erosive response functions, there exist classes of modifications to the response that can have a dramatic effect. In contrast to the classical theory, these types of response render the flat surface linearly stable, while imperceptibly modifying the predicted sputter yield vs incidence angle. We discuss the possibility that such corrections underlie recent reports of a “window of stability” of ion-bombarded surfaces at a range of beam angles for certain ion and surface types, and describe some characteristic aspects of pattern evolution near the transition from unstable to stable dynamics. We point out that careful analysis of the transition regime may provide valuable tests for the consistency of any theory of pattern formation on ion sputtered surfaces.

We study the patterns formed on ion sputtered Si surfaces as a function of ion energy and incidence angle, and identify a region in parameter space where the flat surface is stable. The boundaries between the stable and pattern-forming regions represent mathematical bifurcations. Our data set exhibits at least two different bifurcation types. We discuss the constraints imposed by these observations on the correct model of long wavelength dynamics of ion sputtered surfaces.

Recent work has developed a beautiful model system for studying the energy focusing and heating power of collapsing bubbles. The bubble is effectively one-dimensional and the collapse and heating can be quantitatively measured. Thermal effects are shown to play an essential role in the time-dependent dynamics.

We study how functional constraints bound and shape evolution through an analysis of mammalian voltage-gated sodium channels. The primary function of sodium channels is to allow the propagation of action potentials. Since Hodgkin and Huxley, mathematical models have suggested that sodium channel properties need to be tightly constrained for an action potential to propagate. There are nine mammalian genes encoding voltage-gated sodium channels, many of which are more than ?90% identical by sequence. This sequence similarity presumably corresponds to similarity of function, consistent with the idea that these properties must be tightly constrained. However, the multiplicity of genes encoding sodium channels raises the question: why are there so many? We demonstrate that the simplest theoretical constraints bounding sodium channel diversity—the requirements of membrane excitability and the uniqueness of the resting potential—act directly on constraining sodium channel properties. We compare the predicted constraints with functional data on mammalian sodium channel properties collected from the literature, including 172 different sets of measurements from 40 publications, wild-type and mutant, under a variety of conditions. The data from all channel types, including mutants, obeys the excitability constraint; on the other hand, channels expressed in muscle tend to obey the constraint of a unique resting potential, while channels expressed in neuronal tissue do not. The excitability properties alone distinguish the nine sodium channels into four different groups that are consistent with phylogenetic analysis. Our calculations suggest interpretations for the functional differences between these groups.

The forcibly launched spores of ascomycete fungi must eject through several millimeters of nearly still air surrounding fruiting bodies to reach dispersive air flows. Because of their microscopic size, spores experience great fluid drag, and although this drag can aid transport by slowing sedimentation out of dispersive air flows, it also causes spores to decelerate rapidly after launch.We hypothesize that spores are shaped to maximize their range in the nearly still air surrounding fruiting bodies. To test this hypothesis we numerically calculate optimal spore shapes—shapes of minimum drag for prescribed volumes—and compare these shapes with real spore shapes taken from a phylogeny of >100 species. Our analysis shows that spores are constrained to remain within 1% of the minimum possible drag for their size. From the spore shapes we predict the speed of spore launch, and confirm this prediction through high-speed imaging of ejection in Neurospora tetrasperma. By reconstructing the evolutionary history of spore shapes within a single ascomycete family we measure the relative contributions of drag minimization and other shape determinants to spore shape evolution. Our study uses biomechanical optimization as an organizing principle for explaining shape in a mega-diverse group of species and provides a framework for future measurements of the forces of selection toward physical optima.

Recent experiments have convincingly demonstrated the existence of surface nanobubbles on submerged hydrophobic surfaces. However, classical theory dictates that small gaseous bubbles quickly dissolve because their large Laplace pressure causes a diffusive outflux of gas. Here we suggest that the bubbles are stabilized by a continuous influx of gas near the contact line, due to the gas attraction towards hydrophobic walls [Dammer and Lohse, Phys. Rev. Lett. 96, 206101 (2006); Zhang et al., Phys. Rev. Lett. 98, 136101 (2007); Mezger et al., J. Chem. Phys. 128, 244705 (2008)]. This influx balances the outflux and allows for a metastable equilibrium, which, however, vanishes in thermodynamic equilibrium. Our theory predicts the equilibrium radius of the surface nanobubbles, as well as the threshold for surface nanobubble formation as a function of hydrophobicity and gas concentration.