Bounds for systems of coupled advection-diffusion equations with application to the Poisson-Nernst-Plank equations.

We consider coupled systems of advection-diffusion equations with initial and boundary conditions and determine conditions on the advection terms that allow us to obtain solutions that can be explicitly bounded above and below using the initial and boundary conditions. Given the advection terms, using our methodology one can easily check if such bounds can be obtained and then one can construct the necessary nonlinear transformation to allow the bounds to be determined. We apply this technique to determine bounding quantities for a number of examples. In particular, we show that the three-ion electroneutral Poisson-Nernst-Planck system of equations can be transformed into a system, which allows for the use of our techniques and we determine the bounding quantities. In addition, we determine the general form of advection terms that allow these techniques to be applied and show that our method can be applied to a very wide class of advection-diffusion equations.

A continuum study of layer analysis for single species ion transport inside double-layered graphene sheets with various separations.

We investigate the formation of thin ionic layers driven by electro-osmotic forces, that are commonly found in micro- and nano-channels. Recently, multi-layers have been reported in the literature. However, the relation between classical Debye layers and multi-layers, which is a practically and fundamentally important question, was previously unexplained. Here, we fill this gap by using a continuum approach to investigate the flow of lithium ions inside double-layered graphene sheets. Fluid flow, charge conductivity and thermal stability will be investigated. We show that the separation and strength of forces between the sheets, the external electric field and thermal effects determine the topology of the ionic layers between the graphene sheets.

Dual-Gated Transistor Platform for On-Site Detection of Lead Ions at Trace Levels.

On-site monitoring of heavy metals in drinking water has become crucial because of several high profile instances of contamination. Presently, reliable techniques for trace level heavy metal detection are mostly laboratory based, while the detection limits of contemporary field-based methods are barely meeting the exposure limits set by regulatory bodies such as the World Health Organization (WHO). Here, we show an on-site deployable, Pb2+ sensor on a dual-gated transistor platform whose lower detection limit is 2 orders of magnitude better than the traditional sensor and 1 order of magnitude lower than the exposure limit set by WHO. The enhanced sensitivity of our design is verified by numerically solving PNP (Planck-Nernst-Poisson) model. We demonstrate that the enhanced sensitivity is due to the suppression of ionic flux. The simplicity and the robustness of the design make it applicable for on-site screening, thereby facilitating rapid response to contamination events.

Modelling of particle-laden flow inside nanomaterials.

In this paper, we demonstrate the usage of the Nernst-Planck equation in conjunction with mean-field theory to investigate particle-laden flow inside nanomaterials. Most theoretical studies in molecular encapsulation at the nanoscale do not take into account any macroscopic flow fields that are crucial in squeezing molecules into nanostructures. Here, a multi-scale idea is used to address this issue. The macroscopic transport of gas is described by the Nernst-Planck equation, whereas molecular interactions between gases and between the gas and the host material are described using a combination of molecular dynamics simulation and mean-field theory. In particular, we investigate flow-driven hydrogen storage inside doubly layered graphene sheets and graphene-oxide frameworks (GOFs). At room temperature and with slow velocity fields, we find that a single molecular layer is formed almost instantaneously on the inner surface of the graphene sheets, while molecular ligands between GOFs induce multi-layers. For higher velocities, multi-layers are also formed between graphene. For even larger velocities, the cavity of graphene is filled entirely with hydrogen, whereas for GOFs there exist two voids inside each periodic unit. The flow-driven hydrogen storage inside GOFs with various ligand densities is also investigated.

Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems.

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Our methodology thus provides a standard procedure for solving the PNP system and we illustrate this by solving some practical examples. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.

A mathematical model of the metabolic and perfusion effects on cortical spreading depression.

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism-aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium-ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD--vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization--found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.

Mathematical approaches to modeling of cortical spreading depression.

Migraine with aura (MwA) is a debilitating disease that afflicts about 25%-30% of migraine sufferers. During MwA, a visual illusion propagates in the visual field, then disappears, and is followed by a sustained headache. MwA was conjectured by Lashley to be related to some neurological phenomenon. A few years later, Leão observed electrophysiological waves in the brain that are now known as cortical spreading depression (CSD). CSD waves were soon conjectured to be the neurological phenomenon underlying MwA that had been suggested by Lashley. However, the confirmation of the link between MwA and CSD was not made until 2001 by Hadjikhani et al. [Proc. Natl. Acad. Sci. U.S.A. 98, 4687-4692 (2001)] using functional MRI techniques. Despite the fact that CSD has been studied continuously since its discovery in 1944, our detailed understandings of the interactions between the mechanisms underlying CSD waves have remained elusive. The connection between MwA and CSD makes the understanding of CSD even more compelling and urgent. In addition to all of the information gleaned from the many experimental studies on CSD since its discovery, mathematical modeling studies provide a general and in some sense more precise alternative method for exploring a variety of mechanisms, which may be important to develop a comprehensive picture of the diverse mechanisms leading to CSD wave instigation and propagation. Some of the mechanisms that are believed to be important include ion diffusion, membrane ionic currents, osmotic effects, spatial buffering, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. Discrete and continuum models of CSD consist of coupled nonlinear differential equations for the ion concentrations. In this review of the current quantitative understanding of CSD, we focus on these modeling paradigms and various mechanisms that are felt to be important for CSD.

Periodic orbits of inelastic particles on a ring.

We consider the dynamics of N rigid particles of arbitrary mass that are constrained to move on a frictionless ring. Collisions between particles are inelastic with a constant coefficient of restitution e, and between collisions the particles move with constant velocity. We study sequences of collisions that are self-similar in the sense that the relative positions return to their original relative positions after the collision sequence while the relative velocities are reduced by a constant factor. For a given collision sequence, we develop the analytic machinery to determine the particle velocities and the locations of collisions, and we show that the problem of determining self-similar orbits reduces to solving an eigenvalue problem to obtain the particle velocities and solving a linear system to obtain the locations of interparticle collisions. For inelastic systems, we show that the collision locations can always be uniquely determined. We also show that this is in sharp contrast to the case of elastic systems in which infinite families of self-similar orbits can coexist.

Critical role of friction for a single particle falling through a funnel.

We investigate a single frictional, inelastic, spherical particle falling under gravity through a symmetric funnel. A recent study showed that, for a frictionless particle in such a system, several anomalous phenomena occur: The particle can stay longer, lose more energy, and exert more impulsive force in a funnel with steeper walls. For frictionless particles, such phenomena exist for many small ranges of funnel angles and are a consequence of the many possible repeated patterns in particle trajectories. However, in reality, friction always exists and it is a natural question whether the anomalous phenomena still exist for frictional particles in such systems. We show that, surprisingly, the inclusion of friction in the dynamics actually dramatically enhances the anomalous phenomena. For frictional particles, the anomalous phenomena exist for all funnel angles steeper than 45^{°} and are thus more robust than the frictionless case. Furthermore, instead of many possible complicated repeated patterns in particle trajectories, there is a unique repeated pattern for frictional particles. Moreover, this is the simplest possible repeated pattern. We derive an analytical expression for this unique repeated pattern and provide a theoretical explanation for the anomalous phenomena observed in frictional particle systems. We further show that the friction, no matter how small, plays a critical role in the dynamics, that is, the dynamics of the frictionless particle system is singular.

Restricted diffusion in cellular media: (1+1)-dimensional model.

We consider the diffusion of molecules in a one-dimensional medium consisting of a large number of cells separated from the extra-cellular space by permeable membranes. The extra-cellular space is completely connected and allows unrestricted diffusion of the molecules. Furthermore, the molecules can diffuse within a given cell, i.e., the intra-cellular space; however, direct diffusion from one cell to another cell cannot occur. There is a movement of molecules across the permeable membranes between the intra- and extra-cellular spaces. Molecules from one cell can cross the permeable membrane into the extra-cellular space, then diffuse through the extra-cellular space, and eventually enter the intra-cellular space of a second cell. Here, we develop a simple set of model equations to describe this phenomenon and obtain the solutions using an eigenfunction expansion. We show that the solutions obtained using this method are particularly convenient for interpreting data from experiments that use techniques from nuclear magnetic resonance imaging.

Degenerate orbit transitions in a one-dimensional inelastic particle system.

Continuous transitions between different periodic orbits in a one-dimensional inelastic particle system are investigated. We show that continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. We give a full mechanical description of the system and explain why this is the case. Surprisingly, we also show that there are an infinite set of degenerate transitions of co-dimension 1. We provide a theoretical analysis that gives a simple criteria to classify which transitions are degenerate purely using the discrete set of collisions that occur in the orbits. Our analysis allows us to understand the nature of the degeneracy. We also show that higher degrees of degeneracy can occur, and provide an explanation.

Deflection of a dilute stream of particles.

We consider a two-dimensional system in which a dilute stream of particles collides with an oblique planar wall. Both collisions between particles and collisions between particles and the wall are inelastic. We perform numerical simulations in two dimensions and show that the mean force experienced by the wall can be a nonmonotonic function of the angle between the wall and the particle stream. We show that this occurs because particles that rebound from the wall can collide with incoming particles and be scattered. This kind of particle-particle collision can reduce the force experienced by the wall. We refer to this effect as shielding. Furthermore, we show that the force experienced by the wall may be an increasing, decreasing or nonmonotonic function of the restitution coefficient in particle-particle collisions. We derive an exact solution for the mean force on the wall if the system is dilute, and the theoretical prediction is found to be in good agreement with our numerical results. The theory allows us to explicitly quantify the effects of shielding, and thus to explain a number of interesting features. The theory generally provides a useful upper bound for the mean force.

Driven inelastic-particle systems with drag.

We study steady states of the motion of a large number of particles in a closed box that are excited by a vibrating boundary and experience a linear drag force from the interstitial fluid. The dissipation in such systems arises from two main sources: Inelasticity in particle collisions and the effects of interstitial fluid on the particles. In many applications, order of magnitude estimates suggest that the dissipation due to interstitial fluid effects may greatly exceed that due to inelasticity and one is naturally led to neglect inelastic effects. In this study, we show that, if one adopts a linear drag force and inelastic effects are neglected, a steady state only exists when the vibration speed of the boundary is below a critical value. For vibration speeds above this critical value, no steady state exists since the kinetic energy of the particles grows without bound. We show that, for vibration speeds above the critical value, inelastic effects must be included to obtain a steady state even if order of magnitude estimates suggest they are negligible. Numerical simulations confirm these theoretical predictions. We also show that inclusion of apparently small nonlinear drag terms can also play a similar role in preventing the kinetic energy of the particles growing without bound.

Anomalous behavior of a single particle falling through a funnel.

We show several surprising phenomena that occur in an extremely simple system of a single frictionless, inelastic, spherical particle falling under gravity through a symmetric funnel. One might naively expect that particles would fall through funnels with steeper sides more quickly, exert a smaller total impulse on the funnel walls, and lose less energy. However, we show that there are special ranges of angles of the funnel walls for which exactly the opposite occurs. Typically, the particle will experience a sequence of collisions that is highly sensitive to the location at which it enters the funnel and nearby particle trajectories become widely dispersed. However, in the special angular ranges this is not the case and the particle can experience sequences of collisions that have a highly coherent structure. We provide a theoretical analysis that can predict and explain this surprising behavior.

Traveling waves in coupled reaction-diffusion models with degenerate sources.

We consider a general system of coupled nonlinear diffusion equations that are characterized by having degenerate source terms and thereby not having isolated rest states. Using a general form of physically relevant source terms, we derive conditions that are required to trigger traveling waves when a stable uniform steady-state solution is perturbed by a highly localized disturbance. We show that the degeneracy in the source terms implies that traveling waves have a number of surprising properties that are not present for systems with nondegenerate source terms. We also show that such systems can lead to a pair of waves that initially propagate outwards from the disturbance, slow down, and reverse direction before ultimately colliding and annihilating each other.

Anomalous Richtmyer-Meshkov fingering in dissipative particle systems.

We study the fingering instability induced by a shock that propagates across a perturbed interface that separates two types of discrete particles. If collisions between particles conserve energy, then the relative sizes and growth rates of the fingers are similar to those in the analogous shock-induced fingering instability in fluids. However, we show that energy loss during particle collisions, even when very small, causes the qualitative features of the finger growth to be completely opposite to the fluid case. The fingers formed by light particles grow faster and become longer and narrower than the fingers formed by heavy particles. In addition, the finger composed of light particles collapses into an extremely compact, tortuous filament, and diffusive mixing between particle types at the particle scale is heavily suppressed.

Collapse of periodic orbits in a driven inelastic particle system.

The dynamical behavior of a one-dimensional inelastic particle system with particles of unequal mass traveling between two walls is investigated. The system is driven by adding energy at one of the walls while the other wall is stationary and does not add energy. By deriving analytic solutions for the periodic orbits of this system, we show that there are a countable infinity of critical mass ratios at which the particle dynamics become highly degenerate in the following sense. As the mass ratio passes through these critical points, large numbers of stable periodic orbits can collapse onto a single trivial orbit. We show that the widely studied equal-mass systems represent one of these critical points and are therefore such a degenerate case. We also show that in the elastic limit the number of orbits that collapse onto the single trivial orbit can become arbitrarily large.