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We discuss the foundations of a model based on an extension of the Langmuir approximation for the adsorption–desorption phenomena, in which the phenomenological coefficients depend on the bias potential, in addition to their dependence on the adsorption energy. The theoretical analysis focuses on the effect of these effective coefficients on the electrical response of an electrolytic cell to an external electric field, as predicted by the Poisson–Nernst–Planck model. Kinetic balance equations govern the current densities on the electrodes when the adsorption phenomenon occurs in the presence of an electric bias. The influence of the phenomenological parameters entering the model, as well as of the symmetry of the cell on the cyclic voltammetry, is investigated.

We investigate the solutions of a two-dimensional Schrödinger equation in the presence of geometric constraints, represented by a backbone structure with branches, by taking a position-dependent effective mass for each direction into account. We use Green’s function approach to obtain the solutions, which are given in terms of stretched exponential functions. The results can be linked to the properties of the system and show anomalous spreading for the wave packet. We also analyze the interplay between the backbone structure with branches constraining the different directions and the effective mass. In particular, we show how a fractional Schrödinger equation emerges from this scenario.

We present a theoretical analysis of the current–voltage characteristics of electrolytic cells using their electrical impedance response to a simple periodic external voltage of small amplitude. Th...

The recent experimental observation of twist-bend nematic phases has kindled interest on this new class of materials. The natural twist-bend deformations of the director results in a nanoscale pitch (pseudo-layers) probed by freeze-fracture transmission electron microscopy (FFTEM) and very recently it was verified by resonant soft x-ray scattering (RSoXS) technique. It is pivotal that other mechanisms arise to certify the recent experiments and corroborate the theoretical models. Impedance spectroscopy is a method commonly used to investigate the dynamics of bound or mobile charge in the bulk or interfacial regions of materials in the vast field of condensed matter. Here we use impedance spectroscopy technique to investigate the electrical transport properties of twist-bend nematic phase in a dimeric liquid crystal mixture. The experimental data is analyzed in the framework of an anomalous diffusion equation coupled with Poisson’s equation where bulk and surface contributions are taken into account. Our fitting results agree well with the impedance measurements in the nematic (N) and the twist-bend ( N tb ) phase and indicate that strong subdiffusive regimes take place in the system. We also find a large diffusion anisotropy depending on the imposed orientation, which is associated with motions along or through the pseudo-layers, showing that the impedance technique can be used to explore the molecular arrangement of N tb phases.

We present a short overview of recent investigations of the equilibrium tilt angle profile in a cell limited by two concentric cylinders filled with nematic liquid crystals (NLC), both in presence ...

The molecular organization of a nematic film sandwiched between two planar randomly aligned surfaces is studied by means of detailed Monte Carlo simulations. The formation as well as the evolution of topological defects induced by these particular boundary conditions are investigated. The resulting defect structure is compared with the one induced by hybrid aligned surfaces. The observation of such defects and some features of their structures can be associated with geometric parameters of the film and with properties of the confining surfaces.

Diverse communities are more productive than less diverse ones because of two overyielding mechanisms: a selection effect and a complementarity effect, which operate in different ways. Moreover, ecosystem multifunctionality is expected to increase and become more stable with increasing species diversity. However, it is unclear how the aforementioned processes operate in aquatic ecosystems, where the overyielding mechanisms, ecosystem multifunctionality, and stability have been poorly explored. Here, we conducted an experimental study manipulating three levels of macrophyte richness to test two hypotheses: (i) the positive macrophyte richness effect on community biomass production is the result of the species complementarity effect; (ii) ecosystem multifunctionality and its stability increase with macrophyte richness. We found that macrophyte species increased their biomass production in high richness treatment, evidencing overyielding, which occurred through a significant complementarity effect. Macrophyte richness also enhanced ecosystem multifunctionality, and made it more stable over time, but only in high richness treatment. Our study shows that preserving high macrophyte richness is essential to preserving the ability of aquatic communities and ecosystems to sustain their functioning.

The present study is devoted to the investigation of surface anchoring and finite-size effects on nematic--smectic-$A$--smectic-$C$ $(N\ensuremath{-}\mathrm{Sm}\text{\ensuremath{-}}A\ensuremath{-}\mathrm{Sm}\text{\ensuremath{-}}C)$ phase transitions in free-standing films. Using an extended version of the molecular theory for smectic-$C$ liquid crystals, we analyze how surface anchoring and film thickness affect the thermal behavior of the order parameters in free-standing smectic films. In particular, we determine how the transition temperature depends on the surface ordering and film thickness. We show that the additional orientational order imposed by the surface anchoring may lead to a stabilization of order parameters in central layers, thus modifying the nature of the phase transitions. We compare our results with experimental findings for typical thermotropic compounds presenting a $N\ensuremath{-}\mathrm{Sm}\text{\ensuremath{-}}A\ensuremath{-}\mathrm{Sm}\text{\ensuremath{-}}C$ phase sequence.

We present a Monte Carlo study of the effects of elastic anisotropy on the topological defects which can be formed in nematic films with hybrid boundary conditions. We simulate the polarized microscopy images and analyze their evolution in uniaxial systems for different values of the Frank elastic constants.

Periodic modulations in chiral nematic liquid crystals (CLCs) are used in several optical and photonic devices. A remarkable way of fabricating and controlling such patterns is in a thin wetting layer that grows as temperature is decreased. Here, we study this stripe formation process experimentally and by means of Monte Carlo simulations. We report on samples of CLCs that present, after the wetting transition, a stripe/fringe formation occurring either through a nucleation process or homogeneously across the cell. These patterns depend on the ratio of the pitch to thickness of the layer, elastic anisotropy, and anchoring strength. Although such process is well documented for CLCs under applied field, we show here that similar textural transitions occur in thin layers mediated by the weak anchoring at the CLC-isotropic interface; it happens for values of pitch to thickness ratio much smaller than the ones reported in field driven situation. The Monte Carlo method takes action in simulating the experimentally observed textures by using a pairwise additive potential model for CLCs. A qualitative comparison with the experimental phenomena is possible to be made in which both the stripes formation and their behavior agree with the experiments.

We investigate a diffusion process in heterogeneous media where particles stochastically reset to their initial positions at a constant rate. The heterogeneous media is modeled using a spatial-dependent diffusion coefficient with a power-law dependence on particles' positions. We use the Green function approach to obtain exact solutions for the probability distribution of particles' positions and the mean square displacement. These results are further compared and agree with numerical simulations of a Langevin equation. We also study the first-passage time problem associated with this diffusion process and obtain an exact expression for the mean first-passage time. Our findings show that this system exhibits non-Gaussian distributions, transient anomalous diffusion (sub- or superdiffusion) and stationary states that simultaneously depend on the media heterogeneity and the resetting rate. We further demonstrate that the media heterogeneity non-trivially affect the mean first-passage time, yielding an optimal resetting rate for which this quantity displays a minimum., Comment: 8 pages, 3 figures; accepted for publication Physica A

A model for the binary adsorption phenomenon of cations by a porous adsorbent in an isotropic liquid taking into account a localized adsorption energy is proposed. It consists of a set of algebraic and differential equations aiming at determining the profile of the electric field, the chemical potential, and the concentration of cations in the adsorbent material. The mathematical development of this model is based on the Poisson–Boltzmann equation to describe the electric field profile of a sample limited at one side by a flat surface adsorbent. This geometrical simplification is necessary to reduce the mathematical complexity and yields a more physical, even if still phenomenological, isotherm unlike Langmuir and many others. The proposed model was tested to represent data of binary biosorption equilibrium. To represent adequately the experimental data, it was necessary to use a mixing rule in the adsorption energy-related term. The model was able to accurately describe the case investigated showing results similar to those obtained by means of conventional isotherms.

A phenomenological model to represent the ions adsorption equilibrium is proposed. The model is based on the Poisson–Boltzmann equation for electric field modeling, resulting from the charges present in the bulk. With this approach, in solving the Poisson–Boltzmann equation the Debye–Huckel approximation was applied. Solving the equations of the model involves a system of nonlinear equations where the unknowns are the chemical potential and the electric potential on the adsorbent surface. To solve these equations an iterative procedure is presented. The main advantages of the developed model, when applied to the Debye–Huckel approximation, are intuitiveness and the difficulty in drawing physical meaning of parameters, which is simpler, and in fact delivers similar results to those obtained with the Poisson–Boltzmann model, without the Debye–Huckel simplification, in the cases studied.

We revisit the electrochemical impedance spectroscopy problem in the framework of the Poisson-Nernst-Planck model for a semi-infinite electrolytic cell, using the Gouy-Chapman interface approximation. The drift-diffusion problem is formulated in terms of fractional differential operators having singular and non-singular kernels. These operators are considered as extending the diffusion equations related to the ions’ motion in the sample to the arbitrary order. The solutions to these equations are searched in the small AC linear approximation for the applied voltage. In this way as well, we show that the resulting electrical impedance, taking into account anomalous diffusion effects, naturally predicts a constant-phase elements behavior emerging from the physicochemical parameters, without considering equivalent circuits. The whole approach yields different, non-usual scenarios for the electrical impedance response of the cell, which may be connected to anomalous diffusion behavior of the mobile charges.

This work focuses on the striped pattern formation of cholesteric liquid crystals confined in a hybrid aligned cell. The study is performed by means of Monte Carlo simulations. This pattern formati...

Representation of the oblique helicoidal structure with a nanometric pitch of typical of the twist-bend nematic phase.

We revisit the classical problem of the molecular orientation inside a hybrid cell filled with liquid-crystals with various elastic constant anisotropies by means of Monte Carlo simulations using t...

The twist-bend nematic (N_{TB}) phase is a doubly degenerated heliconical structure with nanometric pitch and spontaneous bend and twist deformations. It is favored by symmetry-breaking molecular structures, such as bent dimers and bent-core molecules, and it is currently one of the burgeoning fields of liquid-crystal research. Although tremendous advances have been reported in the past five years, especially in molecular synthesis, most of its potential applications are held back by the lack of a proper and definitive elastic model to describe its behavior under various situations such as confinement and applied field. In this work we use a recently proposed stable state elastic model and the fact that the mesophase behaves as a lamellar structure to propose a mesoscopic or coarse-grained model for the N_{TB} phase. By means of standard procedures used for smectic and cholesteric liquid crystals, we arrive at a closed-form energy for the phase and apply it to a few situations of interest. The predicted compressibility for several values of the cone angle and the critical field for field-induced deformation agree well with recent experimental data.

We propose that a kinetic equation of fractional order may be used to account for the frequency dispersion of the effective parameters in the framework of the Langmuir approximation for the adsorption–desorption phenomena. A frequency dependence of these parameters naturally arises in this formalism, indicating that it may play a similar role of a generalization of the Langmuir isotherm obtained from a homogeneous distribution of relaxation times. The fractional approach formalism opens the possibility to consider different relaxation regimes characterizing the interfacial behaviour of electrolytic cells, and may be a powerful tool to interpret complex impedance spectroscopy data.

We present an extended theory for the electric response of an electrolytic cell with two different electrodes to a small amplitude external ac stimulus. In this case, the two electrodes limiting the cell are different in the sense that the spatial dependencies of the dynamical quantities characterising the system have not a well defined symmetry, with respect to the electrodes. This requires extending the Poisson-Nernst-Planck model to solve the fundamental equations allowing for the analytical determination of the electric impedance of the cell. The new theoretical framework opens the pathway to describe asymmetric electrolytic cells limited by conductive or adsorbing-desorbing boundaries.

We investigate possible connections between two different implementations of the Poisson-Nernst-Planck (PNP) anomalous models used to analyze the electrical response of electrolytic cells. One of them is built in the framework of the fractional calculus and considers integro-differential boundary conditions also formulated by using fractional derivatives; the other one is an extension of the standard PNP model presented by Barsoukov and Macdonald, which can also be related to equivalent circuits containing constant phase elements (CPEs). Both extensions may be related to an anomalous diffusion with subdiffusive characteristics through the electrical conductivity and are able to describe the experimental data presented here. Furthermore, we apply the Bayesian inversion method to extract the parameter of interest in the analytical formulas of impedance. To resolve the corresponding inverse problem, we use the delayed-rejection adaptive-Metropolis algorithm (DRAM) in the context of Markov-chain Monte Carlo (MCMC) algorithms to find the posterior distributions of the parameter and the corresponding confidence intervals.

We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.

The adsorption phenomenon is often described by means of a kinetic equation, in the form proposed by Langmuir. From the statistical point of view, the analysis is done in terms of adsorption sites, in the framework of Boltzmann or Fermi-like statistics. We show that a more simple description can be done in terms of an effective adsorption energy, taking into account its dependence on the density of adsorbed particles. Some elementary lateral interaction potentials are analyzed to illustrate this dependence. We show that the effective adhesion energy may be written as a power law of the surface coverage ratio. Eventually, we propose a generalization of the Frumkin-Fowler-Guggenheim model as a self-consistent non-linear relation determining the actual equilibrium distribution of surface particles.

We study diffusing particles in a confined liquid whose diffusion coefficient periodically varies across the sample. The limiting substrates may adsorb-desorb particles when memory effects are also considered. We show how the bulk modulation (amplitude and number of turns) affects bulk and surface distributions. The diffusing regimes are also studied in terms of the physical parameters governing the system, and we show that, by controlling the modulation (amplitude and frequency) in such limited system, might lead to superdiffusive, usual, or subdiffusive regimes. In other words, increasing the number of turns and the amplitude of modulation changes how the mean square displacement changes over time. Fewer turns with large negative amplitude favors faster diffusion whereas higher number of turns and large positive amplitudes favor slower diffusion. This simple model might help understanding diffusing behavior in modulated and crowded systems such as in cholesteric and smectic liquid crystals and other lamellar systems.

In this study, we argue that ion motion in electrolytic cells containing Milli-Q water, weak electrolytes, or liquid crystals may exhibit unusual diffusive regimes that deviate from the expected behavior, leading the system to present an anomalous diffusion. Our arguments lie on the investigation of the electrical conductivity and its relationship with the mean square displacement, which may be used to characterize the ionic motion. In our analysis, the Poisson-Nernst-Planck diffusional model is used with extended boundary conditions to simulate the charge transfer, accumulation, and/or adsorption-desorption at the electrode surfaces.

Broken symmetry of any kind in natural or synthetic systems is a burgeoning area of science. In this article, the temporal behavior of the bulk density, electric potential inside the sample, and the adsorbed ionic particles density in an electrolytic cell are shown to be non-symmetric and strongly affected by the surfaces dynamics. These parameters are analytically calculated by considering electrodes surfaces with non-identical adsorption-desorption phenomena. The system is submitted to an applied external voltage in the limit of small amplitude, where the time scales involved allow strong surface effects. The influence of the surfaces on the ionic distribution is performed in the framework of the Poisson-Nernst-Planck model by taking into account non-identical boundary conditions on the electrodes and on the electric potential, described by distinct kinetic equations. Such approach may be directly applied to explain strong surface dependence on the impedance spectroscopy, in hybrid aligned liquid crystal cells, fuel cells, biological and condensed matter systems in general.

We analyze the asymptotic behavior of the impedance (or immittance) spectroscopy response of an electrolytic cell in a finite-length situation obtained from the Poisson-Nernst-Planck (PNP) diffusional model and extensions by taking into account different surface effects. The analysis starts with the case characterized by perfect blocking electrodes and proceeds by considering non-blocking conditions on electrodes surface. We argue that the imaginary part of the impedance may be directly related to the boundary condition on the electrode surface, such as charge accumulation and/or transfer by electrochemical reaction or adsorption-desorption processes. We also compare the theoretical predictions with experimental data obtained for a weak electrolytic solution of KClO 3 .

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach.

Diffusion processes occurring in a myriad of systems sparkle great interest in understanding their general properties and applications. In this work, we investigate a broad set of diffusive systems that can be governed by a generalized diffusion equation and subjected to a surface that can promote sorption and, consequently, desorption, thus releasing the particles to the bulk. The general bulk equation used here can reproduce different diffusive regimes, among them, those described by the Cattaneo equation or by a fractional, anomalous diffusion. The equation related to the processes on the surface incorporates non-Debye relaxations which can be used to model non-exponential relaxations commonly found in biological or fractal systems. The solutions are obtained by using the Green function approach and show a rich class of behavior that can be related to anomalous diffusion.

General diffusion processes involve one or more diffusing species and are usually modelled by Fick’s law, which assumes infinite propagation velocity. In this article, searching for the effect of finite propagation speeds in a system with two reacting species, we investigate diffusing and reacting particles governed by a hyperbolic diffusion equation, that is, the Cattaneo equation, which describes a diffusion process with finite propagation velocity, in the presence of a constant external field and reaction terms. These reaction terms are linear and may be related to irreversible and reversible processes, including memory effects, depending on the choices of the reaction rates. We obtain exact solutions for the equilibrium concentrations and explore the rich variety of behaviours exhibited by the species involved in reaction processes. Our results may shine new light into systems with more than one kind of diffusing and reacting particles, as is the case in several industrial and biological process, when finite speeds and memory effects are involved.

The electric response of a capacitive deionisation cell submitted to a periodic external electric field is investigated. The case in which the applied potential has a nonzero average value on one period (polarised cell) is considered. The theoretical analysis of the experimental data, relevant to nearly symmetric electrodes, is done in the framework of the Poisson-Nernst-Planck model. The current densities on the electrodes are described by kinetic equations related to the adsorption phenomenon in the presence of a bias potential. We propose a new form for the Langmuir isotherm in which the effective adsorption coefficients depend on the bias potential according to the Boltzmann statistics. This kinetic equation extends the Butler-Volmer equation for non-blocking electrodes also to the blocking ones. The equation proposed here is such that for dc external voltage the total current across the electrodes vanishes.

Recent discoveries of advanced photocontrolled materials have kindled a great deal of interest on their use as command surfaces that switch easy axis under light radiation. One noticeable point when using switchable surfaces on any application is how the dynamical process propagates to the bulk directors. In this paper, we theoretically study the effect of a relaxing easy axis over time on a nematic sample when finite anchoring energy and surface viscosity are included. We first consider the case where just one of the substrates decay over time in an initially distorted director organization. Next, we assume that both substrates can be switched simultaneously. From the calculated director we obtained the optical profile and finally the molecular response time of the material. The response time depends on both the materials and the surfaces properties including its decay time. Our results might be used for understanding and engineering liquid crystal displays and other electro-optical devices with photocontrolled alignment layers.

Surface driven pattern formation is an intriguing phenomenon in the liquid crystal field. Owing to its ability to transmit torque, one can generate different patterns by propagating distortions on the optical wavelength scale in the sample from the surface. Here, we theoretically investigate (from the elasticity point of view) twist deformations induced by a rotating easy axis at one surface, by considering the anchoring energy and surface viscosity of nematic and chiral nematic samples. The model is solved analytically in the limit of strong anchoring and numerically for a low anchoring strength situation. Such rotation could be induced, in principle, by light-controlling the orientation of an azobenzene monolayer coated at one of the glass substrates or by an in-plane rotating field. We discuss the role of the surface parameters and the different distortions, and calculate light transmission using the Jones method. Three different regimes are identified: free twist, stick-slip twist, and constrained twist. The results obtained here may be relevant for liquid crystal active waveplates and for determining surface viscosity and the azimuthal anchoring energy.

We present a detailed Monte Carlo study of the effects of molecular biaxiality on the defect created at the centre of a nematic droplet with radial anchoring at the surface. We have studied a lattice model based on a dispersive potential for biaxial mesogens [Luckhurst et al., Mol. Phys. 30, 1345 (1975)] to investigate how increasing the biaxiality influences the molecular organisation inside the confined system. The results are compared with those obtained from a continuum theory approach. We find from both approaches that the defect core size increases by increasing the molecular biaxiality, hinting at a non universal behaviour previously not reported.