Apparent Power Law Research Articles

Theoretical study of kinetic arrest, shear elastic modulus, and yielding in simple biphasic colloidal mixtures.

Based on integrating microscopic statistical mechanical theories for structure and ideal kinetic arrest at the naive mode coupling level, we study dynamic localization, the linear elastic shear modulus, applied stress induced modulus softening, and the absolute yielding of simple biphasic binary mixtures composed of equal diameter hard and attractive spheres. The kinetic arrest map is a rich function of total packing fraction, strength of attraction, and mixture composition. The gel to attractive ideal glass transition, the degree of glass melting re-entrancy, and the crossover boundary separating repulsive glasses from attractive glasses vary with the mixture composition. Exponential and/or apparent (high) power law dependences of the elastic shear modulus on the total packing fraction are predicted with effective exponents or exponential prefactors that are sensitive to mixture composition and location in the kinetic arrest map. An analysis of the effective mean square force on a tagged particle that induces dynamic localization reveals a compensation effect between structural correlations and degree of particle localization, resulting in the emergence of a weaker dependence of the shear modulus on mixture composition at very high attraction strengths. Based on a microrheologically inspired formulation of how external stress weakens particle localization and the shear modulus, we analyze mechanical-induced modulus softening and absolute yielding, defined as a discontinuous solid-to-fluid stress-induced transition that can occur in either one or two steps. Estimates of the corresponding yield strains predict that the binary mixture becomes more brittle with increasing sticky particle composition and/or attraction strength.

Quantifying stream-loss recovery in a spring using dual-tracer injections in the Snake Creek drainage, Great Basin National Park, Nevada, USA

Simultaneous short-pulse injections of two tracers (sodium bromide [Br–] and fluorescein dye) were made in a losing reach of Snake Creek in Great Basin National Park, Nevada, USA, to evaluate the quantity of stream loss through permeable carbonates that resurfaces at a spring approximately 10 km down drainage. A revised hydrogeologic cross section for a possible flow path of the infiltrated Snake Creek water is presented, and the results may inform water management in the region. First arrival and peak concentration of the two tracers occurred at 9.5 and 12.7 days after injection, respectively. Fracture transport simulations indicate that Br– preferentially diffuses into immobile regions of the aquifer, and this diffusive flux is likely responsible for the major differences in the breakthrough curves. When considering the diffusive tracer flux, total apparent Br– and fluorescein dye recoveries were 16.9–22.1% and 21.7–24.3%, respectively. These findings imply that consideration of diffusive flux and long-term monitoring in fracture-dominated flow may support accurate quantification of tracer recovery. In addition, the apparent power law slopes of the breakthrough tails for both tracers were steeper at early times than have been attributed to heterogeneous advection or channeling in meter-scale tests, but the late-time Br– power law slope becomes less steep than has been attributed to diffusive exchange. These deviations may reflect fracture transport patterns that occur at larger scales.

Microscopic activated dynamics theory of the shear rheology and stress overshoot in ultradense glass-forming fluids and colloidal suspensions

We formulate a particle and force level, activated dynamics-based statistical mechanical theory for the continuous startup nonlinear shear rheology of ultradense glass-forming hard sphere fluids and colloidal suspensions in the context of the elastically collective nonlinear Langevin equation approach and a generalized Maxwell model constitutive equation. Activated structural relaxation is described as a coupled local-nonlocal event involving caging and longer range collective elasticity which controls the characteristic stress relaxation time. Theoretical predictions for the deformation-induced enhancement of mobility, the onset of relaxation acceleration at remarkably low values of stress, strain, or shear rate, apparent power law thinning of the steady-state structural relaxation time and viscosity, a nonvanishing activation barrier in the shear thinning regime, an apparent Herschel–Buckley form of the shear rate dependence of the steady-state shear stress, exponential growth of different measures of a yield or flow stress with packing fraction, and reduced fragility and dynamic heterogeneity under deformation were previously shown to be in good agreement with experiments. The central new question we address here is the defining feature of the transient response—the stress overshoot. In contrast to the steady-state flow regime, understanding the transient response requires an explicit treatment of the coupled nonequilibrium evolution of structure, elastic modulus, and stress relaxation time. We formulate a new quantitative model for this aspect in a physically motivated and computationally tractable manner. Theoretical predictions for the stress overshoot are shown to be in good agreement with experimental observations in the metastable ultradense regime of hard sphere colloidal suspensions as a function of shear rate and packing fraction, and accounting for deformation-assisted activated motion appears to be crucial for both the transient and steady-state responses.

Accuracy of Rockfall Volume Reconstruction from Point Cloud Data—Evaluating the Influences of Data Quality and Filtering

Rockfall processes are now commonly studied through monitoring campaigns using repeat lidar scanning. Accordingly, several recent studies have evaluated how the temporal resolution of data collection and various data-processing decisions can influence the apparent rockfall volumes estimated using typical rockfall database creation workflows. However, there is a lack of studies that consider how data quality and associated data-processing decisions influence rockfall volume estimation. In this work, we perform a series of tests based on an existing reference rockfall database from the Front Range of Colorado, USA, to isolate the influences of data resolution (point spacing), individual point precision, and the filter threshold applied to change results, on the volume estimates obtained for rockfalls. While the effects of individual point precision were found to be limited for typical levels of gaussian noise (standard deviation per coordinate direction ≤ 0.02 m), data resolution and change filter threshold were found to have systematic impacts on volume estimates, with the volume estimates for the smallest rockfalls decreasing substantially with increases in point spacing and change filter threshold. Because these factors disproportionately impact volume estimates for smaller rockfalls, when these factors change, the slope of the apparent power law that describes the relative frequency-volume distribution of rockfalls changes. Evidence is presented that suggests that this phenomenon can explain discrepancies between power law slopes presented in the literature based on studies focused on different scales of rockfall activity. Overall, this study demonstrates the impacts of raw data attributes on rockfall volume estimation and presents an additional effect that tends to bias rockfall frequency–magnitude power law relationships towards underestimation of the relative prevalence of small rockfalls.

Rheology of emulsions with polymer solutions as the continuous phase

We investigate the effect of the addition of polymers on the flow properties of emulsions. Surfactant-stabilised 80 v% oil-in-water emulsions, exhibiting a yield stress, with either xanthan gum (a stiff, rodlike polymer) or polyethylene oxide (PEO, a flexible, elastic polymer) in the continuous phase (concentrations between 0.005 wt% and 0.5 wt%) are prepared and investigated using shear rheology and confocal microscopy. The flow properties of the emulsions are very robust, and only at high concentrations of polymer (≥ 0.2 wt%), significant changes in the flow properties are observed. In the case of xanthan gum, higher shear stresses are measured. For high concentrations the yield stress is masked by the high zero shear viscosity and shear thinning behaviour of the xanthan gum giving rise to an apparent second power law regime. In the case of PEO, an increase in the shear thinning exponent is observed, together with a decrease in the yield stress. The interaction of PEO with the surfactant SDS (sodium dodecyl sulfate) at higher PEO concentrations strongly affects the emulsion rheology, perhaps by reducing the friction coefficient between the oil droplets.

Simulations of the solar modulation of Helium isotopes constrained by observations

The observation of galactic cosmic ray (GCR) Helium isotopes (3He2 and 4He2) at the Earth had been done precisely with the PAMELA and AMS02 space detectors and reported from July 2006 to December 2007 and from May 2011 to May 2017, respectively. These observations span time frames that include solar maximum activity and the magnetic field reversal epoch. A comprehensive, three-dimensional numerical modulation model for the transport of GCRs in the heliosphere is utilized to simulate the modulation of galactic Helium isotopes from the end of the minimum activity in the previous A < 0 cycle, through solar maximum, and toward minimum activity in the current A > 0 cycle. A particular objective is to reproduce the main features of the 3He2 to 4He2 ratio observed at rigidities between 2.15 GV and 15.3 GV. We find that the model reproduces the apparent single power law rigidity of this ratio and its time dependence in good agreement with observations. The simulated results indicate that in addition to the significant change of particle drift with solar activity, the rigidity dependence of the diffusion coefficients below 4 GV change differently before solar maximum activity than afterwards. This influences the unfolding of the modulation of GCRs with changing solar activity differently when they have different very local interstellar spectra and mass-to-charge ratios.

Refined burned-area mapping protocol using Sentinel-2 data increases estimate of 2019 Indonesian burning

Abstract. Many nations are challenged by landscape fires. A confident knowledge of the area and distribution of burning is crucial to monitor these fires and to assess how they might best be reduced. Given the differences that arise using different detection approaches, and the uncertainties surrounding burned-area estimates, their relative merits require evaluation. Here we propose, illustrate, and examine one promising approach for Indonesia where recurring forest and peatland fires have become an international crisis. Drawing on Sentinel-2 satellite time-series analysis, we present and validate new 2019 burned-area estimates for Indonesia. The corresponding burned-area map is available at https://doi.org/10.5281/zenodo.4551243 (Gaveau et al., 2021a). We show that &gt;3.11 million hectares (Mha) burned in 2019. This burned-area extent is double the Landsat-derived official estimate of 1.64 Mha from the Indonesian Ministry of Environment and Forestry and 50 % more that the MODIS MCD64A1 burned-area estimate of 2.03 Mha. Though we observed proportionally less peatland burning (31 % vs. 39 % and 40 % for the official and MCD64A1 products, respectively), in absolute terms we still observed a greater area of peatland affected (0.96 Mha) than the official estimate (0.64 Mha). This new burned-area dataset has greater reliability than these alternatives, attaining a user accuracy of 97.9 % (CI: 97.1 %–98.8 %) compared to 95.1 % (CI: 93.5 %–96.7 %) and 76 % (CI: 73.3 %–78.7 %), respectively. It omits fewer burned areas, particularly smaller- (&lt;100 ha) to intermediate-sized (100–1000 ha) burns, attaining a producer accuracy of 75.6 % (CI: 68.3 %–83.0 %) compared to 49.5 % (CI: 42.5 %–56.6 %) and 53.1 % (CI: 45.8 %–60.5 %), respectively. The frequency–area distribution of the Sentinel-2 burn scars follows the apparent fractal-like power law or Pareto pattern often reported in other fire studies, suggesting good detection over several magnitudes of scale. Our relatively accurate estimates have important implications for carbon-emission calculations from forest and peatland fires in Indonesia.

Apparent Power Laws Can Occur without Criticality.

Power laws often lead to the conclusion that self-organized criticality is at work. This is not the case, and power laws can also occur away from criticality or can occur for other reasons.

Diffusion of Short Semiflexible DNA Polymer Chains in Strong and Moderate Confinement.

In many technological applications, DNA is confined within nanoenvironments that are smaller than the size of the unconfined polymer in solution. However, the dependence of the diffusion coefficient on molecular weight and characteristic confinement dimension remains poorly understood in this regime. Here, convex lens-induced confinement (CLiC) was leveraged to examine how the diffusion of short DNA fragments varied as a function of slit height by using single-molecule fluorescence tracking microscopy. The diffusion coefficient followed approximate power law behavior versus confinement height, with exponents of 0.27 ± 0.01, 0.32 ± 0.02, and 0.42 ± 0.06 for 692, 1343, and 2686 base pair chains, respectively. The weak dependence on slit height suggests that shorter semiflexible chains may adopt increasingly rodlike conformations and therefore experience weaker excluded-volume interactions as the confinement dimension is reduced. The diffusion coefficient versus molecular weight also exhibited apparent power law behavior, with exponents that varied slightly (from -0.89 to -0.85) with slit height, consistent with hydrodynamic interactions intermediate between Rouse and Zimm model predictions.

A fatigue model based on M-integral in notched elastic–plastic material

In this paper, an innovative fatigue model is investigated based on the concept of M-integral in notched elastic–plastic material. The contribution of notch and plastic zone damage to the lifetime of material are taken into account in the present fatigue model. The new form of fatigue damage evolution rate (dAD/dN) and fatigue driving force (ΔM) are introduced, where AD deontes the equivalent damage area of notch, plastic zone and cracks, N is the number of cycles, and ΔM corresponds to the M-integral range per load cycle. The fatigue experimental evaluations of a typically elastic–plastic material (e.g., No. 45 steel) with a circular notch have been carried out to validate the effectiveness of the present fatigue model. For experimental study, the change of the total potential energy (CTPE) is introduced to measure the value of M-integral. The results demonstrate that dAD/dN shows an apparent power law relation with ΔM in notched elastic–plastic material. The slope n and intercept λ of lg(dAD/dN)-lg(ΔM) curve has linear correlation with the initial notch radius R, but not with applied stress σ. Moreover, the model can clearly describe the two-stage process from the initiation of microcracks to the growth of macrocracks in notched body. It is concluded that the proposed fatigue model based on M-integral can accurately predict the fatigue lifetime of the notched elastic–plastic material.

Shelf Seas Baroclinic Energy Loss: Pycnocline Mixing and Bottom Boundary Layer Dissipation

AbstractObservations of turbulent kinetic energy dissipation rate () from a range of historical shelf seas data sets are viewed from the perspective of their forcing and dissipation mechanisms: barotropic to baroclinic tidal energy conversion, and pycnocline and bottom boundary layer (BBL) dissipation. The observations are placed in their geographical context using a high resolution numerical model (NEMO AMM60) in order to compute relevant maps of the forcing (conversion). We analyze, in total, 18 shear microstructure surveys undertaken over a 17 year period from 1996 to 2013 on the North West European shelf, consisting of 3,717 vertical profiles of shear microstructure: 2,013 from free falling profilers and 1,704 from underwater gliders. A robust positive relationship is found between model‐derived barotropic to baroclinic conversion, and observed pycnocline integrated . A fitted power law relationship of approximately one‐third is found, giving a simple new parameterization. We discuss reasons for this apparent power law and where the “missing” dissipation may be occurring. We conclude that internal wave related dissipation in the bottom boundary layer provides a robust explanation and is consistent with a commonly used fine‐scale pycnocline dissipation parameterization.

Influence of thermal fluctuations on active diffusion at large Péclet numbers

Three-dimensional Wavelet Monte Carlo dynamics simulations are used to study the dynamics of passive particles in the presence of microswimmers—both represented by neutrally buoyant spheres—taking into account the often-omitted thermal motion alongside the hydrodynamic flows generated by the swimmers. Although the Péclet numbers considered are large, we find the thermal motion to have a significant effect on the dynamics of our passive particles and can be included as a decorrelation factor in the velocity autocorrelation with a decay time proportional to the Péclet number. Similar decorrelation factors come from swimmer rotations, e.g., run and tumble motion, and apply to both entrainment and far field loop contributions. These decorrelation factors lead to active diffusivity having a weak apparent power law close to Pe0.2 for small tracer-like particles at Péclet numbers appropriate for E. coli swimmers at room temperature. Meanwhile, the reduced hydrodynamic response of large particles to nearby forces has a corresponding reduction in active diffusivity in that regime. Together, they lead to a non-monotonic dependence of active diffusivity on particle size that can shed light on similar behavior observed in the experiments by Patteson et al. [“Particle diffusion in active fluids is non-monotonic in size,” Soft Matter 12, 2365–2372 (2016)].

The role of collective elasticity on activated structural relaxation, yielding, and steady state flow in hard sphere fluids and colloidal suspensions under strong deformation.

We theoretically study the effect of external deformation on activated structural relaxation and aspects of the nonlinear mechanical response of glassy hard sphere fluids in the context of elastically collective nonlinear Langevin equation theory. This microscopic force-based approach describes activated relaxation as a coupled local-nonlocal event involving caging and longer range collective elasticity, with the latter becoming more important and ultimately dominant with increasing packing fraction under equilibrium conditions. The central new question we address is how this physical picture of activated relaxation, and the relative importance of local caging vs collective elasticity physics, depends on external deformation. Theoretical predictions are presented for deformation-induced enhancement of mobility, the onset of relaxation speed up at remarkably low values of stress, strain, or shear rate, apparent power law thinning of the steady state structural relaxation time and viscosity, a non-vanishing activation barrier in the shear thinning regime, an apparent Herschel-Bulkley form of the rate dependence of the steady state shear stress, exponential growth of different measures of a dynamic yield or flow stress with the packing fraction, and reduced fragility and dynamic heterogeneity under deformation. The results are contrasted with experiments and simulations, and qualitative or better agreement is found. An overarching conclusion is that deformation strongly reduces the importance of longer range collective elastic effects relative to the local caging aspect for most, but not all, physical questions, with deformation-dependent fragility and dynamic heterogeneity phenomena being qualitatively sensitive to collective elasticity. Overall, nonlinear rheology is predicted to be a more local problem than quiescent structural relaxation, albeit with deformation-modified activated processes still important.

Anomalous dynamics on the ergodic side of the many-body localization transition and the glassy phase of directed polymers in random media

Using the non-interacting Anderson tight-binding model on the Bethe lattice as a toy model for the many-body quantum dynamics, we propose a novel and transparent theoretical explanation of the anomalously slow dynamics that emerges in the bad metal phase preceding the Many-Body Localization transition. By mapping the time-decorrelation of many-body wave-functions onto Directed Polymers in Random Media, we show the existence of a glass transition within the extended regime separating a metallic-like phase at small disorder, where delocalization occurs on an exponential number of paths, from a bad metal-like phase at intermediate disorder, where resonances are formed on rare, specific, disorder dependent site orbitals on very distant generations. The physical interpretation of subdiffusion and non-exponential relaxation emerging from this picture is complementary to the Griffiths one, although both scenarios rely on the presence of heavy-tailed distribution of the escape times. We relate the dynamical evolution in the glassy phase to the depinning transition of Directed Polymers, which results in macroscopic and abrupt jumps of the preferred delocalizing paths when a parameter like the energy is varied, and produce a singular behavior of the overlap correlation function between eigenstates at different energies. By comparing the quantum dynamics on loop-less Cayley trees and Random Regular Graphs we discuss the effect of loops, showing that in the latter slow dynamics and apparent power-laws extend on a very large time-window but are eventually cut-off on a time-scale that diverges at the MBL transition.

A collective elastic fluctuation mechanism for decoupling and stretched relaxation in glassy colloidal and molecular liquids.

We propose a microscopic theory for the decoupling of self-diffusion and structural relaxation in glass-forming liquids within the Elastically Collective Nonlinear Langevin Equation (ECNLE) activated dynamics framework. Our central hypothesis is that the heterogeneity relevant to this problem is static fluctuations of local density on the scale of 3-4 particle diameters and how this changes local packing correlations. These fluctuations modify the degree of dynamical cage expansion that mechanistically couples intracage large amplitude hopping and longer range collective elasticity in ECNLE theory. Decoupling only emerges in the deeply supercooled regime where the strongly temperature dependent elastic barrier becomes non-negligible relative to its noncooperative local analog. The theory makes predictions for various aspects of the decoupling phenomenon, including apparent fractional power law Stokes-Einstein behavior, that appear to be consistent with experiments and simulations on hard sphere fluids and molecular liquids. Of central importance is a microscopic connection between the barrier fluctuation variance and most probable barrier height. Sensible results are also obtained for the nonexponential stretching of a generic relaxation time correlation function and its temperature evolution. Nonuniversality can arise from the relative importance of the local and collective barriers (related to fragility) and the precise magnitude of the length scale that defines the transition from local cage to elastic physics. Comparison is made with a traplike model based on a Gaussian distribution of barriers.

Thermosolutal hydromagnetic convection of power law fluids in an enclosure with periodic active zones

A finite volume technique based double diffusive natural convection of power-law fluid in an enclosure has been studied numerically in the presence of magnetic field due to different physical parameters such as Rayleigh number (Ra), Lewis number (Le), Hartmann number (Ha) and Prandtl number (Pr). The flow is generated by the combined mechanism of a time periodic thermal and solutal gradient acts along the vertical walls. The combined buoyancy effect due to thermal and species diffusion plays a key role on the flow characterisation. The study dealt with the effect of pertinent parameters on fluid flow, heat and mass transfer in case of a laminar flow regime. The apparent power law indices and magnetic field strengths are varied using the non-Newtonian power law forms. The relevant results are presented in terms of streamlines, isotherms, isoconcentrations, average Nusselt number, Sherwood number and total entropy generation with the variation of power law index (n) from 0.2 to 1.8, Ra from 104 to 105,Ha from 0 to 20, Pr from 1 to 5 and buoyancy ratio (N) from -1 to 1. The entropy generation and Bejan number are determined to analyze the thermodynamic optimization of the conjugate double diffusive convection. It is observed that the average Nusselt and Sherwood number shows linear variation with the increase of amplitudes of the imposed sinusoidal temperature and concentration gradient and the fluctuation is found to be trivial. Also the results indicate that the power law index variation influences largely the rate of heat and mass transfer and average entropy generation. The magnetic field strength increases the strength of the fluid flow and also strongly supported by the increased values of n. It is found that with the decrement of flow behavior index n the buoyancy effect and the hydrodynamic behavior of the fluid is getting increased.

Thymic involution and rising disease incidence with age

For many cancer types, incidence rises rapidly with age as an apparent power law, supporting the idea that cancer is caused by a gradual accumulation of genetic mutations. Similarly, the incidence of many infectious diseases strongly increases with age. Here, combining data from immunology and epidemiology, we show that many of these dramatic age-related increases in incidence can be modeled based on immune system decline, rather than mutation accumulation. In humans, the thymus atrophies from infancy, resulting in an exponential decline in T cell production with a half-life of ∼16 years, which we use as the basis for a minimal mathematical model of disease incidence. Our model outperforms the power law model with the same number of fitting parameters in describing cancer incidence data across a wide spectrum of different cancers, and provides excellent fits to infectious disease data. This framework provides mechanistic insight into cancer emergence, suggesting that age-related decline in T cell output is a major risk factor.

High-energy Variability of PSR J1311-3430

Abstract We have studied the variability of the black-widow-type binary millisecond pulsar PSR J1311−3430 from optical to gamma-ray energies. We confirm evidence for orbital modulation in the weak off-pulse ≳200 MeV emission, with a peak at , following pulsar inferior conjunction. The peak has a relatively hard spectrum, extending above ∼1 GeV. XMM-Newton and Swift UV observations also show that this source’s strong X-ray flaring activity is associated with optical/UV flares. With a duty cycle of ∼7%–19%, this flaring is quite prominent with an apparent power-law intensity distribution. Flares are present at all orbital phases, with a slight preference for . We explore possible connections of these variabilities with the intrabinary shock and magnetic activity on the low-mass secondary.

On the apparent power law in CDM halo pseudo-phase space density profiles

Abstract We investigate the apparent power-law scaling of the pseudo-phase space density (PPSD) in cold dark matter (CDM) haloes. We study fluid collapse, using the close analogy between the gas entropy and the PPSD in the fluid approximation. Our hydrodynamic calculations allow for a precise evaluation of logarithmic derivatives. For scale-free initial conditions, entropy is a power law in Lagrangian (mass) coordinates, but not in Eulerian (radial) coordinates. The deviation from a radial power law arises from incomplete hydrostatic equilibrium (HSE), linked to bulk inflow and mass accretion, and the convergence to the asymptotic central power-law slope is very slow. For more realistic collapse, entropy is not a power law with either radius or mass due to deviations from HSE and scale-dependent initial conditions. Instead, it is a slowly rolling power law that appears approximately linear on a log–log plot. Our fluid calculations recover PPSD power-law slopes and residual amplitudes similar to N-body simulations, indicating that deviations from a power law are not numerical artefacts. In addition, we find that realistic collapse is not self-similar; scalelengths such as the shock radius and the turnaround radius are not power-law functions of time. We therefore argue that the apparent power-law PPSD cannot be used to make detailed dynamical inferences or extrapolate halo profiles inwards, and that it does not indicate any hidden integrals of motion. We also suggest that the apparent agreement between the PPSD and the asymptotic Bertschinger slope is purely coincidental.

Structural, dielectric, electrical and magnetic properties of CuFe2O4 nanoparticles synthesized by honey mediated sol–gel combustion method and annealing effect

In this work, CuFe2O4 nanoparticles were synthesized by natural source of glucose and fructose (i.e., honey)—mediated sol–gel auto-combustion method. Grain size, cation distribution and crystal phase were further tuned through annealing at higher temperature 500, 700, 900 and 1100 °C. The structural investigation was performed using powder X-ray Diffraction, Raman Spectroscopy, Fourier Transform Infrared Spectroscopy, Scanning Electron Microscopy, Energy Dispersive X-ray Spectroscopy and X-ray Photoelectron Spectroscopy. X-ray diffraction study confirmed the phase transformation from cubic to tetragonal as a function of annealing temperature. Magnetic properties were investigated by using vibrating sample magnetometer under an applied magnetic field of 10 kOe at room temperature. The highest value of saturation magnetization (Ms) was 26 emu/g for ferrite nanoparticles annealed at 1100 °C, whereas the lowest value was 11 emu/g for annealed at 700 °C. The highest and lowest coercivity (Hc) was 1389 and 65 Oe for ferrite nanoparticles annealed at 900 and 1100 °C, respectively. Detailed study of modulus and impedance spectroscopy revealed the contribution of grain and grain boundary on electrical transport mechanism and relaxation process. Further, dependence of relaxation time, resistance and capacitance at grain and grain boundary on grain size, cation distribution and annealing temperature was noticed. The asymmetry of peak in imaginary part of modulus spectra indicates that the relaxation process is non-Debye type. At lower frequency, ac conductivity is frequency independent, whereas, at high frequency, it follows an apparent power law, σ(ω) α ωs. Dielectric parameters (real and imaginary part, dielectric loss) with variation of frequency (1 Hz to 10 MHz) are investigated and dependence with frequency and annealing temperature is observed.