Diffusion Phenomena Research Articles

Nano friction behaviour between magnetic materials and copper considering the inter-diffusion effect

Copper, permalloy, cobalt, and silicon are the materials that have been widely utilised in magnetic devices. When the size of interest is down to the nanoscale, the inter-diffusion between certain materials becomes influential. This paper studies the nanoscale friction characteristics between frictional pairs with and without inter-diffusion properties through the atomic force microscope. The distinct evolution features of nanoscale friction force when inter-diffusion is involved are discovered experimentally, which is also confirmed through theoretical analysis. Firstly, through the thin film deposition method, four pairs of contact materials (Cu–Ni81Fe19, Si–Ni81Fe19, Cu–Co, Cu–Si) are designed for friction tests, in which diffusion occurs at the interface of Cu–Ni81Fe19 pair. Then, the effects of sliding velocity and loading force on the nano friction of each pair are measured. It is found that regardless of the diffusion phenomenon: (1) the adhesion force values exhibit a notable correlation to the values of the friction force; (2) the friction force in all four material pairs consistently increases with the growth of the normal loading force, although the growth rate may differ. In terms of the sliding velocity effect, the friction forces of immiscible materials (Si–Ni81Fe19, Cu–Co, and Cu–Si) are found to increase with the increasing sliding velocity. However, the friction force of Cu–Ni81Fe19, decreases with the increasing sliding velocity. Furthermore, a compositive friction model considering both the velocity and the normal force effect was proposed, which shows good agreement with the experimental results and explains the nano friction behaviour of both miscible and immiscible metals.

Restricted diffusion characteristics in oscillating gradient spin echo with mesoscopic phantom

In diffusion magnetic resonance imaging, oscillating gradient spin echo (OGSE) has an extremely short diffusion time if motion probing gradient (MPG) is applied to the waveform. Further, it can detect microstructural specificity. OGSE changes sensitivity to spin displacement velocity based on the MPG phase. The current study aimed to investigate the restricted diffusion characteristics of each OGSE waveform using the capillary phantom with various b-values, frequencies, and MPG phases. We performed OGSE (b-value = 300, 500, 800, 1200, 1600, and 2000 s/mm2) for the sine and cosine waveforms using the capillary phantom (6, 12, 25, 50, and 100 μm and free water) with a 9.4-T experimental magnetic resonance imaging system and a solenoid coil. We evaluated the axial and radial diffusivity (AD, RD) of each structure size. The output current of the MPG was assessed with an oscilloscope and analyzed with the gradient modulation power spectra by fast Fourier transform.In sine, the sidelobe spectrum was enhanced with increasing frequency, and the central spectrum slightly increased. The difference in RD was detected at 6 and 12 μm; however, it did not depend on the structure scale at 50 or 100 μm and free water. In cosine, the diffusion spectrum was enhanced, whereas the central spectrum decreased with increasing frequency. Both AD and RD in cosine had a frequency dependence, and AD and RD increased with a higher frequency regardless of structure size. AD and RD in either sine or cosine had no evident b-value dependence. We evaluated the OGSE-restricted diffusion characteristics. The measurements obtained diffusion information similar to the pulsed gradient spin echo. Hence, the cosine measurements indicated that a higher frequency could capture faster diffusion within the diffusion phenomena.

Conditional entropy and weak fluctuation correlation in nonequilibrium complex systems

The weak correlation between spatiotemporal fluctuations in nonequilibrium complex systems is shown to govern the fluctuation distribution, maximizing the conditional entropy associated with such fluctuations. The result is illustrated in diffusion phenomena observed in living cells. A generic feature of the weak correlation is briefly mentioned.

Grain boundary diffusion and grain boundary phase transition in tungsten in the temperature range of activated sintering

Abstract Grain boundary self- and solute (cobalt) diffusion in tungsten was found [Lee et al., Scr. Metall, 1988; Lee et al., Col. de Physique, 1990] to exhibit discontinuities in the Arrhenius behavior at the homologous temperatures of 0.36 < T/T m < 0.4 that surprisingly match the activation sintering temperature of W (T m is the melting point). In the present work, this unusual grain boundary diffusion phenomenon is discussed in terms of a fundamental grain boundary phase transition in W. The experimental data are analysed with respect to predicted segregation-induced grain boundary phase transformation. Competing co-segregation of impurity elements (carbon and phosphor) might induce a discontinuous grain boundary segregation and invoke a grain boundary phase transition which modifies the grain boundary mobilities of substitutional atoms. The improved understanding of grain boundary phase transitions is expected to provide a breakthrough in interpreting the exact mechanism of W-activated sintering.

Formation of alginate/ Alyssum homolocarpum seed gum (AHSG) microgels through electrospraying technique to encapsulate and release curcumin

This study aimed to produce microgels by electrospraying of alginate/AHSG (Alyssum homolocarpum seed gum) dispersions at different mixing ratios (1:0, 3:1, 1:1) for encapsulation of curcumin. Addition of AHSG to alginate increased the ζ-potential while it reduced the viscosity of the dispersions. Increased alginate to AHSG ratio resulted in smaller microgels and more uniform morphology. Results confirmed that curcumin was physically entrapped within the microgels matrix. As the alginate to AHSG ratio decreased, the microgel's encapsulation efficiency and thermal stability (∆H) increased from 87.76 to 95.85 % and 23.84 to 40.33 (J/g), respectivly. The lowest curcumin release rate during storage was for 3:1 alginate:AHSG microgel (41.4 %). The microgels provided more protection for curcumin against UV irradiation. The release profiles of curcumin from microgels during in vitro digestion were controlled by the Fickian diffusion phenomenon. Overall, these results indicated that electrosprayed alginate/AHSG microgels enhanced the photostability and improved the controlled release of curcumin throughout the gastrointestinal tract.

Feasibility of molecular dynamics simulation for process parameter guidance of silicon nitride thin films by PECVD

Due to the limitations of the characterization approaches and the relative complex and expensive characteristics of CVD method itself, experimental-based studies are limited, and simulation of SiNx deposition based on molecular dynamics (MD) simulation provides a more convenient and clear solution. In this work, Tersoff potential function containing the interatomic potential for two-body and three-body interactions was utilized to simulate of the deposition process for SiNx thin films by the PECVD. A quick and accurate method basing on Alpha-shapes was proposed to extract surface atoms and the surface roughness of the thin film samples at substrate temperatures of 353, 493, and 653 K were found to be 0.804, 0.794, and 0.721 nm respectively. By analyzing the radial distribution function, it can get that there are small changes in the Si-Si and Si-N bonds at three temperatures. We also found the diffusion phenomenon at the interface, and the atomic diffusion phenomenon at the substrate interface makes the phenomenon of elemental mutation at the interface less rapid. The simulation results confirm that the substrate temperature during deposition has an effect on the adsorption and atomic drift rate of the substrate, which in turn affects the surface morphology of the film.

Intense pulsed light annealing of solution-based indium–gallium–zinc–oxide semiconductors with printed Ag source and drain electrodes for bottom gate thin film transistors

In this study, an intense pulsed light (IPL) annealing process for a printed multi-layered indium–gallium–zinc–oxide (IGZO) and silver (Ag) electrode structure was developed for a high performance all-printed inorganic thin film transistor (TFT). Through a solution process using IGZO precursor and Ag ink, the bottom gate structure TFT was fabricated. The spin coating method was used to form the IGZO semiconductor layer on a heavily-doped silicon wafer covered with thermally grown silicon dioxide. The annealing process of the IGZO layer utilized an optimized IPL irradiation process. The Ag inks were printed on the IGZO layer by screen printing to form the source and drain (S/D) pattern. This S/D pattern was dried by near infrared radiation (NIR) and the dried S/D pattern was sintered with intense pulsed light by varying the irradiation energy. The performances of the all-printed TFT such as the field effect mobility and on–off ratio electrical transfer properties were measured by a parameter analyzer. The interfacial analysis including the contact resistance and cross-sectional microstructure analysis is essential because diffusion phenomenon can occur during the annealing and sintering process. Consequently, this TFT device showed noteworthy performance (field effect mobility: 7.96 cm2/V s, on/off ratio: 107). This is similar performance compared to a conventional TFT, which is expected to open a new path in the printed metal oxide-based TFT field.

Observation of Long-Lived Plasma Formations in a Toroidal air Vortex under Normal Conditions

Purpose. Experimentally investigate the difference between the characteristic lifetime of plasma formations in a stationary and rotating toroidal air volume under normal ambient conditions.Methods. Experimental study of plasmoids was carried out by videotaping with a high-speed camera with subsequent frame-by-frame processing of the recording. Generation of plasmoids was performed by mechanical rupture of the plasma cord of the welding arc in the field of permanent magnets. The creation of toroidal rotation of the plasma formation in air occurs when it passes through a hole in a solid screen. Numerical estimation of mechanical forces acting on ions in the air vortex is performed by order of magnitude using experimental data.Results. Two different in design laboratory installations for observation of plasma formations in air under normal external conditions have been created. The lifetime of such "long-lived" plasmoids in air is equal to about two milliseconds, which significantly exceeds the lifetime of a plasma-forming welding arc in the free state under the same conditions. The increase in plasmoid lifetime is considered and explained within the framework of mechanical ideas about spatial separation of heavy and light ions inside the plasma under the action of inertia forces as a result of rotational motion of air. The proposed mechanical model of the phenomenon is obviously simplified and does not take into account diffusion, recombination and other related phenomena, since the main and only difference between the observed "longlived" and "ordinary" plasmoids lies in the nature of the mechanical motion of the medium.Conclusion: Two experimental setups allowing to generate and observe the behavior of plasma formations with a characteristic lifetime of the order of two milliseconds are created and described. The results of observations indirectly confirm a simple mechanical hypothesis about the connection between the lifetime of a free plasma formation and the rotational motion of the gas inside it.

Semigroup theory and analysis of solutions for a higher order non‐Lipschitz problem with advection in ℝN$$ {\mathrm{\mathbb{R}}}^N $$

In studying heterogeneous diffusion phenomena beyond the classical order 2 Laplacian, higher order operators emerge as powerful tools. When such operators are applied, the resultant profiles often showcase an oscillatory nature that may be noticeable near the zero stationary solution. These oscillations can pose challenges, especially when they negate the compactness of the support for initial distributions that are both smooth and possess compact support. The presented analysis in this work seeks to delve into the nature of solutions within open finite domains, with an ambition to scale up the findings to the realm of through the use of an extension operator. The foundation for establishing the existence and uniqueness of these solutions leans on the semigroup representation theory. Asymptotic solutions have been carved out via the Hamilton–Jacobi transport equation. To cement the theoretical foundation, numerical simulations have been incorporated shedding light on the accuracy of our initial assumptions within the asymptotic framework.

Laminar burning velocity of Ammonia/Air mixtures at high pressures

Ammonia (NH3) is one solution for decarbonizing the economy. This study used high-speed schlieren imaging to investigate the NH3/air flame morphology and propagation characteristics (laminar burning velocity (LBV) and flame thickness) inside a constant volume combustion chamber (CVCC), at conditions relevant to different types of combustors used in power generation. Experiments were carried out for equivalence ratios (ϕ) from 0.7 to 1.3, initial pressures from 1 atm to 10 atm, and an initial temperature of 298 K using a variable gap ignition system. A novel passivation method that minimized the effect of NH3 adsorption on stainless steel surfaces was used during the CVCC filling to reduce the uncertainty in the actual ϕ before ignition. Results showed that the low speed of NH3/air flames increased the effect of buoyancy on the flame shape, especially at high initial pressures or near the flammability limits when flames adopted a bean-shaped structure. A comparison of Rayleigh and Peclet numbers (indicators of buoyancy and diffusion phenomena in flames, respectively) supported these findings. Maximum LBV was measured at ϕ = 1.1 and decreased non-linearly with the initial pressure. Similarly, flame thickness decreased with pressure, and it was thinnest at ϕ = 1.1 for all the examined conditions. Numerical simulations using Cantera produced similar LBV values as the experiment. A sensitivity analysis indicated that reactions involving NH2 and NO have an important role on LBV. Reactions leading to H, O, and OH radicals generation had the strongest effects on LBV enhancement, while LBV-inhibiting reactions had more accentuated effects at high pressures at which LBV is lower.

Attention-based fusion of multiple graphheat networks for structural to functional brain mapping

Over the last decade, there has been growing interest in learning the mapping from structural connectivity (SC) to functional connectivity (FC) of the brain. The spontaneous fluctuations of the brain activity during the resting-state as captured by functional MRI (rsfMRI) contain rich non-stationary dynamics over a relatively fixed structural connectome. Among the modeling approaches, graph diffusion-based methods with single and multiple diffusion kernels approximating static or dynamic functional connectivity have shown promise in predicting the FC given the SC. However, these methods are computationally expensive, not scalable, and fail to capture the complex dynamics underlying the whole process. Recently, deep learning methods such as GraphHeat networks and graph diffusion have been shown to handle complex relational structures while preserving global information. In this paper, we propose a novel attention-based fusion of multiple GraphHeat networks (A-GHN) for mapping SC-FC. A-GHN enables us to model multiple heat kernel diffusion over the brain graph for approximating the complex Reaction Diffusion phenomenon. We argue that the proposed deep learning method overcomes the scalability and computational inefficiency issues but can still learn the SC-FC mapping successfully. Training and testing were done using the rsfMRI data of 1058 participants from the human connectome project (HCP), and the results establish the viability of the proposed model. On HCP data, we achieve a high Pearson correlation of 0.788 (Desikan-Killiany atlas with 87 regions) and 0.773 (AAL atlas with 86 regions). Furthermore, experiments demonstrate that A-GHN outperforms the existing methods in learning the complex nature of the structure-function relation of the human brain.

Cattaneo-Christov heat and mass fluxes model of Casson fluid employing non-Fourier double diffusion theories with ion slip and Hall effects

It is vitally important to comprehend non-Newtonian fluids flow behaviour from an industrial perspective. Non-Newtonian fluids are needed for a number of industrial and technical processes, including the production of paper, photographic films, and polymer sheet extrusion. There are several industrial applications for heat and mass transfer. However, thermal and solute relaxation time phenomena can't be predicted by classical heat and mass transfer laws (Fourier and Fick laws). This work aims to apply the modified Ohm law to thermal and mass transfer which are based on Fick's and generalized Fourier principles, respectively. The current study examines the three-dimensional Darcy-Forchheimer flow of non-Newtonian fluid “Casson fluid” via porous medium with ion slip and Hall effects across a stretched sheet. Thermophoresis and Brownian motion aspects are also considered. The diffusion phenomena are modelled using the Boungrino model. The modified Buongiorno model (MBM) for nanofluids is used to investigate the enhancement of heat transport. Together with practical nanofluid properties, it covers the mechanics of thermo-migration and random motion of nanoparticles. By adding the appropriate similarity transformations, this collection of (PDEs) that reflect the mathematical model is transformed into an (ODEs) system, which it is subsequently resolved utilizing the Lobatto IIIA technique's powerful computing capabilities via Matlab software. Data visualisations and numerical examples are presented considering many physical restrictions affect the fluctuation of velocities, temperatures, concentration, dimensionless shear stress, Nusselt number, and Sherwood number. According to certain findings, the existence of Hall and ion slip effects, and other factors contribute to an increase in horizontal velocity distribution. Furthermore, Fourier's law produces a wider temperature distribution than the Cattaneo-Christov heat flux model.

Computational heat transfer analysis of liquid food in a heat exchanger with varying stirrer settings in a nonclassical continuum framework

Rapid heat diffusion and stirring of food in channels and tanks for more desirable and safely regulated food delivery is of particular interest in food industry. In the present article, we investigate heat diffusion in a heat exchanger with varying stirrer settings in a nonclassical framework. Interestingly, the liquid food flows in all previous investigations are modeled using the classical continuous paradigm. Therefore, the purpose of this article is to offer a mathematical model that uses a nonclassical continuum framework to mimic the behavior of liquid food flows. This is influenced by the fact that during the stirring process, food particle microrotations affect the heat diffusion phenomenon, necessitating consideration of this in the modeling aspect of such studies. This non-Newtonian physical process is modeled based on the well-established theory of Cosserat continuum. The mathematical model presented here is taking into consideration a newly established constitutive relation by the author for the strain-based viscosity models in non-Newtonian mechanics. The applied constitutive relation incorporates the microrotational effects into the current rheology of rate-dependent viscosity model as a result of the kinematics at the macroscopic level. It offers a more practical method for examining the heat diffusion process in a heat exchanger. To the best of the author’s knowledge, this approach is first of its kind in analyzing heat diffusion process within non-Newtonian mechanics. The presented strain-based viscosity model is described in the form of PDEs along with prescribed boundary conditions. An in-house code is developed based on finite element method. The code is implemented using FreeFEM++. Liquid food flow in a heating process is analyzed numerically for heat transfer characteristics with varying Cosserat number and other physical aspects. Simulations are performed in a channel connecting a tank with different stirrer settings. Computed results are analyzed in a limiting case for accuracy and a strong agreement is achieved with the available solutions from the literature. Some interesting features of the liquid food flow and heat transfer are presented and discussed. It is observed that when liquid food particles rotate faster, the temperature difference between the walls also increases. The temperature difference between the top and bottom walls of the liquid food tank increases with faster speed of stirrers. Rise in the Cosserat number results in a dispersed vortices throughout the liquid food tank. The temperature differential between the liquid food tank’s top and bottom walls rises as a result of an increase in the microrotational motions of the food particles. Moreover, the microrotational motion of the liquid food particles rises with an increase in Cosserat number, enabling effective particle mixing.

Unravelling the thermal behavior and kinetics of unsaturated polyester resin supplemented with organo-nanoclay.

The integration of nanoclays within polymeric systems to develop high-performance materials is an emerging research field that has garnered significant attention. In this context, an organically modified montmorillonite (OMMT) is utilized as a reinforcing agent for unsaturated polyester resin (UPR) with loads of 1%, 3%, and 5 wt%. The modification of montmorillonite nanoclay (MMT) using a quaternary ammonium compound is performed through an effective repetitive modification process under reflux conditions. The curing behavior of the unsaturated polyester resin containing organically modified clay catalyzed with methyl ethyl ketone peroxide (MEKP) initiator and promoted by cobalt naphthenate accelerator is investigated using dynamic differential scanning calorimetry (DSC) followed by kinetic analysis using isoconversional methods. The dynamic DSC curing curves showed a bimodal exothermic peak, where two independent reactions, namely, redox and thermal decomposition of the initiator occurred. In this study, novel insights into the curing reaction of the studied UPR and UPR/OMMT systems have been revealed through the application of the Trache-Abdelaziz-Siwani (TAS) and Sbirrazzuoli (VYA/CE) isoconversional methods. These methods have enabled the elucidation of the intricate mechanisms and phenomena that impact the curing reaction, including the dilution effect in the redox reaction and the diffusion phenomenon at the end of the thermal decomposition reaction. The incorporation of nanoclay into unsaturated polyester resin (UPR) resulted in a reduction in the activation energy for both the redox and thermal reactions. Specifically, the energetic barrier decreased from 93.85 and 101.58 kJ mol-1 for pristine UPR to 60.71 and 72.93 kJ mol-1 for UPR/OMMT-5 in the redox and thermal reactions, respectively. The addition of OMMT caused a significant decrease in the pre-exponential factor. The values of UPR/OMMT-5 were 2.75 × 105 and 5.50 × 106 for the redox and thermal decomposition reactions, respectively, compared to 1.41 × 1012 and 5.13 × 1013 for UPR. The thermogravimetric analysis demonstrated that UPR/OMMT systems were more stable than UPR.

COMBUSTION OF ALUMINUM PARTICLES IN A STREAM OF STEAM

Several models for aluminum combustion have been proposed, thus an overview of the combustion characteristics of aluminum powders is presented, taking into account both early and recent research activities and distinguishing between micron-sized and nano-sized aluminum. A collection of models for the two classes is discussed and compared, considering different environments in terms of oxidizers (H<sub>2</sub>O, O<sub>2</sub>, CO<sub>2</sub>), diluents (N<sub>2</sub>, Ar), temperature and pressure. In particular, the oxidation and the combustion of aluminum in steam are addressed, highlighting the oxidizing capabilities of various species. Different models are implemented for the complete description of aluminum combustion in the micron range, since smaller particles (1-10 μm) burn differently from bigger ones: both kinetics-based and diffusion-based processes are analyzed, stressing the role of reaction paths as well as diffusive phenomena. A single mechanism (kinetics-based) is identified and modeled for the combustion of nano-sized aluminum, outlining the limits and hypotheses for the application. For both sizes, only homogeneous reactions are considered and a model for the growth of an oxide cap on the surface of the particle is implemented. The results obtained are compared to the available numerical simulations and experimental evidence, aiming at the approximation of the burning time through the D<sup>n</sup> law. Also, the profile over time of the temperature, the radius of the particle and the concentrations of the different chemical species involved in the reactions are analyzed.

Impact of Anomalous Diffusion Phenomenon on Molecular Information Delivery in Bounded Environment.

Through this paper, a three-dimensional molecular communication (MC) inside a cuboid container is considered. Instead of normal diffusion phenomenon, the anomalous diffusion phenomenon is incorporated which enhances the practicability of the model. The Fick's law is re-defined for the considering rectangular coordinate system in which information carrying molecules (ICMs) diffuse anomalously in the environment. The impact of flow of the fluid along the +x direction in the environment is also considered. Moreover, considering free propagator phenomenon, the expressions of spatio-temporal probability density function (PDF) of the ICMs is derived for the considered model. Further, the novel closed-form expressions for first arrival time density (FATD) of the ICM, survival probability (SP) at any time, and its corresponding first arrival probability (FAP) are also derived. Furthermore, the considered MC model is also analyzed in terms of minimum bit-error-rate (BER) using log-likelihood ratio test (LLRT) optimal detector. The derived expressions are verified using MATLAB based particle-based and Monte-Carlo simulations.

Exploring the phase behavior of C8-BTBT-C8 at ambient and high temperatures: insights and challenges from molecular dynamics simulations.

C8-BTBT-C8 is one promising candidate for the development of high-performance electronic devices based on thin-film technologies. Its monoclinic polymorph has a well-established role in thin-film growth. Yet, quite little information is available about its dynamics on the molecular scale, and the structures of the mesophases which form at high temperature (about 100 K above ambient temperature). The present study is devoted to the analysis of such phases, with the ultimate goal of developing molecular models. Already at ambient temperature, our molecular dynamics simulations reveal a rich conformational behavior of the alkyl side chains, with gauche conformations as leading structural defects. Heating promotes the formation of a stacking faulted mesophase (380 K), and a smectic phase, at 385 K, upon side chain melting. Although more disordered, this phase bears several analogies with the smectic A phase, experimentally observed at 382.5 K. At higher temperatures, the increase in configurational disorder is brought by molecular diffusion and other phenomena, finally leading to an isotropic molten phase. Our in-depth analysis, complemented by hot-stage polarizing microscopy data, provides interesting insights into this material, highlighting the challenges associated with the modeling of soft semiconducting systems.

Lévy-walk-like Langevin dynamics with random parameters.

Anomalous diffusion phenomena have been widely found in systems within an inhomogeneous complex environment. For Lévy walk in an inhomogeneous complex environment, we characterize the particle's trajectory through an underdamped Langevin system coupled with a subordinator. The influence of the inhomogeneous environment on the particle's motion is parameterized by the random system parameters, relaxation timescale τ, and velocity diffusivity σ. We find that the two random parameters make different effects on the original superdiffusion behavior of the Lévy walk. The random σ contributes to a trivial result after an ensemble average, which is independent of the specific distribution of σ. By contrast, we find that a specific distribution of τ, a modified Lévy distribution with a finite mean, slows down the decaying rate of the velocity correlation function with respect to the lag time. However, the random τ does not promote the diffusion behavior in a direct way, but plays a competition role to the superdiffusion of the original Lévy walk. In addition, the effect of the random τ is also related to the specific subordinator in the coupled Langevin model, which corresponds to the distribution of the flight time of the Lévy walk. The random system parameters are capable of leading to novel dynamics, which needs detailed analyses, rather than an intuitive judgment, especially in complex systems.

Exploring Solution Strategies for Volterra Integro‐Differential Lane–Emden Equations in Astrophysics Using Haar Scale 3 Wavelets

The current research introduces a novel approach to address the computational challenges associated with solving the Lane–Emden‐type equations by transforming them from their conventional differential form to the corresponding integro‐differential form. These equations have wide‐ranging applications in physical sciences, including modeling diffusion phenomena and thermal gradients. We utilize the Volterra integro‐differential (VID) form to resolve computational challenges due to singularity issues. Through the Scale 3 Haar Wavelet (S3‐HW) algorithm, we transform the VID equations into algebraic form and obtain solutions using the Gauss‐elimination method. The quasilinearization technique is implemented whenever a nonlinearity is encountered. Comparative analysis against various techniques demonstrates the superior accuracy and efficiency of our method. Despite challenges such as the discontinuity of Scale 3 Haar Wavelets and singularity issues of Lane–Emden‐type equations, our algorithm paves the way for extending its application to a wide range of physical problems.

Important role of pore-filling mechanism in separating naproxen from water by micro-mesoporous carbonaceous material.

Commercial micro-mesoporous carbonaceous material (MCM; 56.8% mesopores) was applied for investigating the removal phenomenon of naproxen drug in aqueous solutions through batch adsorption experiments. Results demonstrated that the adsorption capacity of MCM to naproxen was slightly affected by different pHeq (2.0-11) and ionic strength (0-1M NaCl). Adsorption kinetics, isotherms, thermodynamics, and mechanisms were evaluated at pH7.0. Adsorption kinetics indicated the rate constants for adsorption (0.2 × 10-3 L/(mg × min) and desorption (0.076/min) and the adsorption equilibrium constant (2.6 × 10-3 L/mg). Adsorption isotherm showed that MCM exhibited a high-affinity adsorption capacity to naproxen (even at low concentrations) and its Langmuir maximum adsorption capacity (Qmax ) was 252.7mg/g at 25°C. Adsorption thermodynamics proved that the adsorption process was endothermic and physisorption (ΔH° = 9.66 kJ/mol). The analysis result of pore size distribution demonstrated that the internal pore structure of MCM was appropriate for adsorbing naproxen molecules. Pore-filing mechanism (pore diffusion phenomenon) was confirmed by a considerable decrease in BET-surface area (585 m2 /g) and total pore volume (0.417 cm3 /g) of MCM after adsorbing naproxen (~1000 mg/L and pH7.0) at 5min (341 and 0.256), 60 min (191 and 0.205), 120 min (183 and 0.193), 360 min (144 and 0.175), and 24 h (71.6m2 /g and 0.123 cm3 /g, respectively). The pore diffusion occurred rapidly (even at the initial adsorption period of 5min). The FTIR technique was applied to identify the existence of C-H···π and n-π interaction. π-π interaction (evaluated through ID /IG ratio and C=C band) played a minor contribution in adsorption mechanisms. The ID /IG ratio (determined by the Raman technique) of MCM before adsorption (1.195) was similar to that after adsorption (1.190), and the wavenumber (C=C band; its FTIR spectrum) slightly shifted from 1638 to 1634 cm-1 after adsorption. A decrease in the Qmax value of MCM from 249 to 217 (H2 O2 -oxidized MCM) or to 224 mg/g (HNO3 -oxidized MCM) confirmed the presence ofπ-π interaction. Electrostatic attraction was a minor contribution. MCM can serve as a promising material for removing naproxen from water environment through a pore-filling mechanism. PRACTITIONER POINTS: Pore-filling mechanism was proposed by comparing textural properties of MCM before and after adsorbing naproxen. C-H···π and n-π interactions were identified via FTIR technique. π-π interaction was observed by FTIR and Raman techniques. Oxidation of MCM with HNO3 or H2 O2 was a helpful method to explore π-π interaction. Electrostatic attraction was explained through studies: effects of pH and NaCl along with desorption.