Behavior Of Solutions Research Articles

Longtime behavior of homoenergetic solutions in the collision dominated regime for hard potentials

Longtime behavior of homoenergetic solutions in the collision dominated regime for hard potentials

Mitigating Sodium Ordering for Enhanced Solid Solution Behavior in Layered NaNiO2 Cathodes

AbstractThe O‐type layered nickel oxides suffer from undesired cooperative Jahn–Teller distortion stemming from Ni3+ ions and undergo multiple biphasic structural transformations during the insertion/extraction of large Na+ ions, posing a significant challenge to stabilize the structural integrity. We present here a systematic investigation of the impact of substituting 5 % divalent (Mg2+) or trivalent (Al3+ or Co3+) ions for Ni3+ to alleviate Na+ion ordering and perturb the Jahn–Teller effect to enhance structural stability. We gauge a fundamental understanding of the Mg−O and Na−O or Mg−O−Na bonding interactions, noting that the ionicity of the Mg−O bond deshields the electronic cloud of oxygen from Na+ ions. Furthermore, calculations of the Van Vleck distortion modes reveal a relaxation of NiO6 octahedra from Jahn–Teller distortion and a reduced electron density at the interlayer with Mg2+ substitution. Long‐range (operandoX‐ray diffraction) and short‐range (magic angle spinning nuclear magnetic resonance) structural analyses provide insights into reduced ordering, allowing a stable continuous solid solution. Overall, Mg‐substitution results in a high‐capacity retention of ~96 % even after 100 cycles, showcasing the potential of this strategy for overcoming the structural instabilities and enhancing the performance of sodium‐ion batteries.

Solution Behavior of Glyco-Copoly(l-Glutamic Acid)s in Dilute Saline Solution.

A small series of copoly(α,l-glutamic acid/dl-allylglycine)s with the same chain length and allylglycine content (∼10 mol %) but different spatial distribution of allylglycine units was synthesized and subsequently glycosylated via thiol-ene chemistry. Dilute aqueous copolypeptide solutions (0.1 wt %, physiological saline) were analyzed by circular dichroism spectroscopy, dynamic light scattering, and cryogenic transmission electron microscopy. The copolypeptides adopted a random coil or α-helix conformation, depending on solution pH, and the glycosylated residues either distorted or enhanced the folding into an α-helix depending on their location and spatial distribution along the chain. However, regardless of their secondary structure and degree of charging, all partially glycosylated copolypeptides self-assembled into 3D spherical structures, supposedly driven by a hydrophilic effect promoting microphase separation into glucose-rich and glutamate-rich domains.

Synthesis of Perylene Diimide Modified by POSS and its Assembly Behavior in Solution and SBS Matrix

In this work, Perylene diimide derivatives 1,7- POSS-PDI-C12 (PPC) were synthesized using 1,7-Br-PDI- C12, 18-crown-6 ether as phase transfer agent, K2CO3 as catalyst and THF as solvent in a reaction system. Subsequently, the POSS-SH group was successfully introduced into the perylene imide bay site in high yield. The chemical structure and molecular weight of the products were characterized by 1H-NMR, 13C-NMR, MALDI-TOF MS and FT-IR. The self-assembly behavior in different solvents was studied using the research solution self-assembly method. PPC remained in a single monomolecular state due to the bulky POSS group, which introduced large steric hindrance in the molecular packing, leading to the weakening of the π–π interactions. Then, the Composite thin films containing PPC in SBS was prepared. PPC do not exhibit significant force-chromic behavior owing to the dispersion of single- molecule states. This shows that in the construction of mechanochromic materials, chromophore molecules need to have a certain degree of aggregation in the matrix, but the aggregation degree should not be too high.

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Four-Component Statistical Copolymers by RAFT Polymerization

This manuscript serves as the starting point for in-depth research of multicomponent, statistical, methacrylate-based copolymers that potentially mimic the behavior of proteins in aqueous solutions. These synthetic macromolecules are composed of specially chosen comonomers: methacrylic acid (MAA), oligoethylene glycol methyl ether methacrylate (OEGMA475), 2-(dimethylamino)ethyl methacrylate (DMAEMA) and benzyl methacrylate (BzMA). Monomer choice was based on factors such as the chemical nature of pendant functional groups, the polyelectrolyte/polyampholyte and amphiphilic character and the overall hydrophobic–hydrophilic balance (HLB) of the obtained quaterpolymers. Their synthesis was achieved via a one-pot reversible addition fragmentation chain transfer (RAFT) polymerization in two distinct compositions and molecular architectures, linear and hyperbranched, respectively, in order to explore the effects of macromolecular topology. The resulting statistical quaterpolymers were characterized via 1H-NMR and ATR-FTIR spectroscopies. Their behavior in aqueous solutions was studied by dynamic (DLS) and electrophoretic light scattering (ELS) and fluorescence spectroscopy (FS), producing vital information concerning their self-assembly and the structure of the formed aggregates. The physicochemical studies were extended by tuning parameters such as the solution pH and ionic strength. Finally, the quaterpolymer behavior in FBS/PBS solutions was investigated to test their colloid stability and biocompatibility in an in vivo-mimicking, biological fluid environment.

Limiting behavior of quasilinear wave equations with fractional-type dissipation

Abstract In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

AbstractThe aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

Analyzing the time-fractional (3 + 1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions

The paper focuses on investigating the time-fractional (3 + 1)-dimensional cubic and quantic nonlinear Schrödinger equation. We adopt the novel Kudryashov method to generate a distinct class of optical solutions for the current conformable fractional derivative problem. Our method explores various solution forms, including dark, wave, mixed dark-bright, and singular solutions. The soliton solutions we construct are visually represented to illustrate the influence of the fractional order derivative. Further, we elucidate the influence of solution parameters on the wave envelope, providing clear interpretations through 2D graphics presentations. The results underscore the efficacy of our approach in discovering exact solutions for nonlinear partial differential equations, especially in cases where alternative methods prove ineffective. The significance of the present paper lies in its contribution to advancing the understanding of the behavior of optical solutions in nonlinear systems, providing valuable insights for both theoretical and practical applications. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons.

Modeling the transmission dynamics of COVID-19 with genetically resistant humans

The emergence of a coronavirus disease (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus (SARS-COV 2) has caused an unprecedented global societal and public health crisis. A mathematical model that incorporates an individual’s ability to resist COVID-19 infection is considered to study the dynamics of COVID-19. Detection and testing for COVID-19 in individuals who has ability to resist the infection, proper preventive measures, and timely medical care are essential in managing and controlling COVID-19 outbreak. These allows us to make quantifiable predictions about the effects of testing utilization on the COVID-19 prevalence. First, we analyze the disease-free state of the COVID-19 model, calculate the baseline control reproduction number, and obtain the local stability of disease-free equilibrium. The model is calibrated using data obtained from Nigeria Centre for Disease Control and key parameters of the model are estimated. Sensitivity analysis is carried out to investigate the influence of the parameters in curtailing the disease. Numerical simulations are then used to explore the behavior of the model solutions involving the novel variable (genetically resistant humans) and control measures. Our results suggest that strict intervention effort (early detection of genetically resistant individuals, testing and isolation of the infected individuals) is required for quick suppression of the disease.

Unraveling the Molecular Weight Dependence of High Magnetic Field to Manipulate the Semiconducting Polymer Molecular Orientation.

The magnetic alignment of molecules, which exploits the anisotropy of diamagnetic susceptibility, provides a clean and versatile approach to the structural manipulation of semiconducting polymers. Here, the magnetic-alignment dynamics of two molecular-weight (MW) batches of a diketopyrrolopyrrole (DPP)-based copolymer (PDVT-8) were investigated. Microstructural characterizations revealed that the magnetically aligned, high-MW (Mn = 53.7 kDa) PDVT-8 film exhibited a higher degree of backbone chain alignment and film crystallinity compared with the low-MW (Mn = 17.6 kDa) PDVT-8 film grown via the same magnetic alignment method. We found that as the MW increases, the degree of preaggregation of the polymer molecules in solution significantly increases and the aggregation mode changes from H-aggregation to J-aggregation through a cooperative assembly mechanism. These events improved the responsiveness of high-MW polymer molecules to magnetic fields. Field-effect transistors based on the magnetic aligned high-MW PDVT-8 films exhibited a 6.8-fold increase in hole mobility compared to the spin-coated films, along with a mobility anisotropy ratio of 12.6. This work establishes a significant correlation among chain aggregation behavior in solution, polymer film microstructures, magnetic responsiveness, and carrier transport performance in donor-acceptor polymer systems.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Modular synthesis of dendritic ionic triphiles and their behavior in three-phase liquid–liquid–liquid system

Triphilic molecules possess unique features due to the presence of chemically distinct domains − oleophilic, hydrophilic, and fluorophilic. In bulk, they can form intriguing crystal structures, while in solutions, they can act as multi-purpose emulsifiers or form assemblies such as multi-compartment micelles applicable in drug delivery. However, their chemico-physical properties are still relatively unexplored. Namely, they have never been studied in a three-phase liquid–liquid–liquid system consisting of organic and fluorous solvents and water. Here we present an adaptive, modular synthesis of a novel type of dendritic ionic triphiles and a study of their solution behavior, with a focus on a three-phase toluene/water/perfluoro(methylcyclohexane) (PMC) system. The synthesis is based on the quaternization of various alkylimidazoles by a previously published type of carbosilane dendritic wedges bearing polyfluorinated chains of varied lengths at their periphery. This approach enables the facile combination of different organic and fluorous blocks and, thus, easy tailoring of final structures according to the results of this work. The distribution of triphiles in the selected three-phase system was mapped in a ternary diagram representing the mass proportions of individual domains. The distribution between toluene and PMC changes gradually depending on the sizes of the corresponding domains and significantly differs from the distribution in a biphasic system where water is absent (i.e., the fluorous partition coefficient). However, at a certain size of the fluorous domain, the molecules completely lose their affinity to the fluorous phase and gain affinity to water even though being still partially soluble in PMC. The results can help understand the structure-affinity relationship of triphilic compounds and can help find a substance with balanced affinities to all three solvents. Moreover, some of the prepared compounds manifested interesting surface activity, namely the emulsification capability on both phase interfaces or the formation of stable nanoparticles in an aqueous environment.

Synthesis of phosphonic acid and amide functionalized core–shell magnetic composite and its mechanisms for synergistic adsorption of thorium from strongly acidic solution

To develop adsorbents for the preeminent extraction of thorium in strongly acidic solutions is very imperative, due to strong inorganic acids are frequently used in the fuel cycle of thorium. Herein, a robust core–shell magnetic nanocomposite Fe3O4@SiO2/P (MBA-VPA-AM) possessing phosphonic acid and amide was developed via distillation-precipitation copolymerization of N, N'-methylene diacrylamide (MBA), vinylphosphonic acid (VPA) with acrylamide (AM) employing Fe3O4 decorated by SiO2 (Fe3O4@SiO2) as magnetic core. Due to the synergistic effect of phosphonic acid and amide ligand, the sorbent could capture thorium from concentrated HNO3 media efficiently, and the maximum adsorption capacity (qmax) at 4 mol L−1 of HNO3 solution reached up to 232.5 mg g−1, superior to that of all magnetic sorbents reported previously. Furthermore, the sorbent displayed good selectivity towards thorium in the presence of competitive metal ions including La3+, Nd3+, and Ce3+. The excellent thorium sorption behavior in strong hydrogen nitrate solution was merely ascribed to the complexation of thorium with phosphonic acid ligand, and the amide ligand could not complex with thorium, evidenced by XPS, FT-IR and DFT method. However, the presence of amide ligand could enhance the hydrophilicity of the sorbent and the free movement degree of phosphonic acid ligand to a certain extent, both of which could help to promote the thorium sorption on the sorbent in strong HNO3 media. Besides, due to its core–shell structure and superparamagnetism, the sorbent not only exhibited excellent acidic resistance in strong acid medias, but could be magnetically recouped, which could avoid tedious centrifugation or filtration of traditional adsorbents. This work provides a novel strategy to develop magnetic sorbent for the preeminent entrapment of thorium from strong acid solutions.

Linear solvent strength model on porous graphitic carbon stationary phase using high temperature liquid chromatographic method for allopurinol related substances analysis

A high-performance liquid chromatography (HPLC) method was developed for the analysis of Allopurinol and its Ph.Eur. impurities using a porous graphitic carbon (PGC) stationary phase. Retention behavior of solutes was studied across a wide temperature range (30–90 °C) and various gradient times (5–20 min). Analysis of the data revealed distinct retention mechanisms between reversed-phase and PGC phases. However, it was proved that the retention of Allopurinol and its Ph.Eur. impurities on PGC stationary phase can be effectively modeled using the linear solvent strength (LSS) theory. This allows for the utilization of LSS-based method development software to optimize methods under these conditions. By using commercial chromatographic modeling software, separation of Allopurinol and Ph.Eur. impurities was optimized within a large design space. At the optimized operating conditions (pH = 2.0, tG = 6 min, T = 60 °C), all solutes were separated within 6 min with baseline resolution. Comparison between predicted and experimentally measured chromatograms further confirmed the applicability of LSS theory in developing analytical methods for PGC-based HPLC systems. The presented approach offers a general framework for method development on PGC phases.

Galerkin approximation for multi-term time-fractional differential equations

Fractional differential equations (FDEs) are utilized as a precise model for describing a wide range of biological and physical processes, benefiting from the inherent symmetry feature in natural phenomena. The introduction of multi-term time-fractional order equations serves to tackle intricate phenomena spanning various physical domains. This study solves multi-term time-fractional DEs using the quadratic B-spline Galerkin method. In Galerkin approach, the quadratic B-spline is utilized as both a test and a basis function. The fractional portion of time is evaluated using the Caputo definition. In addition, the Gauss quadrature formula is used to evaluate the integration of complex function which enhance the accuracy. Lax-Richtmyer's stability criterion is employed to evaluate the stability of the proposed scheme. The E2 and E∞ norms are computed and displayed in the tables to illustrate the robustness and efficiency of the method. We compute simulation for various nodal points as well as various mesh sizes to analyze the solution behavior. The obtained results are compared with the exact and available results, indicating that the proposed scheme is more efficient. The graphical solution of each example is also demonstrated which verifies that exact and approximate solutions graphs closed with each other.

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

Nonlinear elliptic problem in exterior domains: exact boundary behavior of blow-up positive solutions

Nonlinear elliptic problem in exterior domains: exact boundary behavior of blow-up positive solutions

A Study on the Effect of Dilution on Refractive Index of Acetone

This r study investigates the impact of dilution on the refractive index of acetone-water solutions, aiming to study the relationship between solution concentration and optical properties. The experiment involves preparing a series of acetone-water solutions with varying concentrations and measuring their refractive indices using a spectrometer. The abstract will outline the methodology employed, including the preparation of solutions, experimental setup, and data collection procedures. It will also highlight the use of refractive index measurements in describing the characteristics of solutions and their applicability in physics, chemistry, and materials science. The study's expected findings, including the possibility of departures from ideal behaviour and the relationship between solution concentration and refractive index, will be covered in the abstract. It will also stress how crucial it is to understand how dilution affects refractive index for real-world uses, like those in the food, chemical, and pharmaceutical sectors. A greater understanding of solution chemistry and optical phenomena will result from this research's insightful observations of the optical behaviour of acetone-water solutions. Key Words: optics, spectrometer, refractive index, acetone

A small angle X-ray scattering study comparing protic ionic liquid solvent effects on the solution behaviors of hen egg white and human lysozyme

Proteins demand customized solvents for maintaining their stability. However, the specific impact of different ions on different proteins remains unclear. This study explores the effects of protic ionic liquid (PIL)-based solvents on two structurally similar proteins, namely hen egg white and human lysozyme. By employing size exclusion chromatography small-angle X-ray scattering (SEC-SAXS) and capillary SAXS, the effects of protein concentration (3 to 60 mg/mL), IL type (five PILs) and concentration (1, 10 and 20 mol% with water) and protein type on the aggregation behavior and radius of gyration (Rg) of the two lysozymes were studied. HEWL was mostly monomeric in buffer and solubilized beyond 20 mg/mL, while HL displayed slight aggregation in buffer but had increased aggregation at over 10 mg/mL. Further capillary SAXS data show that both proteins became more aggregated at concentrations exceeding 20 mg/mL and over 1 mol% PILs. The nitrate anion generally showed the “salting in” effect, while formate and glycolate anions induced the “salting out” effect on the two proteins. Overall, anion selection and protein solution behaviors at protein concentration of over 20 mg/mL and at 20 mol% ILs can be critical for maintaining the lysozyme stability.