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This chapter presents the pathway towards the construction of Poisson–Nernst– Planck anomalous (PNPA) models proposed to connect the anomalous diffusion phenomenon with the impedance spectroscopy. The first part is dedicated to analysing the conceptual links between a PNPA model and equivalent electrical circuits containing constant-phase elements (CPEs) in the low frequency domain. It is demonstrated, on analytical grounds, that the effect of a CPE in an equivalent electrical circuit may be represented by an appropriate term added to the boundary conditions of PNP or PNPA models. The second part recalls the fundamental equations of the PNPA models, including also reaction terms and Ohmic boundary conditions to account for more complex bulk and interfacial behaviour. It is shown that the formulation based on the fractional diffusion equations establishes on general theoretical grounds a connection between the PNPA models with an entire framework of continuum models and equivalent circuits with CPEs to analyse impedance data. PNPA Models and Equivalent Circuits As we have seen in Chapter 9, the continuum models frequently used to analyse the data are essentially based on diffusion-like equations for the ions, satisfying the Poisson's equation requirement for the electric potential (Poisson–Nernst–Planck or PNP model), or on equivalent electrical circuits [312]. There are various distributed circuit elements that can be incorporated into equivalent circuits [342]. However, a careful analysis is necessary before reaching to general conclusions about the data, since the incorrect choice of the equivalent circuit can lead to deceptive conclusions about the process that occurs in the sample [343]. Even more powerful and useful general models, such as ordinary (PNP) or anomalous diffusion (PNPA) ones, are not free from ambiguities [344]. On one hand, the PNPA models aim at incorporating behaviours that may not be well described in terms of usual diffusive PNP models. On the other hand, an important extension used in the framework of equivalent circuits is the CPE, whose presence can be connected with the necessity of describing unusual effects in many solid electrode–electrolyte interfaces. For instance, it has been pointed out that simple elements cannot describe frequency dispersion often found in the solid electrode– electrolyte interfacial region [345]. This behaviour can be related to surface disorder and roughness [346–349], electrode porosity [350], and electrode geometry [351].

The extended Frank elastic energy density is used to investigate the existence of a stable periodically modulate structure that appears as a ground state exhibiting a twist-bend molecular arrangement. For an unbounded sample, we show that the twist-bend nematic phase $N_{TB}$ is characterized by a heliconical structure with a pitch in the nano-metric range, in agreement with experimental results. For a sample of finite thickness, we show that the wave vector of the stable periodic structure depends not only on the elastic parameters but also on the anchoring energy, easy axis direction, and the thickness of the sample., Comment: 11 pages

A phenomenological model representing ion adsorption equilibrium is proposed. It is an extension of the model used to represent surface adsorption, and it is intended to interpret bulk adsorption data. The fundamental equations of the model were established for fluid and solid phases mass balance and for the spatial distribution of ions across the sample. The electric field distribution in the system, modeled on Poisson's equation, and the chemical potential behavior were determined as a function of fluid phase equilibrium concentration. The concentration of ions as a function of electric potential is given by Boltzmann distribution. The proposed model was successfully tested to represent equilibrium adsorption of metallic ions in zeolite, showing results similar to conventional isotherms.

A confined liquid with dispersed neutral particles is theoretically studied when the limiting surfaces present different dynamics for the adsorption-desorption phenomena. The investigation considers different non-singular kernels in the kinetic equations at the walls, where the suitable choice of the kernel can account for the relative importance of physisorption or chemisorption. We find that even a small difference in the adsorption-desorption rate of one surface (relative to the other) can drastically affect the behavior of the whole system. The surface and bulk densities and the dispersion are calculated when several scenarios are considered and anomalous-like behaviors are found. The approach described here is closely related to experimental situations, and can be applied in several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers.

One-dimensional diffusion problems in bounded porous media characterized by the presence of nonlocal interactions are investigated by assuming a Darcy’s constitutive equation of convolution integral type. A power law attenuation function is implemented: Analogies and differences of the flow-rate-pressure law with respect to other nonlocal and fractal models are outlined. By means of the continuity relationship, the fractional diffusion equation is then derived. It involves spatial Riemann-Liouville derivatives with a noninteger order consisting of between 1 and 2. The solution is obtained numerically using fractional finite differences, and results are presented in both the transient and the steady-state regimes. Eventually, the physical meaning of fractional operators is discussed and potential applications of the analysis are suggested.

The reaction process occurring on a solid surface where active sites are present is investigated. The phenomenon is described by a linear kinetic equation capable of accounting for memory effects in the adsorption–desorption and a first order reaction process. In order to broaden the formulation of the problem, the surface is in contact with a system defined in a half space where the dynamics is governed by a fractional diffusion equation, meaning, in principle, that the approach can be applied to complex systems such as biological fluids. Our results prove that the anomalous behavior has great importance on the reaction and, consequently, on the densities rates of particles at the surface and on the distribution of particles in the bulk. The results are particularly relevant for heterogeneous catalysis.

An analytical expression for the relaxation time of a nematic liquid crystal is obtained for the first time by considering the influence of surface viscosity, anchoring energy strength and cell gap, validated numerically by using the so-called relaxation method. This general equation for the molecular response time ( τ 0 ) was derived for a vertical aligned cell and by solving an eigenvalue equation coming from the usual balance of torque equation in the Derzhanskii and Petrov formulation, recovering the usual equations in the appropriate limit. The results show that τ ∼ d b , where b = 2 is observed only for strongly anchored cells, while for moderate to weak anchored cells, the exponent lies between 1 and 2, depending on both, surface viscosity and anchoring strength. We found that the surface viscosity is important when calculating the response time, specially for thin cells, critical for liquid crystal devices. The surface viscosity’s effect on the optical response time with pretilt is also explored. Our results bring new insights about the role of surface viscosity and its effects in applied physics.

Cholesteric liquid crystals (CLCs) are stimuli driven, photonic bandgap materials that can be used for several optical applications, among which color tuning devices and tunable mirrors. Although many methods for controlling the reflection spectra have been proposed, few of them are able to control the position of the reflection notch and its bandwidth altogether. Here we propose a surface driven method to control the reflection notch center position and width based on an active surface. We theoretically formulated the equations involved and solved them to obtain the dynamical configuration of the liquid crystal and the resulting reflection spectra. Both notch position and potential broadening can be obtained using this method. We also identified the range of parameters which results in one of these regimes or a combination of both. Our results can be helpful for designing surface driven optical devices with CLC materials.

The bulk and the surface-like elastic constants of a nematic liquid crystal are calculated for an ensemble of particles interacting via anisotropic dispersion forces using the pseudo-molecular method. The geometrical anisotropy of the molecules is also taken into account in the calculations by choosing a molecular volume of ellipsoidal shape. Analytical expressions for the elastic constants are obtained as a function of the eccentricity in the molecular volume shape. The method allows one to explore the dependence on the molecular orientation with respect to the intermolecular vector by analyzing the magnitude and the behaviour of macroscopic elastic parameters defining the nematic phase.

The electrical current profile in a liquid sample containing ionic impurities, submitted to a time-dependent external voltage, is determined in the framework of the Poisson–Nernst–Planck (PNP) diffusional model. The boundary conditions at the limiting electrodes include the situations of blocking electrodes, generalized Chang–Jaffe (specific adsorption), and the adsorption–desorption of positive and negative ions at the interface, governed by a kinetic equation in the Langmuir’s approximation. The current–voltage characteristics of the confined system are obtained in the presence of an arbitrary time-dependent applied potential of small amplitude (linear approximation). The calculations are potentially useful for modeling and interpreting experimental situations found in biological cells and liquid crystals samples.

A connection between the impedance spectroscopy response of an anomalous Poisson–Nernst–Planck (PNPA) diffusional model and of equivalent circuits containing constant phase elements (CPEs) is established for a typical electrolytic cell. The analysis is carried out in the limit of low frequency in order to highlight the surface effects and to explore how they can be connected to the presence of CPEs in the circuit. It is shown that, depending on the choice of the equivalent circuit, the action of these elements can be the same as the one obtained using integro–differential boundary conditions to describe anomalous diffusive processes in the framework of PNPA models. The predictions are also compared with experimental data obtained from an electrolytic solution.

We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching them from the resting to the movement. The results show that in the asymptotic limit of small and long times, the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for intermediate times depends on the rates present in the reaction terms. In this scenario, we show that, in the asymptotic limits, the distributions for this process are given by in terms of power laws which may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a situation characterized by different diffusive regimes, which emerges when the diffusive term is a mixing of linear and nonlinear terms.

The recently discovered twist-bend nematic phase, Ntb, is a non-uniform equilibrium nematic phase that presents a spontaneous bend with a precession of the nematic director, n, on a conical helix with a tilt angle θ and helical pitch P. The stability of the Ntb phase has been recently demonstrated from the elastic point of view by extending the Frank elastic energy density of the nematic phase to include the symmetry element of the helical axis, t. In the present article, we investigate the influence of an external bulk field (magnetic or electric) on the Ntb phase. Using symmetry arguments we derive the expression for the flexoelectric polarisation in twist-bend nematic phases. We show that, besides the standard contribution related to the spatial variation of the nematic director, two new contributions connected with the existence of the helical axis appear. In the ground state, where the nematic deformation is a pure heliconical deformation, the new contribution vanishes identically, and the to...

From the elastic point of view, the twist-bend nematic phase, (NTB), has a ground state characterised by a spontaneous bend with a precession of the director n along the direction of an axis t. The...

We investigate the director relaxation of a nematic liquid crystal cell by considering the molecular moment of inertia, surface viscosity and anchoring energies at the boundaries. The problem solved here corresponds to a more complete description of the nematic director reorientation, since it incorporates ingredients that are usually neglected due to the mathematical difficulty raised by the moment of inertia and surface viscosity. In order to face this problem, we apply the half space approximation for the sample with the surfaces characterized by a time dependent easy direction, which can be mechanically or optically induced in the system. The results show that both, the molecular moment of inertia and the surface viscosity, have influence on the relaxation process of the director. In particular, the presence of the molecular moment of inertia introduces a finite phase velocity and, consequently, leads to a non-instantaneous response time for the distortion propagation in a liquid crystal cell.

The electrical impedance response of six nematic liquid crystals samples is analyzed by means of an extension of the Poisson-Nernst-Planck diffusional model. This formulation includes fractional derivatives of distributed order governing the behavior of the bulk density of mobile charges, whose solutions are subjected to integro-differential equations accounting for the boundary conditions. The role of the intrinsic surface lengths involved in the adsorption-desorption phenomena at the electrodes is emphasized, in order to analyze the frequency dependence of the real and imaginary parts of the electrical impedance experimentally obtained.

Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.

The electrical responses of a nematic liquid crystal cell are investigated by means of the elastic continuum theory. The nematic medium is considered as a parallel circuit of a resistance and a capacitance and the electric current profile across the sample is determined as a function of the elastic constants. In the reorientation process of the nematic director, the resistance and capacitance of the sample are determined by taking into account the elastic anisotropy. A nonmonotonic profile for the current is observed in which a minimum value of the current may be used to estimate the elastic constants values. This scenario suggests a theoretical method to determine the values of the bulk elastic constants in a single planar aligned cell just by changing the direction of applied electrical field and measuring the resulting electrical current.

A Poisson–Boltzmann approach is used to determine the double-layer integral and differential capacitances in a finite-length situation for an electrolytic cell. By means of simple analytical calculations, it is shown how these quantities exhibit the bell-like and the camel-like shapes as a function of the applied voltage, when the thickness of the sample or the surface charge is varied. The problem is formulated in such a way that an applied voltage dependent Debyeʼs screening length arises as a fundamental length governing the electrical behavior of the system.

We investigate the relaxation of the director in a nematic liquid crystal cell by considering the presence of different surface viscosities and anchoring energies on the boundaries. To face a more general problem, we also model the surfaces as characterised by time dependent easy directions which can be mechanically or optically induced in the system. In order to connect the analytical results with measurable physical quantities, the temporal behaviour of the optical path difference is established incorporating the contribution of the surface viscosity.

We study the optical transmittance of ternary mixtures of water, glycerin and ferrofluids. These mix- tures are subject to pulsed magnetic field and placed between crossed polarizers. After the magnetic field is switched off, the decay process is compared with q- exponential, stretched exponential, Mittag-Leffler, and one-sided Levy stable functions. For short time, the ex- perimental results are interpreted in terms of stretched exponentials. The parameters of this non-exponential relaxation are investigated as functions of temperature and the water, glycerin and ferrofluid concentrations.

A Fermi-like distribution for charged particles is used to investigate adsorption and external electric field effects in a typical confined sample of an isotropic liquid. In the absence of adsorption, expressions for the reduced potential across the sample and for the specific differential capacitance of the double layer are established as a function of the applied dc voltage. In the presence of adsorption, without external field, the fundamental equations of the electrostatic model are analytically solved in the limit of small potential. Simple expressions for the reduced and chemical potential as well as for the specific differential capacitance of the double layer are established. The behavior of these quantities are analyzed as a function of the adsorption energy, the total density of ions in the bulk, and the potential drop between the electrode surface and the bulk.

We present a Monte Carlo study of the effects of elastic anisotropy on the textures of nematic films with specific random hybrid boundary conditions. The polarized microscopy images and their evolution are analyzed in uniaxial systems for different values of the elastic constants.

Two empirical, but plausible, previously published independent generalizations of the standard Poisson−Nernst−Planck (PNP) continuum diffusion model for mobile-charge conduction in liquids and solids are discussed, their responses are compared, and their physical appropriateness and usefulness for data fitting are investigated. They both involve anomalous diffusion of PNPA type with power-law frequency-response elements involving fractional exponents. Both models apply to finite-length regions of material between completely blocking electrodes and, for simplicity, deal primarily with positive and negative charge carriers of equal valence numbers and mobilities. The charge carriers may be either ionic or electronic. The first PNPA model, model A, involves the common separation of the expression for ordinary PNP impedance into an interface diffusion part and a high-frequency limiting conductance and capacitance, followed by the replacement of all normal diffusion elements in the former by anomalous ones. It...

The influence of surface viscosity and anchoring energy on the reorientation process of a nematic liquid crystal cell is theoretically investigated. The cell is a slab of thickness, d, whose limiting surfaces are characterised by different anchoring strengths and present easy directions parallel to the bounding surfaces, changing with time due to some external action. The exact space-time profile of the director angle is obtained by means of integral transform techniques and a Green function approach. From this formalism, the time dependence of the optical path difference is exactly determined and its behaviour is analysed in connection with the presence of surface viscosity and different anchoring energies. The problem is also exactly solved in the presence of a constant electric field. It is shown that the compatibility problem between the time derivative of the director field on the surface and in the bulk can be avoided.

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.

The molecular orientation of a nematic sample limited by two concentric cylinders is theoretically investigated. The surfaces are assumed as having an inhomogeneous distribution of easy directions. The results are found in the one-constant approximation, in the presence of an external electric field, for weak anchoring of the molecules at the surfaces. In order to validate our approximations, the analytical results are compared with the ones obtained by numerically solving the non-linear bulk differential equation for the tilt angle profile subjected to non-linearized boundary conditions. We show that the agreement improves as the extrapolation length increases.

The equilibrium tilt angle profile in a cell limited by two concentric cylinders filled with nematic liquid crystals is determined for strong homeotropic anchoring at the surfaces. The anchoring condition is such that the nematic director is perpendicular to the cylinder axes and a radial nonuniform electric field is applied to investigate a Freedericksz transition. The distortions induced by the field remain in the plane perpendicular to the cylinder axes, and a threshold field is analytically determined indicating a transition from a pure splay to a splay-bend conformation of the director. It is shown that this transition can be induced by the thickness of the region between the two cylinders, and can be detected even in the absence of an external field. If the maximum value of the tilt angle is assumed as an order parameter, its behavior near to the transition can be used to obtain the critical exponent, which is the same as the one obtained in the mean field approximation. These results are indications that nontrivial consequences may occur when complex fluids are subject to non-planar geometries.

We argue that nonlocal effects represented by fractionary terms in the kinetic energy can be relevant to achieve a satisfactory phenomenological description of the thermal behavior of the specific heat of non-crystalline solids at very low temperature. We propose a simple model formed by the direct sum of two Hamiltonians, one of which is obtained by incorporating fractional derivatives in the kinetic energy of a conventional Hamiltonian, and the other one accounts for the presence of phonons in the system. Some experimental data are used to support the proposed description.

A family of entropy indices constructed in the framework of Tsallis entropy formalism is used to investigate ecological diversity. It represents a new perspective in ecology because a simple equation can incorporate all aspects of a diversity, from richness to dominance and can be also related to a measure of species rarity. In addition, a generalized Kullback-Leibler distance, constructed in the framework of a nonextensive formalism, is recalled and used as a measure of b diversity between two systems. These tools are applied to data relative

The influence of the ions on the electrochemical impedance of a cell is calculated in the framework of a complete model in which the fractional drift-diffusion problem is analytically solved. The resulting distribution of the electric field inside the sample is determined by solving Poisson's equation. The theoretical model to determine the electrical impedance we are proposing here is based on the fractional derivative of distributed order on the diffusion equation. We argue that this is the more convenient and physically significant approach to account for the enormous variety of the diffusive regimes in a real cell. The frequency dependence of the real and imaginary parts of the impedance are shown to be very similar to the ones experimentally obtained in a large variety of electrolytic samples.

The tilt angle profile in a nematic cell limited by two concentric cylindrical surfaces with inhomogeneous distribution of easy axes is investigated in the one-constant approximation. The results are presented in terms of the Green function approach by considering the strong anchoring case and the presence of an external electric field for small distortions.

We investigate an N -dimensional fractional diffusion equation with radial symmetry by taking a spatial and time dependent diffusion coefficient into account, i.e., D ˜ ( r , t ) = D ( t ) r − η with D ( t ) = D δ ( t ) + D ¯ ( t ) . The equation is considered in a confined region and subjected to time dependent boundary conditions which may be related to inhomogeneous characteristics of the surfaces confining the system. The results show an anomalous spreading of the solutions and an unusual behavior of the survival probability.

The elastic constants of a nematic liquid crystal are calculated by means of a pseudo-molecular method that considers the quadrupolar intermolecular interaction with a screening length governing the range of the forces. The bulk as well as the surface-like elastic constants are determined as functions of the eccentricity of the interaction volume of ellipsoidal shape. It is shown that the elastic constants become negative for some values of the eccentricity, and, therefore, that the screened quadrupolar interaction could be the source of instabilities in the nematic phase and should be taken into account to interpret threshold phenomena in these systems.

We obtain time dependent solutions for a fractional diffusion equation containing a nonlocal term by considering the spherical symmetry and using the Green function approach. The nonlocal term incorporated in the diffusion equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution. In addition, a rich class of anomalous diffusion processes may be described from the results obtained here.

Several indices have been created to measure diversity, and the most frequently used are the Shannon-Wiener (H) and Simpson (D) indices along with the number of species (S) and evenness (E). Controversies about which index should be used are common in literature. However, a generalized entropy (Tsallis entropy) has the potential to solve part of these problems. Here we explore a family of diversity indices (S q ; where q is the Tsallis index) and evenness (E q ), based on Tsallis entropy that incorporates the most used indices. It approaches S when q = 0, H when q → 1 and gives D when q = 2. In general, varying the value of the Tsallis index (q), S q varies from emphasis on species richness (q 1). Similarly, E q also works as a tool to investigate diversity. In particular, for a given community, its minimum value represents the maximum deviation from homogeneity (E q = 1) for a particular q (herein named q*). It is remarkable that our analysis indicates that q* and its corresponding evenness, E q *, are negatively affected by S when using simulated data. They may represent an index related to species rarity. Furthermore, S q* (i.e. the value of S q for a specific q*) is positively affected by richness that is an important property of any diversity index. In general, our findings indicate that the indices H, D, S, S q* , E and E q* are only part of a whole set of possibilities. In addition, the ecological properties of E q* and S q* , proposed here for the first time, show promise in ecology.

We investigate the solutions of the Burgers equation ∂ t u ( x , t ) = D ∂ x 2 u ( x , t ) − ∂ x [ F ( x , t ) u ( x , t ) ] − κ u ( x , t ) ∂ x u ( x , t ) + Φ ( x , t ) , where F ( x , t ) is an external force and Φ ( x , t ) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F ( x , t ) = k 1 ( t ) − k 2 ( t ) x into account. For this case, the solution obtained extends the usual one present in the Ornstein–Uhlenbeck process and depending on the choice of k 1 ( t ) and k 2 ( t ) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ ( x , t ) = Φ 1 ( t ) + Φ 2 ( t ) x and Φ ( x , t ) = Φ 3 x − Φ 4 / x 3 are incorporated in the previous analysis and exact solutions are obtained for both cases.

This work is devoted to investigate the solutions of the one-dimensional diffusion equation by taking the nonlinear external force F ( x , t ; ρ ) = − k ( t ) x + K / x + κ x | x | α − 1 [ ρ ( x , t ) ] η into account. Our investigation is first performed by considering the case α = 0 and η = 1 , which results in a Burgers like equation with a spatial and time dependent external force. After, we consider the case α ≠ 0 and η = α + 1 and show that the solution found may be expressed in terms of the q-exponential functions present in the Tsallis formalism. In addition, we also discuss the stationary solution for α and η arbitraries.

We consider nonlocal effects, obtained by incorporating fractional derivatives in the kinetic energy of a conventional Hamiltonian, to analyze physical properties of non-crystalline solids at very low temperature. By using thermal Green functions, we deduce some experimentally observable quantities such as the particle momentum distribution function, the particle energy distribution function, and the specific heat. The agreement between the results obtained for the specific heat and the experimental data suggests that the approach presented here may be useful as a phenomenological model to investigate thermal properties of non-crystalline solids at low temperature.

We obtain solutions for a fractional diffusion equation by taking the spherical symmetry into account and using the Green function approach. These solutions are found in a confined region by considering a spatial and time dependent boundary conditions, i.e, inhomogeneous surfaces. In our analysis, we also consider the diffusion coefficient given by D ( r ¯ ) = D r - η , the presence of the external force F ¯ ( r ) = K / r 1 + η r ˆ and a source (or absorbent) term. They show us an anomalous behavior due to the presence of the fractional derivative and the surface which for this case is inhomogeneous.

A model for the selective adsorption phenomenon in an isotropic liquid accounting for a van der Waals interaction between the ions and the surface is presented, in the framework of the Poisson-Boltzmann theory. The fundamental equations governing the electric field distribution are exactly solved for low and high potential regimes.

Monte Carlo simulations of lattice spin models are employed to investigate the effects of an external field on the creation and evolution of inversion walls in nematic films with conical boundary conditions. The simulations, based on the well known Lebwohl-Lasher potential, allow to produce simulated polarized microscopy images which are followed during their evolution when the field is applied.

The orientation of the nematic director field under the action of an external time‐dependent field is theoretically investigated as a mixed Dirichlet–Neumann boundary‐value problem. This mathematical problem represents the situation in which a nematic liquid crystal sample is limited by two inhomogeneous flat surfaces, separated by a distance d, on which the anchoring is weak. By considering the one‐constant approximation and a parabolic approximation for the surface energy, the initial conditions and boundary‐value problem for the profile of the tilt angle can be analytically solved even in the case in which the surfaces are not identical, which represents the more general situation. The results are valid for small deviations from the homeotropic orientation and for θ−Θ≪1, where θ is the actual tilt angle and Θ characterizes the easy direction imposed by the surface, and can be relevant to investigation of the molecular orientation in a nematic cell submitted to a small external voltage.

We investigate the solutions of a generalized diffusion equation which contains space and time fractional derivatives by taking an absorbent (or source) term and a linear external force into account. They are expressed in terms of the Fox H functions and, for some special cases, they are directly related to the Levy distributions. We also analyze the first passage time distribution for the case characterized by the absence of absorbent (or source) term and external force for a semi-infinite interval with absorbent boundary conditions. We also discuss the connection of the results presented here with the anomalous diffusion.

The adsorption–desorption phenomenon is investigated with the help of a modified kinetic equation at the interface fluid-substrate. This equation incorporates memory effects in the process by means of three different kernels, whose importance is associated to the predominance of chemisorption or physisorption in the adsorption–desorption processes. The time evolution of the density of adsorbed particles is exactly calculated. The results are shown to be relevant for adsorption phenomenon in general and, in particular, for the phenomenon of adsorption with oscillation, for which a comparison with the experimental data from adsorption of organosilane on oxides surfaces reveals a good agreement.

Density measurements were performed near the smectic C (SmC)-nematic (N1)-nematic (N2)-isotropic (I) phase transitions for HOBA liquid crystal. A discontinuity in the density values at the SmC–N1 and N2–I phase transitions was observed. From a thermodynamic point of view these phase transitions are considered to be first order. An additional small change in the density data was observed in the vicinity of the N1–N2 phase transition. This experimental result confirms the existence of these nematic phases previously reported in the literature. Furthermore, the thermal expansion coefficient (β) was also calculated, exhibiting a large peak at the transition. This behavior can be interpreted as an indication that the transition N1/–N2 is weakly first order.