Fractional diffusion in the multiple-trapping regime and revision of the equivalence with the continuous-time random walk

Defect related anomalous mobility of small polarons in dielectric oxides at the example of congruent lithium niobate

Polarons play a major role in the description of optical, electrical and dielectrical properties of several ferroelectric oxides. The motion of those particles occurs by elementary hops among the material lattice sites. In order to compute macroscopic transport parameters such as charge mobility, normal (i.e. Fickian) diffusion laws are generally assumed. In this paper we show that when defect states able to trap the polarons for long times are considered, significant deviations from the normal diffusion behaviour arise. As an example of this behavior, we consider here the case of lithium niobate (LN). This can be considered as a prototypical system, having a rich landscape of interacting polaron types and for which a significant wealth of information is available in literature. Our analysis considers the case of a stoichiometric, defect-free lithium niobate containing a certain concentration of small electron polarons hopping on regular Nb sites, and compares it to the material in congruent composition, which is generally found in real-life applications and which is characterized by a large concentration of antisite NbLi defects. While in the first case the charge carriers are free polarons hopping on a regular Nb sublattice, in the second case a fraction of polarons is trapped on antisite defects. Thus, in the congruent material, a range of different hopping possibilities arises, depending on the type of starting and destination sites. We develop a formalism encompassing all these microscopic processes in the framework of a switching diffusion model which can be well approximated by a mobile-immobile transport model providing explicit expressions for the polaron mobility. Finally, starting from the Marcus-Holstein's model for the polaron hopping frequency we verify by means of a Monte Carlo approach the diffusion/mobility of the different polarons species showing that, while free polarons obey the laws for normal diffusion as expected, bound polarons follow an anomalous diffusion behaviour and that in the case of the congruent crystal where mixed free and bound polaron transport is involved, our expressions indeed provide a satisfactory description.

Capturing ion trapping and detrapping dynamics in electrochromic thin films

Ion trapping has been found to be responsible for the performance degradation in electrochromic oxide thin films, and a detrapping procedure was proved to be effective to rejuvenate the degraded films. Despite of the studies on ion trapping and detrapping, its dynamics remain largely unknown. Moreover, coloration mechanisms of electrochromic oxides are also far from clear, limiting the development of superior devices. Here, we visualize ion trapping and detrapping dynamics in a model electrochromic material, amorphous WO3. Specifically, formation of orthorhombic Li2WO4 during long-term cycling accounts for the origin of shallow traps. Deep traps are multiple-step-determined, composed of mixed W4+-Li2WO4, amorphous Li2WO4 and W4+-Li2O. The non-decomposable W4+-Li2WO4 couple is the origin of the irreversible traps. Furthermore, we demonstrate that, besides the typical small polaron hopping between W5+ ↔ W6+ sites, bipolaron hopping between W4+ ↔ W6+ sites gives rise to optical absorption in the short-wavelength region. Overall, we provide a general picture of electrochromism based on polaron hopping. Ion trapping and detrapping were demonstrated to also prevail in other cathodic electrochromic oxides. This work not only provides the ion trapping and detrapping dynamics of WO3, but also open avenues to study other cathodic electrochromic oxides and develop superior electrochromic devices with great durability.

Unconventionally fast transport through sliding dynamics of rodlike particles in macromolecular networks

Transport of rodlike particles in confinement environments of macromolecular networks plays crucial roles in many important biological processes and technological applications. The relevant understanding has been limited to thin rods with diameter much smaller than network mesh size, although the opposite case, of which the dynamical behaviors and underlying physical mechanisms remain unclear, is ubiquitous. Here, we solve this issue by combining experiments, simulations and theory. We find a nonmonotonic dependence of translational diffusion on rod length, characterized by length commensuration-governed unconventionally fast dynamics which is in striking contrast to the monotonic dependence for thin rods. Our results clarify that such a fast diffusion of thick rods with length of integral multiple of mesh size follows sliding dynamics and demonstrate it to be anomalous yet Brownian. Moreover, good agreement between theoretical analysis and simulations corroborates that the sliding dynamics is an intermediate regime between hopping and Brownian dynamics, and provides a mechanistic interpretation based on the rod-length dependent entropic free energy barrier. The findings yield a principle, that is, length commensuration, for optimal design of rodlike particles with highly efficient transport in confined environments of macromolecular networks, and might enrich the physics of the diffusion dynamics in heterogeneous media.

Challenges of modeling nanostructured materials for photocatalytic water splitting

Understanding the water splitting mechanism in photocatalysis is a rewarding goal as it will allow producing clean fuel for a sustainable life in the future. However, identifying the photocatalytic mechanisms by modeling photoactive nanoparticles requires sophisticated computational techniques based on multiscale modeling. In this review, we will survey the strengths and drawbacks of currently available theoretical methods at different length and accuracy scales. Understanding the surface-active site through Density Functional Theory (DFT) using new, more accurate exchange-correlation functionals plays a key role for surface engineering. Larger scale dynamics of the catalyst/electrolyte interface can be treated with Molecular Dynamics albeit there is a need for more generalizations of force fields. Monte Carlo and Continuum Modeling techniques are so far not the prominent path for modeling water splitting but interest is growing due to the lower computational cost and the feasibility to compare the modeling outcome directly to experimental data. The future challenges in modeling complex nano-photocatalysts involve combining different methods in a hierarchical way so that resources are spent wisely at each length scale, as well as accounting for excited states chemistry that is important for photocatalysis, a path that will bring devices closer to the theoretical limit of photocatalytic efficiency.

Time-Fractional Phase Field Model of Electrochemical Impedance

In this paper, electrochemical impedance responses of subdiffusive phase transition materials are calculated and analyzed for one-dimensional cell with reflecting and absorbing boundary conditions. The description is based on the generalization of the diffusive Warburg impedance within the fractional phase field approach utilizing the time-fractional Cahn–Hilliard equation. The driving force in the model is the chemical potential of ions, that is described in terms of the phase field allowing us to avoid additional calculation of the activity coefficient. The derived impedance spectra are applied to describe the response of supercapacitors with polyaniline/carbon nanotube electrodes.

Testing short distance anisotropy in space

The isotropy of space is not a logical requirement but rather is an empirical question; indeed there is suggestive evidence that universe might be anisotropic. A plausible source of these anisotropies could be quantum gravity corrections. If these corrections happen to be between the electroweak scale and the Planck scale, then these anisotropies can have measurable consequences at short distances and their effects can be measured using ultra sensitive condensed matter systems. We investigate how such anisotropic quantum gravity corrections modify low energy physics through an anisotropic deformation of the Heisenberg algebra. We discuss how such anisotropies might be observed using a scanning tunnelling microscope.

Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker–Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

Frequency dispersion in the fractional Langmuir approximation for the adsorption–desorption phenomena

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.

Diffusion behavior of peptide amphiphiles containing different numbers of alkyl tails at a hydrophobic solid-liquid interface: single molecule tracking investigation

Using the single molecule tracking technique, the diffusion behavior of peptide amphiphiles (PAs) with different numbers of alkyl tails at a hydrophobic solid-liquid interface has been investigated. The effect of the number of alkyl tails of PAs on molecular trajectories at the hydrophobic solid-liquid interface has been systematically studied. PA molecules display an intermittent motion consisting of immobilization and hopping processes, which has been well simulated by the continuous time random walk (CTRW) model. The results reveal that the hydrophobic interaction between the PAs and hydrophobic surface plays an important role in the diffusion behavior of PAs. Increasing the number of alkyl tails in PAs systematically reduces the mobility of PAs on the hydrophobic surface. Moreover, the diffusion behavior of PAs at the hydrophobic interface also shows pH dependence. A decrease in pH is beneficial to the motion of all PAs on the hydrophobic surface, which can be ascribed to the protonation of PAs in acidic solutions. Therefore, the hydrophobic interaction is crucial to the transport of peptide amphiphiles at hydrophobic interfaces which would be important for the design of peptides in biological applications.

Recent advances in the use of TiO2 nanotube powder in biological, environmental, and energy applications

The use of titanium dioxide nanotubes in the powder form (TNTP) has been a hot topic for the past few decades in many applications. The high quality of the fabricated TNTP by various synthetic routes may meet the required threshold of performance in a plethora of fields such as drug delivery, sensors, supercapacitors, and photocatalytic applications. This review briefly discusses the synthesis techniques of TNTP, their use in various applications, and future perspectives to expand their use in more applications.

Electrochemical Impedance Imaging via the Distribution of Diffusion Times

We develop a mathematical framework to analyze electrochemical impedance spectra in terms of a distribution of diffusion times (DDT) for a parallel array of random finite-length Warburg (diffusion) or Gerischer (reaction-diffusion) circuit elements. A robust DDT inversion method is presented based on complex nonlinear least squares regression with Tikhonov regularization and illustrated for three cases of nanostructured electrodes for energy conversion: (i) a carbon nanotube supercapacitor, (ii) a silicon nanowire Li-ion battery, and (iii) a porous-carbon vanadium flow battery. The results demonstrate the feasibility of nondestructive "impedance imaging" to infer microstructural statistics of random, heterogeneous materials.

Degradation Dynamics for Electrochromic WO3 Films under Extended Charge Insertion and Extraction: Unveiling Physicochemical Mechanisms

Degradation of electrochromic thin films under extended charge insertion and extraction is a technically important phenomenon for which no in-depth understanding is currently on hand. Here, we report on an explorative study of sputter-deposited WO₃ films in a Li-ion-conducting electrolyte by use of cyclic voltammetry, in situ optical transmittance, and impedance spectroscopy. A cycling-dependent decrease of the charge capacity could be accurately modeled by a power-law function, and impedance spectroscopy gave evidence for anomalous diffusion as well as a higher charge-transfer resistance during deintercalation than during intercalation. Thus, a consistent conceptual picture emerged for the degradation dynamics; it includes the growth of an interfacial barrier layer and also embraces anomalous diffusion coupled with dispersive power-law chemical kinetics.

Eliminating Electrochromic Degradation in Amorphous TiO2 through Li-Ion Detrapping

The quest for superior and low-cost electrochromic (EC) thin films, for applications in smart windows, remains strong because of their large importance for energy-efficient buildings. Although the development of new EC materials for improved devices is important, diminishing or reversing degradation is another key issue, and electrical rejuvenation of degraded EC materials can offer new opportunities. Here we demonstrate that cathodically coloring EC thin films of TiO2, which normally suffer from ion-trapping-induced degradation of charge capacity and optical modulation upon extensive electrochemical cycling, can recover their initial EC performance by a rejuvenation procedure involving Li(+) ion detrapping. Thus, the initial performance can be regained, and refreshed TiO2 films exhibit the same degradation features as as-deposited films upon prolonged electrochemical cycling. The rejuvenation was carried out by using either galvanostatic or potentiostatic treatments. Our study may open avenues to explore the recovery not only of EC materials and devices based on those but also for other ion-exchange-based devices.

Galvanostatic Ion Detrapping Rejuvenates Oxide Thin Films

Ion trapping under charge insertion-extraction is well-known to degrade the electrochemical performance of oxides. Galvanostatic treatment was recently shown capable to rejuvenate the oxide, but the detailed mechanism remained uncertain. Here we report on amorphous electrochromic (EC) WO3 thin films prepared by sputtering and electrochemically cycled in a lithium-containing electrolyte under conditions leading to severe loss of charge exchange capacity and optical modulation span. Time-of-flight elastic recoil detection analysis (ToF-ERDA) documented pronounced Li(+) trapping associated with the degradation of the EC properties and, importantly, that Li(+) detrapping, caused by a weak constant current drawn through the film for some time, could recover the original EC performance. Thus, ToF-ERDA provided direct and unambiguous evidence for Li(+) detrapping.

Eliminating degradation and uncovering ion-trapping dynamics in electrochromic WO3 thin films

There is keen interest in the use of amorphous WO3 thin films as cathodic electrodes in transmittance-modulating electrochromic devices. However, these films suffer from ion-trapping-induced degradation of optical modulation and reversibility on extended Li(+)-ion exchange. Here, we demonstrate that ion-trapping-induced degradation, which is commonly believed to be irreversible, can be successfully eliminated by constant-current-driven de-trapping; that is, WO3 films can be rejuvenated and regain their initial highly reversible electrochromic performance. Pronounced ion trapping occurs when x exceeds ∼0.65 in LixWO3 during ion insertion. We find two main kinds of Li(+)-ion-trapping site (intermediate and deep) in WO3, where the intermediate ones are most prevalent. Li(+) ions can be completely removed from intermediate traps but are irreversibly bound in deep traps. Our results provide a general framework for developing and designing superior electrochromic materials and devices.

Microscopic dynamics research on the "mature" process of dye-sensitized solar cells after injection of highly concentrated electrolyte

After injection of electrolyte, the internal three-dimensional solid-liquid penetration system of dye-sensitized solar cells (DSCs) can take a period of time to reach "mature" state. This paper studies the changes of microscopic processes of DSCs including TiO2 energy-level movement, localized state distribution, charge accumulation, electron transport, and recombination dynamics, from the beginning of electrolyte injection to the time of reached mature state. The results show that the microscopic dynamics process of DSCs exhibited a time-dependent behavior and achieved maturity ∼12 h after injecting the electrolyte into DSCs. Within 0-12 h, several results were observed: (1) the conduction band edge of TiO2 moved slightly toward negative potential direction; (2) the localized states in the band gap of TiO2 was reduced according to the same distribution law; (3) the transport resistance in TiO2 film increased, and electron transport time was prolonged as the time of maturity went on, which indicated that the electron transport process is impeded gradually; (4) the recombination resistance at the TiO2/electrolyte (EL) interface increases, and electron lifetime gradually extends, therefore, the recombination process is continuously suppressed. Furthermore, results suggest that the parameters of EL/Pt-transparent conductive oxide (TCO) interface including the interfacial capacitance, electron-transfer resistance, and transfer time constant would change with time of maturity, indicating that the EL/Pt-TCO interface is a potential factor affecting the mature process of DSCs.

dc electric field effect on the anomalous exponent of the hopping conduction in the one-dimensional disorder model

The dc electric field effect on the anomalous exponent of the hopping conduction in the disorder model is investigated. First, we explain the model and derive an analytical expression of the effective waiting time for the general case. We show that the exponent depends on the external field. Then we focus on a one-dimensional system in order to illustrate the features of the anomalous exponent. We derive approximate expressions of the anomalous exponent of the system analytically. For the case of a weak field, the anomalous exponent is consistent with that of diffusive systems. This is consistent with the treatments of Barkai et al. [Phys. Rev. E 63, 046118 (2001)] and our result supports their theory. On the other hand, for the case of a strong field and a strong disorder, the time evolution of the exponent clearly differs from that in the weak field. The exponent is consistent with the well-known expression of the anomalous exponent in the multiple trapping model at mesoscopic time scales. In the long-time limit, a transition of the anomalous exponent to the same value of the weak field occurs. These findings are verified by the Monte Carlo simulation.

Non-Debye relaxation in the dielectric response of nematic liquid crystals: surface and memory effects in the adsorption-desorption process of ionic impurities

We demonstrate theoretically that the presence of ions in insulating materials such as nematic liquid crystals may be responsible for the dielectric spectroscopy behavior observed experimentally. It is shown that, at low frequencies, an essentially non-Debye relaxation process takes place due to surface effects. This is accomplished by investigating the effects of the adsorption-desorption process on the electrical response of an electrolytic cell when the generation and recombination of ions is present. The adsorption-desorption is governed by a non-usual kinetic equation in order to incorporate memory effects related to a non-Debye relaxation and the roughness of the surface. The analysis is carried out by searching for solutions to the drift-diffusion equation that satisfy the Poisson equation relating the effective electric field to the net charge density. We also discuss the effect of the mobility of the ions, i.e., situations with equal and different diffusion coefficients for positive and negative ions, on the impedance and obtain an exact expression for the admittance. The model is compared with experimental results measured for the impedance of a nematic liquid crystal sample and a very good agreement is obtained.

Anomalous diffusion and memory effects on the impedance spectroscopy for finite-length situations

The contribution of ions to the electrical impedance of an electrolytic cell limited by perfect blocking electrodes is determined by considering the role of the anomalous diffusion process and memory effects. Analytical solutions for fractional diffusion equations together with Poisson's equation relating the effective electric field to the net charge density are found. This procedure allows the construction of general expressions for the electrochemical impedance satisfying the Kramers-Kronig relations when the diffusion of ions in the cell is characterized by the usual, as well as by anomalous, behavior.

Anomalous-diffusion approach applied to the electrical response of water

We investigate the electrical response of Milli-Q deionized water by using a fractional diffusion equation of distributed order with the interfaces (i.e., the boundary conditions at the electrodes limiting the sample) governed by integrodifferential equations. We also consider that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies Poisson's equation. In addition, the good agreement between the experimental data and this approach evidences the presence of anomalous diffusion due to the surface effects in this system.

Building one-dimensional oxide nanostructure arrays on conductive metal substrates for lithium-ion battery anodes

Lithium ion battery (LIB) is potentially one of the most attractive energy storage devices. To meet the demands of future high-power and high-energy density requirements in both thin-film microbatteries and conventional batteries, it is challenging to explore novel nanostructured anode materials instead of conventional graphite. Compared to traditional electrodes based on nanostructure powder paste, directly grown ordered nanostructure array electrodes not only simplify the electrode processing, but also offer remarkable advantages such as fast electron transport/collection and ion diffusion, sufficient electrochemical reaction of individual nanostructures, enhanced material-electrolyte contact area and facile accommodation of the strains caused by lithium intercalation and de-intercalation. This article provides a brief overview of the present status in the area of LIB anodes based on one-dimensional nanostructure arrays growing directly on conductive inert metal substrates, with particular attention to metal oxides synthesized by an anodized alumina membrane (AAM)-free solution-based or hydrothermal methods. Both the scientific developments and the techniques and challenges are critically analyzed.

TiO2 nanotubes and their application in dye-sensitized solar cells

The present article reviews the current status of using TiO(2) nanotubes in Grätzel-type, dye-sensitized solar cells and extends the overview with the latest results and findings. Critical factors in tube geometry (length, diameter, top morphology), crystal structure (amorphous, anatase, rutile) as well as factors affecting dye loading or electron mobility are addressed. The highest solar cell efficiencies today for pure nanotube systems reach approximately 4% while for some mixed systems, around 7% has been reported. For both systems significant room for enhancement is anticipated and some key points and strategies for improvement are outlined.

Fluid imbibition in paper fibers: precursor front

We employ nuclear magnetic resonance imaging to study water penetration in cylindrical blocks of unsized paper prepared under different molding pressures. From the measured kinetics of the imbibition profiles, we determine the dependence of the effective transport diffusivity upon degree of saturation of the pores by the penetrating fluid. In general, the transport process is found to be non-Fickian and we discuss different methods of data analysis adapted to this situation. The effective transport diffusivity vividly captures the presence of a precursor front, consisting of fluid in partially filled pores, with a much higher effective diffusivity than that of fluid in largely saturated pores.

Exact effective transport dynamics in a one-dimensional random environment

We study effective transport under linear equilibrium adsorption characterized by a spatially random retardation factor. In a stochastic framework, we present a methodology to quantify explicitly the impact of spatial disorder on effective transport dynamics. We derive an exact effective transport equation, which is equivalent to transport under linear kinetic adsorption characterized by a spectrum of adsorption times. The distribution of adsorption times is given explicitly in terms of the spatial disorder distribution. Furthermore, we find that effective transport is formally equivalent to a decoupled continuous time random walk. This observation and the explicit nature of the presented results allow for a mapping of the static disorder distribution on the transition time distribution.

Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination

The fractional diffusion equation that is constructed replacing the time derivative with a fractional derivative, (0)D(alpha)(t) f = C(alpha) theta(2) f/theta x(2), where (0)D(alpha)(t) is the Riemann-Liouville derivative operator, is characterized by a probability density that decays with time as t(alpha -1) (alpha < 1) and an initial condition that diverges as t -->0 [R. Hilfer, J. Phys. Chem. B 104, 3914 (2000)]. These seemingly unphysical features have obstructed the application of the fractional diffusion equation. The paper clarifies the meaning of these properties adopting concrete physical interpretations of experimentally verified models: the decay of free-carrier density in a semiconductor with an exponential distribution of traps, and the decay of ion-recombination isothermal luminescence. We conclude that the fractional diffusion equation is a suitable representation of diffusion in disordered media with dissipative processes such as trapping or recombination involving an initial exponential distribution either in the energy or spatial axis. The fractional decay does not consider explicitly the starting excitation and ultrashort time-scale relaxation that forms the initial exponential distribution, and therefore it cannot be extrapolated to t = 0.

Non-Poisson dichotomous noise: higher-order correlation functions and aging

We study a two-state symmetric noise, with a given waiting time distribution psi (tau) , and focus our attention on the connection between the four-time and two-time correlation functions. The transition of psi (tau) from the exponential to the nonexponential condition yields the breakdown of the usual factorization condition of high-order correlation functions, as well as the birth of aging effects. We discuss the subtle connections between these two properties and establish the condition that the Liouville-like approach has to satisfy in order to produce a correct description of the resulting diffusion process.

Aging and rejuvenation with fractional derivatives

We discuss a dynamic procedure that makes fractional derivatives emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first moment, and divergent second moment, namely, with the power index mu in the interval 2<mu<3 , yield a generalized master equation equivalent to the sum of an ordinary Markov contribution and a fractional derivative term. We show that the order of the fractional derivative depends on the age of the process under study. If the system is infinitely old, the order of the fractional derivative, o , is given by o=3-mu . A brand new system is characterized by the degree o=mu-2 . If the system is prepared at time - t(a) <0 and the observation begins at time t=0 , we derive the following scenario. For times 0<t<< t(a) the system is satisfactorily described by the fractional derivative with o=3-mu . Upon time increase the system undergoes a rejuvenation process that in the time limit t>> t(a) yields o=mu-2 . The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative.