Soft Ferromagnetic Materials Research Articles

Peculiarities of Hematite Reduction Using Waste Activated Sludge (WAS) Carbonization Products

In the present study, XRD, SEM/EDS, Raman, EMR/EPR spectroscopy, and vibrating sample magnetometry (VSM) were used to analyze the reduction of hematite by the carbonization products of waste activated sludge (WAS) at 500–1000 °C. The reduction process includes the following steps: α-Fe2O3 → Fe2O3 + Fe3O4 (Ttr~500 °C) → Fe3O4 (Ttr~600–700 °C) → FeO → Feamorph. (Ttr~1000 °C). The prevalence of certain phase compositions at different hematite reduction temperatures makes it possible to predict the areas viable for the application of reduced oxides: adsorbents (after Ttr~500 °C) → soft ferromagnetic materials (after Ttr~600–700 °C) → electrically engineered amorphous iron (after Ttr~1000 °C).

Effect of magnetic field on macroscopic hysteresis and microscopic magnetic domains for different ferromagnetic materials

The macroscopic hysteresis of ferromagnetic materials is influenced by their microscopic magnetic domains, establishing a correspondence between the two that enables a fundamental understanding of the magnetization process of ferromagnetic materials. Based on this, this study employs the phase field method to investigate the macroscopic hysteresis and microscopic magnetic domain evolution of soft, hard, and rectangular ferromagnetic materials. The hysteresis loops of the three materials present narrow, wide and rectangular characteristics respectively, which are quantitatively consistent with the experimental data. Subsequently, the influence of the magnetic field on the macroscopic hysteresis is analyzed. The results show that the hysteresis loop area increases when the magnetic field amplitude increases. Furthermore, the proportion of each type of energy under different magnetic fields is discussed, the dominant energy terms are generally consistent for the same category of ferromagnetic materials. Additionally, the corresponding magnetic domain evolution under different magnetic field is displayed, with soft magnetic materials exhibiting no vortex structures, hard ferromagnetic materials presenting a 45° vortex structure, and rectangular magnetic material showcasing a 90° vortex structure. Finally, the asymmetric characteristics of the major hysteresis loop and minor hysteresis loops are discussed. The change of the magnetic field path leads to the corresponding change of the major and minor hysteresis loops. At the same magnetic field, distinctions in magnetic domain configurations between the major and minor hysteresis loops are evident.

Entanglement between Micro-Magnetism, electromagnetism and the Tensor Magnetic Phase Theory (TMPT) – Symmetry, conservation and invariance laws analysis at low frequency

Ferromagnetic materials show magnetic structures with domains and walls. Thanks to decades of research regarding the origin and behaviour of magnetic domains, we now possess a general foundation which has been verified experimentally in single crystals and powders. The governing equations at the microscopic scale were built in the 1960 s when Brown published calculations of the magnetic moments distribution inside domain walls. This micromagnetic theory uses the so called LLG ‘Landau-Lifshitz-Gilbert’ equation and can include the damping effects. The LLG equation requires a coupling to the field derived from energy contributions: exchange, anisotropy, magnetostriction, stray-field … and the anti-eddy field. At the macroscopic scale, such behaviours are lumped in a homogenized magnetization law for the electromagnetic field equations inside larger polycrystals. Therefore, the inhomogeneous magnetic material nature is always ignored. The Tensor Magnetic Phase Theory (TMPT) describes the magnetic structure and the magnetization thanks to a tensor variable at a mesoscopic scale. The material structuring is then explained thanks to an energy balance which will be discussed.This paper presents the results of investigations on entangled relationships between the micromagnetic theory, the electromagnetic theory and what is called the Tensor Magnetic Phase Theory (TMPT), which statistically describes the magnetic structures of soft ferromagnetic materials. It examines a connection between the TMPT and the LLG by deriving the main energy terms. Then, the TMPT must stay compatible and coupled to the Maxwell equations at low frequencies with volume and surface connections. Additionally, this paper investigates the way to derive the domains structuring and magnetization laws through the Lagrange principle and the corresponding conservation laws with invariants linked to the Nœther theorem. Finally, the TMPT must be discussed while checking its coherence and formulation when changing the reference frame.

Magnetostrictive energy conversion ability of Iron Cobalt Vanadium alloy sheet: Experimental and theoretical evaluation

The increasing demand for autonomous devices has made the concept of energy harvesting a significant industrial and academic point of interest. In this domain, an ideal magnetostrictive material for converting mechanical vibrations into electrical energy in a cost-effective way (i.e. competing with the price of primary batteries) remains to be determined. Iron-Cobalt-Vanadium (Permendur, Co49-Fe49-V2) is a promising candidate: it is a soft ferromagnetic material with high magnetization saturation, high magnetostrictive coefficients, low price, and good availability. In this study, the experimental magnetic characterization and simulation of Permendur sheets under tensile stress were performed, and their energy conversion capabilities were assessed. The conversion ability was predicted using thermodynamic Ericsson cycles from reconstructed anhysteretic curves. A maximum of 10.45 mJ cm−3 energy density was obtained under a tensile stress of 480 MPa and a magnetic excitation of 5.5 kA m−1. Then, an additional estimation was proposed to account for the hysteresis losses. For this, major hysteresis loops at different stress levels were considered, yielding an energy density of 3.52 mJ cm−3. Finally, experimental Ericsson cycles were performed to prove the feasibility of the conversion and corroborate the energy level predictions.

Attraction between like-poles of two magnets mediated by a soft ferromagnetic material

Attraction between like-poles of two magnets mediated by a soft ferromagnetic material

Optical, microstructural, and magnetic hyperthermia properties of green-synthesized Fe3O4/carbon dots nanocomposites utilizing Moringa oleifera extract and watermelon rinds

Cancer therapy with targeted-localized heating is one of the breakthroughs in the biomedical field. The incorporation of magnetic nanoparticles and fluorescent nanoparticles was one of the crucial issues for magnetic hyperthermia applications. This research investigates the magnetic hyperthermia properties of green-synthesized Fe3O4/CDs. Fe3O4 nanoparticles were synthesized using the coprecipitation method with Moringa oleifera leaf extract as a reducing agent and stabilizer. In contrast, CDs were synthesized using a hydrothermal method with watermelon rind waste as a carbon source. X-ray diffraction analysis confirmed the presence of cubic inverse spinel and a reduction in crystallite size with increasing CDs concentration. Transmission electron microscopy revealed particle size distributions of 9.7 nm for Fe3O4 and 7.5 nm for Fe3O4/CDs. Scanning electron microscopy showed that CDs were distributed on the surface of Fe3O4. The detection of characteristic FeO, CC, CO, and CO-C bonds indicated the presence of CDs on the surface of Fe3O4. Ultraviolet-visible spectroscopy spectra absorption peaks at 282 nm for CDs and 193 nm for Fe3O4. Photoluminescence spectra indicated shifts from 509 to 505 nm in excitation wavelength for both CDs and Fe3O4/CDs, situated within the green region of visible light. The vibrating sample magnetometer revealed that the nanocomposites displayed characteristics of soft ferromagnetic materials. Furthermore, the specific absorption rate (SAR) value of Fe3O4/CDs was found to be dependent on magnetization. The SAR value of Fe3O4/CDs decreased as the concentration of CDs increased, and the frequency and strength of the AMF increased. Therefore, these results can promote Fe3O4/CDs nanocomposites as a promising candidate for magnetic hyperthermia applications.

DESIGNING PROGRAMMABLE FERROMAGNETIC SOFT METASTRUCTURES FOR MINIMALLY INVASIVE ENDOVASCULAR THERAPY.

Minimally invasive endovascular therapy (MIET) is an innovative technique that utilizes percutaneous access and transcatheter implantation of medical devices to treat vascular diseases. However, conventional devices often face limitations such as incomplete or suboptimal treatment, leading to issues like recanalization in brain aneurysms, endoleaks in aortic aneurysms, and paravalvular leaks in cardiac valves. In this study, we introduce a new metastructure design for MIET employing re-entrant honeycomb structures with negative Poisson's ratio (NPR), which are initially designed through topology optimization and subsequently mapped onto a cylindrical surface. Using ferromagnetic soft materials, we developed structures with adjustable mechanical properties called magnetically activated structures (MAS). These magnetically activated structures can change shape under noninvasive magnetic fields, letting them fit against blood vessel walls to fix leaks or movement issues. The soft ferromagnetic materials allow the stent design to be remotely controlled, changed, and rearranged using external magnetic fields. This offers accurate control over stent placement and positioning inside blood vessels. We performed magneto-mechanical simulations to evaluate the proposed design's performance. Experimental tests were conducted on prototype beams to assess their bending and torsional responses to external magnetic fields. The simulation results were compared with experimental data to determine the accuracy of the magneto-mechanical simulation model for ferromagnetic soft materials. After validating the model, it was used to analyze the deformation behavior of the plane matrix and cylindrical structure designs of the Negative Poisson's Ratio (NPR) metamaterial. The results indicate that the plane matrix NPR metamaterial design exhibits concurrent vertical and horizontal expansion when subjected to an external magnetic field. In contrast, the cylindrical structure demonstrates simultaneous axial and radial expansion under the same conditions. The preliminary findings demonstrate the considerable potential and practicality of the proposed methodology in the development of magnetically activated MIET devices, which offer biocompatibility, a diminished risk of adverse reactions, and enhanced therapeutic outcomes. Integrating ferromagnetic soft materials into mechanical metastructures unlocks promising opportunities for designing stents with adjustable mechanical properties, propelling the field towards more sophisticated minimally invasive vascular interventions.

A Review of Additive Manufacturing of Soft Magnetic Materials in Electrical Machines

This paper presents a review of the main advantages and challenges of Additive Manufacturing (AM) applied in the production of soft magnetic components for electrical machines. Firstly, a general introduction about additive manufacturing is made, considering all of its possibilities of application, then the authors focused on the electrical machine application field, in particular the AM of soft ferromagnetic materials. The soft ferromagnetic materials are fundamental for the production of electrical machines, and currently, there are more and more requests for designed ad hoc geometries, which can be difficult to produce with conventional manufacturing technologies. With this purpose, AM can be used to produce the desired geometries.

A Numerical Comparison between Preisach, J-A and D-D-D Hysteresis Models in Computational Electromagnetics

The incorporation of hysteresis models in computational electromagnetic software is of paramount importance for the accurate prediction of the ferromagnetic devices’ performance. The Preisach and Jiles-Atherton (J-A) models are frequently used for this purpose. The former is more accurate and can represent a broad range of magnetic materials, but it is computationally expensive. The latter is more efficient but can accurately model only soft ferromagnetic materials. In this paper, a recently proposed hysteresis model, referred to as the D’Aloia-Di Francesco-De Santis (D-D-D) model, is shown to have the best trade-off between accuracy and computational burden. For the first time, a numerical comparison between the Preisach, J-A and D-D-D models is provided for a large class of hysteresis loops including soft, semi-hard and hard ferromagnetic materials.

Application of the modified Rayleigh model in the mathematical analysis, of Permalloy minor loops

The descending and ascending branches of magnetic minor loops measured for a Permalloy sample were fitted with polynomials of varying degrees. The aim was to test whether the fitting method previously successfully applied to the minor loops of a soft and hard ferromagnetic material is applicable to a very soft magnet. The quality of the fits was very good, but the resulting polynomial coefficients revealed asymmetrical minor loops, independent of the polynomial degree considered in the numerical calculations. The asymmetry was attributed to a combination of factors: the magnetic softness and the different magnetic responses of the sample to the positive and negative directions of the applied field, the later possibly originating from defects in the specimen disk shape. Some fitted coefficients were directly identified with important magnetic parameters such as remanence and permeability, while the coercivity was calculated from a set of coefficients. All of them were shown to be more confident than those values usually obtained by direct interpolation on the experimental curves. The remanence, coercivity and maximum magnetization were traced as a function of the maximum magnetic field applied to the minor loops, and the resulting curves were fitted using the complementary function of the Hill equation. These fits were performed considering only points extracted from some initial minor loops, leading to saturation values, including the saturation magnetization for this sample. The same numerical treatment was applied to the hysteresis area and this fit was equally reliable, with the resulting Hill parameters showing a correlation with the magnetic softness of the Permalloy.

The impact of NiO on crystallization and thermo-magnetic properties of Li2O–NiO–P2O5 glasses as new magnetic materials

The main goal of this study is to comprehend the effect of the composition change on crystallization characteristics, thermal, physical, and magnetic properties of glasses with the general formula (35 − x) Li2O − (5 + x) NiO − 60 P2O5 (x = 5, 10, and 15 mol%). The XRD results indicated that Li4P4O12, Li6P6O18, Ni2P2O7, Ni2P4O12, and Li3P crystalline phases were formed in the glass ceramics. The NiO/Li2O replacement led to an increase in the coherence of the glass structure. Density values for the glass and glass–ceramic samples ranged from 2.38 to 2.49 g/cm3 and from 2.41 to 2.53 g/cm3, respectively. Coefficients of thermal expansion CTE of the glasses were varied from 157 × 10−7 to 96 × 10−7 °C over the temperature range of 25–250 °C. However, CTE of the glass–ceramic was ranged from 183 × 10−7 to 129 × 10−7 °C over the temperature range of 25–500 °C. VSM analysis for the glasses revealed that the NiO/Li2O replacement led to a decrease in the magnetic parameters of the materials. Contrariwise from VSM hysteresis loops of glass–ceramic, it was found that the magnetic parameters increased with the addition NiO instead of Li2O. The results indicated a great potential of the prepared glasses and glass ceramics as promising soft ferromagnetic materials that can be used in different magnetic applications.

Investigation of soft magnetic material cores in transcranial magnetic stimulation coils and the effect of changing core shapes on the induced electric field in small animals

Transcranial magnetic stimulation (TMS) is a safe, effective and non-invasive treatment for several psychiatric and neurological disorders. Lately, there has been a surge in research utilizing this novel technology in treating other neurological and psychiatric ailments. The application of TMS on several neurological disorders requires the induced electric and magnetic fields to be focused and targeted to a small region in the brain. TMS of a focal cortical territory will ensure modulation of specific brain circuitry without affecting unwanted surrounding regions. This can be achieved by altering the properties of the magnetic core material used for the TMS system. In this study, soft ferromagnetic materials having high permeability, high saturation magnetization and low coercivity have been investigated as TMS coil cores in finite element simulations. Also, magnetic field measurements have been carried out using different cores in the TMS coil. Finite element analysis of the rat head model is carried out using Sim4life software while investigating variations associated with changing the ferromagnetic core material and shape in the coil. Materials proposed for the analysis in this study include Iron Cobalt Vanadium alloy (Fe-Co-V) also known as Permendur, Carbon Steel (AISI 1010) and Manganese Zinc ferrites (MnZn ferrites). Simulation results indicated significant magnetic field distribution variation when introducing a ferromagnetic core in TMS coil, concentrating the magnetic field to the targeted region in the rat head model without stimulating adjacent regions. It was observed that the v-tip sharpened core attained the highest magnetic field and best focality among other cores in simulations and experimentally.

Analytical solutions of the magnetic and stress fields on a soft ferromagnetic material containing a smooth rigid inclusion in two-dimensional space

The analytical investigations on a rigid inclusion embedded in a soft ferromagnetic material in two-dimensional space subjected to uniform magnetic load at infinity have been carried out, with the magnetic insulation boundary of the rigid inclusion. In our study, explicit solutions of the magnetic and stress fields are obtained based on linear magnetoelastic theory and complex variable theory. The relative rigid-body displacement of the rigid inclusion is considered to ensure the boundary condition and constraint satisfied accurately. It is found that the analytic solutions of Kolosov-Muskhelishvili (K-M) potentials have a compact form when the shape of rigid inclusion is characterized by Laurent polynomial with finite terms. As examples, the rigid-body displacement, magnetic flux and stress on the boundary of elliptic and polygonal rigid inclusions are analyzed, respectively. Our results show that the rigid-body displacement caused by a non-strong magnetic load is small, while the orientation and contour curvature of the rigid inclusions have significant effects on the distribution of the stress and magnetic flux fields. In addition, the maximum magnetic flux and maximum stress do not always occur at the maximum curvature point. For polygonal rigid inclusions, the orientations in which the extreme concentration is obtained are closely related to the number of tips of the inclusions.

Experimental and Theoretical Investigation of the Synthesis, Electronic and Magnetic Properties of MnFe2O4 Spinel Ferrite

MnFe2O4 ferrite nanoparticle was synthesized via the sol–gel method, and structural, morphology and magnetic characteristics were investigated. X-ray diffraction analysis showed that the synthesized sample was in a single phase with a spinel-ferrite-like structure (space group Fd-3m). The scanning electron microscopy displayed homogenous spherical grains with an agglomeration of the particles. The chemical composition determined by energy-dispersive spectroscopy shows the absence of any impurities. To understand the role of magnetic interaction in MnFe2O4 spinel ferrites, the structural and magnetic properties of MnFe2O4 have been explored theoretically. Based on the first-principles methods via density functional theory and Monte Carlo simulations, the magnetic hysteresis cycle has been plotted. Using the generalized gradient and GGA-PBE approximation in the full-potential linearized augmented plane wave (FP-LAPW) method, the exchange coupling interactions between magnetic elements and local magnetic moment were evaluated. Furthermore, the theoretical magnetic properties of MnFe2O4 were found to match the experimental ones. They both revealed that MnFe2O4 is a soft ferromagnetic material. The theoretical curve of magnetization versus temperature indicates that the transition occurred at Tc = 580.0 K. This was also in good agreement with the experimental Curie temperature.

The High-Temperature Soft Ferromagnetic Molecular Materials Based on [W(CN)6(bpy)]2−/− System

The synthesis of molecular materials with magnetic properties, in particular ferromagnetic properties, has been the subject of interest in coordination chemistry for decades. In the last three decades, research has accelerated, as it has emerged that creating bridging systems based on cyanido ligands is a good and relatively simple way to create complex polymer structures exhibiting magnetic properties. Based on many years of personal experience in the field of the synthesis of polycyanido systems, supported by comprehensive structural analysis, a simple method of transforming cyanido complexes into soft ferromagnetic materials with a Curie temperature (TC) higher than the thermal decomposition temperature, usually above 150 °C has been developed. Two soft ferromagnetic materials based on zinc and cadmium hexacyanido salts in the system with [W(CN)6(bpy)]2−/− anions are presented. The crystal structures (X-ray single crystal as well as XRD) of the precursors and the properties of the ferromagnetic materials are discussed. Most importantly, a patented method of synthesizing this type of material, based on which we obtain more than 80 soft, high-temperature ferromagnetic compounds, which proves the wide spectrum of this method, is also presented.

Computing Frequency-Dependent Hysteresis Loops and Dynamic Energy Losses in Soft Magnetic Alloys via Artificial Neural Networks

A neural network model to predict the dynamic hysteresis loops and the energy-loss curves (i.e., the energy versus the amplitude of the magnetic induction) of soft ferromagnetic materials at different operating frequencies is proposed herein. Firstly, an innovative Fe-Si magnetic alloy, grade 35H270, is experimentally characterized via an Epstein frame in a wide range of frequencies, from 1 Hz up to 600 Hz. Parts of the dynamic hysteresis loops obtained through the experiments are involved in the training of a feedforward neural network, while the remaining ones are considered to validate the model. The training procedure is accurately designed to, firstly, identify the optimum network architecture (i.e., the number of hidden layers and the number of neurons per layer), and then, to effectively train the network. The model turns out to be capable of reproducing the magnetization processes and predicting the dynamic energy losses of the examined material in the whole range of inductions and frequencies considered. In addition, its computational and memory efficiency make the model a useful tool in the design stage of electrical machines and magnetic components.

Double-doping Effect on Structural and Magnetic Properties of Perovskite Lanthanum Manganite

Polycrystalline samples La(2+4x)/3Sr(1–4x)/3Mn1−xCuxO3 (x = 0, 0.1, 0.15 and 0.25) with double doping in the A site and B site were successfully prepared by the conventional solid-state reaction method with heat treatment up to 1300 °C. All the obtained samples consist of single perovskite phase with a rhombohedral symmetry and space group. The (a) lattice parameter and the volume of the unit cell were found to increase with the increase of Cu doping level (x). The morphology of the prepared samples was studied by scanning electron microscope imaging, and the stoichiometry of the compositions was confirmed. The hysteresis loops at room temperature for x = 0, 0.1 and 0.15 samples are characteristic for soft ferromagnetic materials, with a decrease of the ferromagnetic interaction and a decrease of the magnetic moment with the increase of Cu doping level. No ferromagnetic behavior was detected at room temperature for the x = 0.25 sample. The question of the valency of Cu ions was discussed during the present study. Our results are best interpreted by assuming the coexistence of Cu2+ and Cu3+ ions.

Flux Method Growth and Structure and Properties Characterization of Rare-Earth Iron Oxides Lu1−xScxFeO3 Single Crystals

Perovskite rare-earth ferrites (REFeO3) have attracted great attention for their high ferroelectric and magnetic transition temperatures, strong magnetoelectric coupling, and electric polarization. We report on the flux method growth of rare-earth iron oxide Lu1−xScxFeO3 single crystals through a K2CO3-B2O3-Bi2O3 mixture as a flux solution, and give a detailed characterization of the microstructure, magnetism, and ferroelectric properties. X-ray diffraction (XRD) and energy dispersive X-ray spectroscopy (EDX) measurements revealed that the obtained single crystals can be designated to three different crystal structures of different chemical compositions, that is, Lu0.96Sc0.04FeO3 (perovskite phase), Lu0.67Sc0.33FeO3 (hexagonal phase), and Lu0.2Sc0.8FeO3 (bixbyite phase), respectively. Magnetic measurements indicate that the perovskite Lu0.96Sc0.04FeO3 is an anisotropic hard ferromagnetic material with a high Curie transition temperature, the bixbyite Lu0.2Sc0.8FeO3 is a low temperature soft ferromagnetic material, and the hexagonal Lu0.67Sc0.33FeO3 exhibits multiferroic properties. Lu0.67Sc0.33FeO3 possesses a weak ferromagnetic transition at about 162 K. We further investigate the ferroelectric domain structures in hexagonal sample by scanning electron microscope and the characteristic atomic structures in ferroelectric domain walls by atomically resolved scanning transmission electron microscope. Our successful growth of perovskite Lu1−xScxFeO3 single crystals with distinct crystal structures and stochiometric Lu-Sc substitutions is anticipated to provide a useful ferrites system for furthering exploitation of their multiferroic properties and functionalities.

Characterizing the Fatigue Behavior of Wrought Fe–Co–2V Using Experimental Techniques

Abstract Fe–Co–2V (Hiperco®2 equivalent) is a soft ferromagnetic material that is commonly used for electrical components that require robust magnetic performance. Despite the excellent magnetic properties of Fe–Co–2V, it often exhibits low strength, ductility, and workability due to an ordered B2 microstructure. The mechanical properties exhibit considerable dependency on grain size and degree of order, which are influenced by processing methods. A thorough understanding of Fe–Co–2V’s fatigue performance is required to predict mechanical reliability under operating loads; however, limited fatigue data currently exist for Fe–Co alloys. This work characterizes the fatigue properties of wrought Fe–Co–2V through strain-controlled fatigue testing and fractography. Young’s modulus, ultimate strength, and yield stress were determined through monotonic tension tests. The fatigue behavior was quantified using fully reversed, strain-controlled fatigue testing for applied strain amplitudes ranging from 0.10% to 1.00%. Subsequently, the Coffin–Manson strain-life curve was fit to the experimental data. Failure mechanisms were investigated through fractography with a scanning electron microscope. Inspection of the failure surfaces revealed that crack initiation occurred at defects located on or near the specimen surface with a localized region of crack propagation prior to the transgranular cleavage fracture. Additionally, two material models were calibrated from the experimental static and cyclic experimental testing. The characterization of the fatigue behavior of wrought Fe–Co–2V presented herein will aid in the fatigue analysis of Fe–Co–2V components and the development of analytical fatigue modeling methodologies.

Investigation of structural, mechanical, magnetic properties and hysteresis modelling of Dawasir meteorite

The experimental reports of structural modification, mechanical and magnetic properties of the Saudi meteorite were studied and discussed. The meteorite specimen was studied using the following different characterization techniques like X-ray powder diffraction, scanning electron microscopy, backscattered electron imaging, Energy Dispersive X-Ray Analysis, magnetic property measurements and hardness testing (Rockwell, Vicker, Brinell). The compositional analysis of the meteorite revealed that the sample was mainly composed of an iron-nickel alloy role. The hardness testing showed that the meteorite consisted of a soft material with 22.5 HRC Rockwell hardness. The magnetic measurements of the meteorite specimen indicated that it is a soft ferromagnetic material. The magnetic saturation (Ms) of 0.701 emu/g was observed for the saturation field of (Hs) = 5025 Oe. A sensitive GM counter showed no traces of radioactivity. Mathematical modelling of hysteresis data is also performed using already available models. Few modified models are also proposed and evaluated for the goodness of fit. Finally, most suitable model is proposed as bi-exponential model based on the goodness of fit parameters.