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In this work, we analyse a broad class of generalized Poisson–Boltzmann equations and reveal a common mathematical structure. In the limit of a wide electrode, we show that a broad class of generalized Poisson–Boltzmann equations admits a reduction that affords an explicit connection between the functional form of the corresponding free energy and the associated differential capacitance data. We exploit the relation to we show that differential capacitance curves generically undergo an inflection transition with increasing salt concentration, shifting from a local minimum near the point of zero charge for dilute solutions to a local maximum point near the point of zero charge for concentrated solutions. In addition, we develop a robust numerical method for solving generalized Poisson–Boltzmann equations which is easily applicable to the broad class of generalized Poisson–Boltzmann equations with very few code adjustments required for each model [ABSTRACT FROM AUTHOR]

The relationship between the dynamics and structure of amorphous thin films and nanocomposites near their glass transition is an important problem in soft-matter physics. Here, we develop a simple theoretical approach to describe the density profile and the α-relaxation time of a glycerol-silica nanocomposite (S. Cheng et al. , J. Chem. Phys. , 2015, 143 , 194704). We begin by applying the Derjaguin approximation, where we replace the curved surface of the particle with the planar one; thus, modeling the nanocomposite is reduced to that of a confined thin film. Subsequently, by employing the molecular dynamics (MD) simulation data of Cheng et al. , we approximate the density profile of a supported liquid thin film as a stationary solution of a fourth-order partial differential equation (PDE). We then construct an appropriate density functional, from which the density profile emerges through the minimization of free energy. Our final assumption is that of a consistent, temperature-independent scaled density profile, ensuring that the free volume throughout the entire nanocomposite increases with temperature in a smooth, monotonic fashion. Considering the established relationship between glycerol relaxation time and temperature, we can employ Doolittle-type analysis ("naïve" free-volume model), to calculate the relaxation time based on temperature and film thickness. We then convert the film thickness into the interparticle distance and subsequently the filler volume fraction for the nanocomposites and compare our model predictions with experimental data, resulting in a good agreement. The proposed approach can be easily extended to other nanocomposite and film systems.

Adaptive time stepping methods for metastable dynamics of the Allen-Cahn and Cahn-Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error σ in the limit of small length scale parameter ϵ → 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen-Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity., (© The Author(s) 2021.)

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We present the multicomponent functionalized Cahn-Hilliard (mFCH) free energy as a model which allows competition between bilayers with distinct composition and between bilayers and higher codimensional structures, such as co-dimension two filaments and co-dimension three micelles. We construct symmetric and asymmetric homoclinic bilayer profiles via a billiard limit potential and show that co-dimensional bifurcation is driven by the experimentally observed layer-by-layer pearling mechanism. We investigate the stability and slow geometric evolution of multicomponent bilayer interfaces within the context of an [Formula: see text] gradient flow of the mFCH, addressing the impact of aspect ratio of the amphiphile (lipid or copolymer unit) on the intrinsic curvature and the codimensional bifurcation. In particular we derive a Canham-Helfrich sharp interface energy whose intrinsic curvature arises through a Melnikov parameter associated to amphiphile aspect ratio.

We present a microfield approach for studying the dependence of the orientational polarization of the water in aqueous electrolyte solutions upon the salt concentration and temperature. The model takes into account the orientation of the solvent dipoles due to the electric field created by ions, and the effect of thermal fluctuations. The model predicts a dielectric functional dependence of the form ɛ(c)=ɛ_{w}-βL(3αc/β),β=ɛ_{w}-ɛ_{ms}, where L is the Langevin function, c is the salt concentration, ɛ_{w} is the dielectric of pure water, ɛ_{ms} is the dielectric of the electrolyte solution at the molten salt limit, and α is the total excess polarization of the ions. The functional form gives a remarkably accurate description of the dielectric constant for a variety of salts and a wide range of concentrations.

Differential capacitance (DC) data have been widely used to characterize the structure of electrolyte solutions near charged interfaces and as experimental validation of models for electrolyte structure. Fixing a large class of models of electrolyte free energy that incorporate finite-volume effects, a reduction is identified which permits the identification of all free energies within that class that return identical DC data. The result is an interpretation of DC data through the equivalence classes of nonideality terms, and associated boundary layer structures, that cannot be differentiated by DC data. Specifically, for binary salts, DC data, even if measured over a range of ionic concentrations, are unable to distinguish among models which exhibit charge asymmetry, charge reversal, and even ion crowding. The reduction applies to capacitors which are much wider than the associated Debye length and to finite-volume terms that are algebraic in charge density. However, within these restrictions the free energy is shown to be uniquely identified if the DC data are supplemented with measurements of the excess chemical potential of the system in the bulk state.

The harmonic approximation to transition state theory simplifies the problem of calculating a chemical reaction rate to identifying relevant low energy saddle points in a chemical system. Here, we present a saddle point finding method which does not require knowledge of specific product states. In the method, the potential energy landscape is transformed into the square of the gradient, which converts all critical points of the original potential energy surface into global minima. A biasing term is added to the gradient squared landscape to stabilize the low energy saddle points near a minimum of interest, and destabilize other critical points. We demonstrate that this method is competitive with the dimer min-mode following method in terms of the number of force evaluations required to find a set of low-energy saddle points around a reactant minimum.

Water transport and water management are fundamental to polymer electrolyte membrane fuel cell operation. Accurate measurements of water content within and across the Nafion layer are required to elucidate water transport behavior and validate existing numerical models. We report here a direct measurement of water content profiles across a Nafion layer under wetting and drying conditions, using a novel magnetic resonance imaging methodology developed for this purpose. This method, multi-echo double half k-space spin echo single point imaging, based on a pure phase encode spin echo, is designed for high resolution 1D depth imaging of thin film samples. The method generates high resolution (<8 microm) depth images with an SNR greater than 20, in an image acquisition time of less than 2 min. The high temporal resolution permits water content measurements in the transient states of wetting and drying, in addition to the steady state.

Water management is critical to optimize the operation of polymer electrolyte membrane fuel cells. At present, numerical models are employed to guide water management in such fuel cells. Accurate measurements of water content variation in polymer electrolyte membrane fuel cells are required to validate these models and to optimize fuel cell behavior. We report a direct water content measurement across the Nafion membrane in an operational polymer electrolyte membrane fuel cell, employing double half k-space spin echo single point imaging techniques. The MRI measurements with T2 mapping were undertaken with a parallel plate resonator to avoid the effects of RF screening. The parallel plate resonator employs the electrodes inherent to the fuel cell to create a resonant circuit at RF frequencies for MR excitation and detection, while still operating as a conventional fuel cell at DC. Three stages of fuel cell operation were investigated: activation, operation and dehydration. Each profile was acquired in 6 min, with 6 microm nominal resolution and a SNR of better than 15.

Diabetes is a disease of the glucose regulatory system that is associated with increased morbidity and early mortality. The primary variables of this system are beta-cell mass, plasma insulin concentrations, and plasma glucose concentrations. Existing mathematical models of glucose regulation incorporate only glucose and/or insulin dynamics. Here we develop a novel model of beta -cell mass, insulin, and glucose dynamics, which consists of a system of three nonlinear ordinary differential equations, where glucose and insulin dynamics are fast relative to beta-cell mass dynamics. For normal parameter values, the model has two stable fixed points (representing physiological and pathological steady states), separated on a slow manifold by a saddle point. Mild hyperglycemia leads to the growth of the beta -cell mass (negative feedback) while extreme hyperglycemia leads to the reduction of the beta-cell mass (positive feedback). The model predicts that there are three pathways in prolonged hyperglycemia: (1) the physiological fixed point can be shifted to a hyperglycemic level (regulated hyperglycemia), (2) the physiological and saddle points can be eliminated (bifurcation), and (3) progressive defects in glucose and/or insulin dynamics can drive glucose levels up at a rate faster than the adaptation of the beta -cell mass which can drive glucose levels down (dynamical hyperglycemia)., (Copyright 2000 Academic Press.)