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AuthorChapman, S. Jonathan (5)Erban, Radek (4)Breward, C. J. W. (3)Howell, P. D. (3)Goriely, Alain (2)View MoreJournal

SIAM Journal on Applied Mathematics (14)

KAUST Grant Number
KUK-C1-013-04 (14)

PublisherSociety for Industrial & Applied Mathematics (SIAM) (14)SubjectAsymptotic analysis (5)Approximation of sums (2)Brownian dynamics (2)Discrete-to-continuum (2)Drift diffusion (2)View MoreTypeArticle (14)Year (Issue Date)2015 (2)2014 (2)2013 (4)2012 (2)2011 (2)View MoreItem AvailabilityMetadata Only (14)

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The Mechanics of a Chain or Ring of Spherical Magnets

Hall, Cameron L.; Vella, Dominic; Goriely, Alain (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2013-01) [Article]

Strong magnets, such as neodymium-iron-boron magnets, are increasingly being manufactured as spheres. Because of their dipolar characters, these spheres can easily be arranged into long chains that exhibit mechanical properties reminiscent of elastic strings or rods. While simple formulations exist for the energy of a deformed elastic rod, it is not clear whether or not they are also appropriate for a chain of spherical magnets. In this paper, we use discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres. We find that the mechanical properties of a chain of magnets differ significantly from those of an elastic rod: while both magnetic chains and elastic rods support bending by change of local curvature, nonlocal interaction terms also appear in the energy formulation for a magnetic chain. This continuum model for the energy of a chain of magnets is used to analyze small deformations of a circular ring of magnets and hence obtain theoretical predictions for the vibrational modes of a circular ring of magnets. Surprisingly, despite the contribution of nonlocal energy terms, we find that the vibrations of a circular ring of magnets are governed by the same equation that governs the vibrations of a circular elastic ring. Copyright © by SIAM.

Controlled Topological Transitions in Thin-Film Phase Separation

Hennessy, Matthew G.; Burlakov, Victor M.; Goriely, Alain; Wagner, Barbara; Münch, Andreas (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2015-01) [Article]

© 2015 Society for Industrial and Applied Mathematics. In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. By bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls toward each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model. Using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that this newly created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

Frost Heave in Colloidal Soils

Peppin, Stephen; Majumdar, Apala; Style, Robert; Sander, Graham (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2011-01) [Article]

We develop a mathematical model of frost heave in colloidal soils. The theory accountsfor heave and consolidation while not requiring a frozen fringe assumption. Two solidificationregimes occur: a compaction regime in which the soil consolidates to accommodate the ice lenses, and a heave regime during which liquid is sucked into the consolidated soil from an external reservoir, and the added volume causes the soil to heave. The ice fraction is found to vary inversely with thefreezing velocity V , while the rate of heave is independent of V , consistent with field and laboratoryobservations. © 2011 Society for Industrial and Applied Mathematics.

An Asymptotic Theory for the Re-Equilibration of a Micellar Surfactant Solution

Griffiths, I. M.; Bain, C. D.; Breward, C. J. W.; Chapman, S. J.; Howell, P. D.; Waters, S. L. (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2012-01) [Article]

Micellar surfactant solutions are characterized by a distribution of aggregates made up predominantly of premicellar aggregates (monomers, dimers, trimers, etc.) and a region of proper micelles close to the peak aggregation number, connected by an intermediate region containing a very low concentration of aggregates. Such a distribution gives rise to a distinct two-timescale reequilibration following a system dilution, known as the t1 and t2 processes, whose dynamics may be described by the Becker-Döring equations. We use a continuum version of these equations to develop a reduced asymptotic description that elucidates the behavior during each of these processes.© 2012 Society for Industrial and Applied Mathematics.

A Model for the Operation of Perovskite Based Hybrid Solar Cells: Formulation, Analysis, and Comparison to Experiment

Foster, J. M.; Snaith, H. J.; Leijtens, T.; Richardson, G. (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2014-01) [Article]

This work is concerned with the modeling of perovskite based hybrid solar cells formed by sandwiching a slab of organic lead halide perovskite (CH3NH3PbI3-xClx) photo-absorber between (n-type) acceptor and (p-type) donor materialstypically titanium dioxide and spiro. A model for the electrical behavior of these cells is formulated based on drift-diffusion equations for the motion of the charge carriers and Poisson's equation for the electric potential. It is closed by (i) internal interface conditions accounting for charge recombination/generation and jumps in charge carrier densities arising from differences in the electron affinity/ionization potential between the materials and (ii) ohmic boundary conditions on the contacts. The model is analyzed by using a combination of asymptotic and numerical techniques. This leads to an approximateyet highly accurateexpression for the current-voltage relationship as a function of the solar induced photocurrent. In addition, we show that this approximate current-voltage relation can be interpreted as an equivalent circuit model consisting of three diodes, a resistor, and a current source. For sufficiently small biases the device's behavior is diodic and the current is limited by the recombination at the internal interfaces, whereas for sufficiently large biases the device acts like a resistor and the current is dictated by the ohmic dissipation in the acceptor and donor. The results of the model are also compared to experimental current-voltage curves, and good agreement is shown.

Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory

Hall, Cameron L.; Chapman, S. Jonathan; Ockendon, John R. (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2010-01) [Article]

The system of algebraic equations given by σn j=0, j≠=i sgn(xi-xj )|xi-xj|a = 1, i = 1, 2, ⋯ , n, x0 = 0, appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole. We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n→∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment, but, up to corrections of logarithmic order, it also leads to a differential equation. The continuum approximation is valid only for i neither too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem. © 2010 Society for Industrial and Applied Mathematics.

Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

Erban, Radek; Chapman, S. Jonathan; Kevrekidis, Ioannis G.; Vejchodský, Tomáš (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2009-01) [Article]

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. © 2009 Society for Industrial and Applied Mathematics.

Analysis of Brownian Dynamics Simulations of Reversible Bimolecular Reactions

Lipková, Jana; Zygalakis, Konstantinos C.; Chapman, S. Jonathan; Erban, Radek (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2011-01) [Article]

A class of Brownian dynamics algorithms for stochastic reaction-diffusion models which include reversible bimolecular reactions is presented and analyzed. The method is a generalization of the λ-bcȳ model for irreversible bimolecular reactions which was introduced in [R. Erban and S. J. Chapman, Phys. Biol., 6(2009), 046001]. The formulae relating the experimentally measurable quantities (reaction rate constants and diffusion constants) with the algorithm parameters are derived. The probability of geminate recombination is also investigated. © 2011 Society for Industrial and Applied Mathematics.

Mathematical Modelling of Surfactant Self-assembly at Interfaces

Morgan, C. E.; Breward, C. J. W.; Griffiths, I. M.; Howell, P. D. (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2015-01) [Article]

© 2015 Society for Industrial and Applied Mathematics. We present a mathematical model to describe the distribution of surfactant pairs in a multilayer structure beneath an adsorbed monolayer. A mesoscopic model comprising a set of ordinary differential equations that couple the rearrangement of surfactant within the multilayer to the surface adsorption kinetics is first derived. This model is then extended to the macroscopic scale by taking the continuum limit that exploits the typically large number of surfactant layers, which results in a novel third-order partial differential equation. The model is generalized to allow for the presence of two adsorbing boundaries, which results in an implicit free-boundary problem. The system predicts physically observed features in multilayer systems such as the initial formation of smaller lamellar structures and the typical number of layers that form in equilibrium.

Asymptotic Solution of a Model for Bilayer Organic Diodes and Solar Cells

Richardson, Giles; Please, Colin; Foster, Jamie; Kirkpatrick, James (SIAM Journal on Applied Mathematics, Society for Industrial & Applied Mathematics (SIAM), 2012-11-15) [Article]

Organic diodes and solar cells are constructed by placing together two organic semiconducting materials with dissimilar electron affinities and ionization potentials. The electrical behavior of such devices has been successfully modeled numerically using conventional drift diffusion together with recombination (which is usually assumed to be bimolecular) and thermal generation. Here a particular model is considered and the dark current-voltage curve and the spatial structure of the solution across the device is extracted analytically using asymptotic methods. We concentrate on the case of Shockley-Read-Hall recombination but note the extension to other recombination mechanisms. We find that there are three regimes of behavior, dependent on the total current. For small currents-i.e., at reverse bias or moderate forward bias-the structure of the solution is independent of the total current. For large currents-i.e., at strong forward bias-the current varies linearly with the voltage and is primarily controlled by drift of charges in the organic layers. There is then a narrow range of currents where the behavior undergoes a transition between the two regimes. The magnitude of the parameter that quantifies the interfacial recombination rate is critical in determining where the transition occurs. The extension of the theory to organic solar cells generating current under illumination is discussed as is the analogous current-voltage curves derived where the photo current is small. Finally, by comparing the analytic results to real experimental data, we show how the model parameters can be extracted from the shape of current-voltage curves measured in the dark. © 2012 Society for Industrial and Applied Mathematics.

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