Solution Theory Research Articles

Enhancing CO2 adsorption capacity and selectivity of UiO-67 through external ligand modification

To address the escalating issue of CO2 pollution, the development of innovative CO2 adsorbents has been necessitated, leading to the exploration of UiO-67-based materials. In this study, UiO-67 was synthesized with meticulous selection of the appropriate modulators in optimal ratios. Subsequently, 4-iodopyrazole, a ligand distinguished by its local electric fields and polar sites, was incorporated to fabricate UiO-67-I. This novel material underwent examination employing techniques such as transmission electron microscopy (TEM), X-ray diffraction (XRD), and Brunauer-Emmett-Teller (BET) analysis. The adsorption rates of gases including CO2, CH4, and N2 were measured at various temperatures. Utilizing the ideal adsorbed solution theory (IAST) and the Clausius-Clapeyron equation, the selectivity of CO2 against CH4 and N2 was calculated, along with the CO2 adsorption isotherms. Remarkably, UiO-67-I(50) demonstrated a CO2 adsorption capacity of 1.84 mmol/g at 298 K and 1 bar, representing a significant enhancement attributable to the ligand's enhanced local electric fields and polar sites. The selectivities for CO2 over CH4 and CO2 over N2 were determined to be 14.2 and 84.6, respectively, under these conditions. Moreover, UiO-67-I(50) proved effective in the separation of CO2 from N2 under dynamic flow conditions, showcasing its potential for practical application in gas separation.

Diffusion-induced stress amplification in phase-transition materials for electrodes of lithium-ion batteries

This work sheds the light on the stress amplification in active material particles of electrodes caused by the localized lithium concentration inhomogeneity due to phase transitions. This is a significant issue usually neglected in the literature, as generally the ideal Fickian diffusion model is used to describe the lithium diffusion, which particularly fails when dealing with phase change materials, resulting in the underestimation of the stress. In turn, this results in the underestimation of the fracture behavior of the electrode and the battery degradation, ultimately.To overcome these limitations, this work proposes a mechanical-transport model where the lithium transport is modeled according to the non-ideal solution theory with an equivalent diffusion coefficient depending on the potential of the host material vs Li/Li+. The results show that the proposed model correctly describes the concentration distribution within the particle of phase-change materials (graphite is chosen as a case study), in agreement with the existent experimental measurements. The differences with respect to the traditional Fickian diffusion model is substantial as the stress is up to 85% higher with the proposed model and the concentration distribution can capture the inhomogeneity caused by phase transitions.

Preparation and characterization of UiO-66-(OH)2/MWCNTs composites for CO2/N2 adsorption separation

The increasing CO2 concentration in the atmosphere can lead to climate change, and CO2 capture technology is one of the most direct and effective means of reducing carbon emissions. In this study, a new composite was prepared in situ by loading multi-walled carbon nanotubes (MWCNTs) onto metal–organic frameworks (MOFs) to efficiently capture CO2 in flue gas. The morphological states and pore structures of the composites were analyzed using characterization methods. The CO2 adsorption properties of the composites were tested at 273 K and 298 K with different MWCNTs loadings. The optimal CO2 adsorption quantities of the composite material were measured to be 4.4 and 5.75 mmol/g, respectively, which increased the adsorption capacity by 66.7 % and 55 %, respectively, compared with the parent material at a pressure of 1 bar. The CO2/N2 separation performance of the composites was examined using ideal adsorption solution theory calculations and dynamic adsorption breakthrough experiments. The stability of the composites was investigated by thermogravimetric analysis, acidification experiments, and cyclical adsorption–desorption experimentation. The UiO-66-(OH)2/MWCNTs composites exhibited superior CO2 adsorption–separation performance, thermal stability, acid resistance, and cycling stability. Therefore, the composite has potential applications in CO2 capture separation technology.

Efficient and selective CO2 capture at low concentration from CH4 or N2 using Zn-MOF@AFP composite

The synthesis of structured CO2 adsorbents with high separation selectivity, strong stability, and low cost to meet industrial needs is very challenging. Here, we synthesized an ultra-stable Zn-MOF (Zn2(3-amion-1,2,4-triazolate)2(oxalate), Zn2(Atz)2Ox), and adopted an efficient in-situ growth strategy to produce millimeter-sized spherical Zn2(Atz)2Ox@AFP (Amino-Functionalization Poly(acrylates), AFP) composite. Due to the heterogeneous crystallization induced by the nanoscale crystal seeds and the confined space of AFP, the three-dimensional (3D) Zn2(Atz)2Ox bulk crystals transformed into 2D nanosheets with higher CO2 adsorption efficiency and faster kinetics. Zn2(Atz)2Ox@AFP has a high affinity for CO2 (2.05 mmol g−1 at 0.05 bar and 313 K) while exhibiting an inhibitory effect for N2 (only 0.051 mmol g−1 at 1.0 bar and 313 K), leading to an excellent ideal adsorbed solution theory (IAST) selectivity (2990 for CO2/N2 (5/95, v/v)). Interestingly, Zn2(Atz)2Ox@AFP exhibits a unique gating effect on CH4 adsorption, which further extends its application range to natural gas purification. Dynamic breakthrough experiments confirmed that Zn2(Atz)2Ox@AFP can selectively capture CO2 in the simulated CO2/N2 (5/95) and CO2/CH4 (5/95) mixed gases.

Understanding adsorption selectivity in zirconium-pillared clays for biogas upgradation: the role of metal/clay ratio.

Biogas, as a sustainable energy source, encounters challenges in practical applications due to impurities, notably carbon dioxide (CO2), and nitrogen (N2). This study investigates the effect of metal/clay ratio on the adsorption selectivity of porous zirconium-pillared clay adsorbents for biogas upgradation. Comprehensive analyses including nitrogen adsorption/desorption, X-ray diffraction (XRD), and Fourier-transform infrared spectroscopy (FTIR) were conducted to evaluate the physicochemical properties. Adsorption properties for Zr-pillared clays for biogas components such as CO2, CH4, and N2, at 25 °C under different pressures were investigated. The ideal adsorbed solution theory (IAST) was employed to assess selectivity for three binary gas mixtures (CO2/CH4, CO2/N2, and CH4/N2). Results revealed the substantial impact of Zr/Clay ratio on both adsorption capacity and selectivity of the prepared materials. For instance, the maximum adsorption capacity of gases varies as ZrPILC-4 > ZrPILC-2 > ZrPILC-8 > ZrPILC-1, whereas the adsorption selectivity for CO2/CH4 separation (at 1000kPa) varies as ZrPILC-1 > ZrPILC-2 > ZrPILC-8 > ZrPILC-4. Interestingly, the ZrPILC-8 with maximum surface area (147 m2∙g-1) did not show maximum adsorption capacity for all the three gases, which was attributed to its lower pore volume, and basal spacing, as compared to ZrPILC-4. Amongst all the pillared samples, the ZrPILC-1 exhibited highest selectivity for all binary mixtures (at 1000kPa), signifies increased nonspecific interactions due to its lower surface area. Its separation performance, particularly for CO2/CH4 mixture exceeded that of the parent clay by 1.5 times. A significant increase in the working capacity of the prepared samples underscores the efficacy of these pillared materials in separating biogas components. This study provides valuable insights into effects of Zr/clay ratio for developing robust pillared adsorbents, contributing to the advancement of sustainable biomethane production.

Synergistic effects of an amphiphilic drug(propranolol hydrochloride) with cationic surfactants in an aqueous medium: A physicochemical study

The interactions of amphiphilic drug(propranolol hydrochloride, PPL) with cationic surfactants, such as dodecyltrimethylammoniumbromide(DTAB), tetradecyltrimethylammonium bromide (TTAB), trimethyltetradecylammonium chloride(TTAC) and tetyltrimethylammonium bromide(CTAB), in an aqueous medium were evaluated. The surface tension measurement was used to examine the mixed micellization behavior and surface properties of drug-surfactant mixtures. The nonideal or synergistic behaviors of the mixed systems were explored by using several physical models, such as Clint, Rubingh, Rosen, and Motomura models, based on regular solution theory and pseudophase approximation. In addition, the synergism between PPL and cationic surfactants was also confirmed using ultraviolet–visible spectroscopy. The stoichiometric ratios, binding constants, and free energy changes of the drug–surfactant mixtures were computed using Benesi–Hildebrand (B–H) theory.

Phase diagrams of oxide system FeO–MnO–MgO–SiO2. Report 2. Triple state diagrams FeO–MnO–SiO2, FeO–MgO–SiO2, MgO–MnO–SiO2

The knowledge of the thermodynamics and phase equilibria of silicate melts FeO–MnO–MgO–SiO2 and solid solutions of silicates of the MnO–MgO–SiO2 system allows us to predict the appearance of non-metallic inclusions in steel. The FeO–MnO–MgO–SiO2 system includes six double and four triple state diagrams of oxide systems. Previously, the state diagrams of binary FeO–MnO, FeO–MgO, FeO–SiO2, MnO–MgO and ternary FeO–MnO–MgO oxide systems were studied. In this work, state diagrams of the FeO–MnO–SiO2, FeO–MgO–SiO2 and MgO–MnO–SiO2 systems were constructed. The calculations used the theory of subregular ionic solutions, which takes into account the dependence of the coordination number on the composition of the melt, quite accurately describes diagrams with two immiscible liquids and is consistent with the experimental data presented in the literature. In the work, the energy parameters of this theory were selected to calculate the activities of the components of the oxide melt of each of the systems under study, as well as the parameter of the theory of regular ionic solutions to calculate the activities of the components of the magnesium-manganese orthosilicate solution |Mg2SiO4, Mn2SiO4|. The data obtained on the state diagrams of the systems included in the FeO–MgO–MnO–SiO2 system will make it possible to establish non-metallic inclusions in equilibrium with the liquid metal in the Fe–Mg–Mn–Si–O system

Activation-free preparation of porous carbon fiber derived from phenolic resin for CO2 absorption

BackgroundIt is not uncommon to study the preparation of porous carbon materials for CO2 adsorption, but it is still a challenge to simultaneously achieve the good mechanical strength, porous structure, and efficient adsorption performance of porous carbon fibers. MethodsPorous carbon fibers with desirable morphology and selective CO2 adsorption were prepared using an activation-free method. The fibers were prepared through melt spinning of phenolic resin modified with polyvinyl butyral (PVB) and urea, followed by curing treatment, and then the pore structures were adjusted by controlling the carbonization temperature. Significant findingsThe porous carbon fibers prepared by carbonization at 900 °C not only had excellent CO2 adsorption (3.48 mmol/g) but also possessed desirable tensile strength (132 MPa). Those carbonized at 700 °C showcased CO2/N2 selectivity of 57 (Ideal adsorption solution theory (IAST), CO2: N2 =15:85), and tensile strength of 148 MPa, much higher than most fibers using other methods. The study revealed that the adsorption of CO2 was mainly performed by nitrogen-containing groups and physical function through pores ranging from 0.3 nm to 0.8 nm, and the micropores between 0.3 nm and 0.6 nm were conducive to the selective adsorption of CO2/N2.

A new activity coefficient model for describing adsorbed phase nonidealities

The following study introduces a new adsorbed phase activity coefficient model, the SPLINT model, for describing nonideal multicomponent systems. The derivation is presented and simplification to the thermodynamically robust ABC model in the low‑coverage limit is shown. Both the SPLINT and ABC models are fit to 25 binary systems. Thermodynamic consistency of the data is established through multiple consistency tests including the intersection rule, closed‑loop reduced spreading pressure integration, the Gibbs‑Duhem integral equality, and alignment of pure and binary data. Regressed parameters are used in Real Adsorbed Solution Theory to predict both binary and ternary adsorption. The ABC and SPLINT models accurately describe systems with considerable nonideality when the activity coefficients are symmetric; however, the SPLINT model is required for accurate description of systems with complex, asymmetric activity coefficient behavior. The new model includes one more fitting parameter than the ABC model for a single binary system, but less across multiple data sets and higher‑order systems due to the theoretical basis. The importance of capturing activity coefficient asymmetry is demonstrated with a highly nonideal system across various pressures and temperatures measured by the present authors using the Integral Mass Balance method with HFC‑125 (pentafluoroethane), HFC‑32 (difluoromethane), and zeolite H‑ZSM‑5. The theoretical basis of the SPLINT model is further enforced by presenting two methods for predicting activity coefficients with the SPLINT model: correlating the fitting parameters individually and simultaneously regressing pure isosteric heat data to extract SPLINT fitting parameters. The study concludes with a discussion of pure and binary HFC‑125/HFC‑32 data with zeolite H‑ZSM‑5. A possible binary adsorption mechanism is proposed based on isosteric heat and excess enthalpy predictions and observed system irreversibility.

Asymmetry in Polymer-Solvent Interactions Yields Complex Thermoresponsive Behavior.

We introduce a lattice framework that incorporates elements of Flory-Huggins solution theory and the q-state Potts model to study the phase behavior of polymer solutions and single-chain conformational characteristics. Without empirically introducing temperature-dependent interaction parameters, standard Flory-Huggins theory describes systems that are either homogeneous across temperatures or exhibit upper critical solution temperatures. The proposed Flory-Huggins-Potts framework extends these capabilities by predicting lower critical solution temperatures, miscibility loops, and hourglass-shaped spinodal curves. We particularly show that including orientation-dependent interactions, specifically between monomer segments and solvent particles, is alone sufficient to observe such phase behavior. Signatures of emergent phase behavior are found in single-chain Monte Carlo simulations, which display heating- and cooling-induced coil-globule transitions linked to energy fluctuations. The framework also capably describes a range of experimental systems. Importantly, and by contrast to many prior theoretical approaches, the framework does not employ any temperature- or composition-dependent parameters. This work provides new insights regarding the microscopic physics that underpin complex thermoresponsive behavior in polymers.

Unveiling the power of defect engineering in MOF-808 to enhance efficient carbon dioxide adsorption and separation by harnessing the potential of DFT analysis

Defect engineering in metal–organic frameworks involves intentionally introducing defects to modify their properties. These defects create new active sites, increase surface area, and enhance overall performance in various applications. If properly designed, these intentional defects can create new active sites and increase the specific surface area and thus the overall performance in various applications. The defect engineering strategy in MOF-808 was evaluated using the second ligand agent in different ratios. The exceptional specific surface area obtained compared to the pristine MOF 808 indicates the realization of the improvement potential of MOFs and the increase of their performance capabilities, especially in adsorbing and sequestering carbon dioxide gas in order to achieve a carbon dioxide-free future. A comprehensive set of characterization tests was conducted on the synthesized compounds in order to determine their physical and chemical properties. The Langmuir isotherm and the Weber and Morris kinetic model were chosen as the most suitable models due to their consistent performance, with R2 values surpassing 0.95 across all linker concentrations. In MOF-808 with different linker ratios of 0 %, 5 %, 10 %, 15 %, and 20 %, adsorption capacities of 6.4, 5.9, 6.7, 8.3, and 6.4 mmol/g were determined at 25 °C and 110 kPa, respectively. Notably, MOF-808-15 % exhibited the highest adsorption capacity and a remarkable surface area of 3922 m2/g, indicating a substantial interaction between CO2 molecules and the available open metal sites. Also, MOF-808 variants showed great moisture stability after being exposed to moist air for 7 days. Furthermore, an assessment of this adsorbent’s performance under flue gas conditions, utilizing the Ideal Adsorption Solution Theory (IAST), underscored its outstanding selectivity, with a significant value of 760, representing a 115-fold increase compared to pristine MOF-808-0 %. Density Functional Theory (DFT) shows better adsorption energy for MOF-808-15 % compared to MOF-808-0 %, implying the effect of defects in CO2 adsorption. Isosteric heat of adsorption study shows chemisorption phenomenon for all MOF-808 variants. Lastly, an evaluation of the cyclic stability of these MOFs over ten adsorption–desorption cycles showcased their robust and enduring performance, offering promising prospects for industrial applications.

Remediation of oil-polluted soil using anionic and non-ionic composite biosurfactants

ABSTRACT Petroleum hydrocarbons as pervasive pollutants pose a significant threat to soil ecology and human health. Surfactant washing as an established technique can effectively remediate soils contaminated by hydrocarbons. Biosurfactants, which combine the properties of surfactants and environmental compatibility, have attracted increasing interest. However, due to the high production cost of biosurfactants, their practical application is restricted. This study addressed these limitations by selecting two biosurfactants, β-cyclodextrin (C1) and sodium carboxymethyl cellulose (C2), and developed a promising cleaning agent formula through compounding and the addition of suitable additives. When the volume ratio of C1 to C2 was 8:2 and an 8 g/L mixture of sodium humate and sodium carbonate electrolyte was added, the surfactant system’s surface tension reached a minimum, yielding optimal oil removal. The formation and synergistic behaviour of mixed micelles of surfactants were explained using ideal solution theory and the Rubingh model. By optimising the oil washing process parameters – normal temperature of 25 °C, pH 11, washing time of 2 h, solid–liquid ratio of 1:5, and oscillation frequency of 200 r/min – the oil removal rate achieved 76%. This cleaning agent, characterised by low production cost, straightforward application, environmental compatibility, and rapid, significant cleaning effect, shows potential for field-scale purification of petroleum-contaminated soil.

Selective CO2 adsorption over a Zr-based metal–organic framework functionalized with tris(2-aminoethyl)amine

Selective capture of CO2 from off-gas or even the atmosphere is very important against global warming. In this work, a stable Zr-based MOF (MOF-808) was loaded with –COOH group and was further functionalized with tris(2-aminoethyl)amine (TREN), via a reaction between –NH2 of TREN and –COOH sites. The derived materials were analyzed with X-ray diffraction, scanning electron microscopy, FTIR, X-ray photoelectron spectroscopy, thermogravimetry analysis, elemental analysis, and nitrogen adsorption at 77 K; utilized in the adsorption of CO2 and N2 at relatively low pressure up to 100 kPa. One functionalized MOF-808 prepared in this work (named MOF808-EDTA-TREN(0.3)) showed remarkable performances in the selective capture of CO2 under low pressure. For example, the invented MOF exhibited the highest selectivity (based on ideal adsorbed solution theory) in CO2 capture from N2 and CO2 (SCO2/N2: 519 at 100 kPa) compared with any reported results, thus far, using MOF-808-based materials. Moreover, MOF808-EDTA-TREN(0.3) was recyclable up to the fifth run upon vacuum treatment. The noticeable performances of MOF808-EDTA-TREN(0.3) in CO2 adsorption could be explained by the loaded –NH2 sites of TREN (that are effective in carbamates formation) and bulky TREN that decreases the nitrogen adsorption because of the reduced porosity of the MOF. In brief, loading bulky amines with ample amino groups like TREN can be suggested as an attractive way to modify a MOF for selective CO2 capture from off-gas of various industries.

Dipole Theory of Polyzwitterion Microgels and Gels.

The behavior of polyzwitterions, constituted by dipole-like zwitterionic monomers, is significantly different from that of uniformly charged polyelectrolytes. The origin of this difference lies in the intrinsic capacity of polyzwitterions to self-associate intramolecularly and associate with interpenetrating chains driven by dominant dipolar interactions. Earlier attempts to treat polyzwitterions implicitly assume that the dipoles of zwitterion monomers are randomly oriented. At ambient temperatures, the dipolar zwitterion monomers can readily align with each other generating quadrupoles and other multipoles and thus generating heterogeneous structures even in homogeneous solutions. Towards an attempt to understand the role of such dipolar associations, we present a mean field theory of solutions of polyzwitterions. Generally, we delineate a high-temperature regime where the zwitterion dipoles are randomly oriented from a low-temperature regime where quadrupole formation is significantly prevalent. We present closed-form formulas for: (1) Coil-globule transition in the low-temperature regime, the anti-polyelectrolyte effect of chain expansion upon addition of low molar mass salt, and chain relaxation times in dilute solutions. (2) Spontaneous formation of a mesomorphic state at the borderline between the high-temperature and low-temperature regimes and its characteristics. A universal law is presented for the radius of gyration of the microgel, as a proportionality to one-sixth power of the polymer concentration. (3) Swelling equilibrium of chemically cross-linked polyzwitterion gels in both the high temperature and low-temperature regimes. Addressing the hierarchical internal dynamics of polyzwitterion gels, we present a general stretched exponential law for the time-correlation function of gel displacement vector, that can be measured in dynamic light scattering experiments. The present theory is of direct experimental relevance and additional theoretical developments to all polyzwitterion systems, and generally to biological macromolecular systems such as intrinsically disordered proteins.

Effect analysis of temperature and initial water content on the adsorption and separation of VOCs by ZIF‐8 based on molecular simulation

AbstractVolatile organic compounds (VOCs) have seriously polluted the atmospheric environment and caused harmful effects on human beings, so it is imperative to control oil vapor emissions. Metal organic framework materials, as one of the porous materials, have a good application prospect in the field of gas adsorption separation. In this paper, the effects of temperature and initial water content (IWC) on the adsorption properties of methane (CH4), ethane (C2H6), ethylene (C2H4), propane (C3H8), and propylene (C3H6) by ZIF‐8 were studied by combining the large gauge Monte Carlo (GCMC) and ideal adsorption solution theory. The results show that the higher the temperature is, the lower the adsorption capacity is, the higher the threshold pressure is, and the single molecule interaction energy changes little with temperature. But pore volume is still the main influencing factor. The IWC also reduces the saturated adsorption capacity. Preloaded water molecules can enhance the electrostatic interaction at low pressure, thus affecting the adsorption heat and interaction energy of single molecules. Overall, these findings provide valuable mechanistic insights into the effects of temperature and IWC on the adsorption properties of ZIF‐8 for VOCs.

Elucidating the failure of the Ideal Adsorbed Solution Theory for CO2/H2O mixture adsorption in CALF-20

The use of CALF-20 for CO2 capture from humid flue gases has attracted a vast amount of attention from both academia and industry. The focus of this article is on CO2/H2O mixture adsorption for which published experimental data demonstrate the severe limitations of the Ideal Adsorbed Solution Theory (IAST) in providing a quantitative estimation of the component loadings. We use Configurational-Bias Monte Carlo (CBMC) simulations for elucidating the origins of thermodynamic non-idealities. The CBMC simulations reveal that two distinct regimes prevail during mixture adsorption. For low relative humidities less than say 20 %, the CO2 loadings in the adsorbed phase is significantly higher than anticipated by the IAST. CBMC simulations of intermolecular distances for guest pairs reveal that failure of the IAST can be traced to the phenomenon of segregated adsorption. The H2O-H2O pairs are close together at typical distances of about 3 Å. The CO2-H2O pairs are typically 8 Å apart, occupying adjacent adsorption sites corresponding to the O atoms of the oxalate group. This implies that the CO2 molecules face a less severe competitive adsorption with H2O than is anticipated by the IAST, whose applicability mandates a uniform and homogeneous distribution of adsorbates in pore space. The situation changes dramatically at high values of relative humidities, typically larger than about 50 %; in this scenario, the adsorbed phase is significantly richer in H2O. The CO2 molecules are compelled to share same adsorption site with pairs of H2O molecules that are hydrogen-bonded with each other. Consequently, the competition faced by CO2 is significantly higher than anticipated by the IAST, resulting in significantly lower CO2 uptakes.The important message that emerges from this investigation is the need to incorporate the Real Adsorbed Solution Theory (RAST) for quantitative modelling of fixed-bed adsorbers in CO2 capture with CALF-20.

Mitigating low-temperature corrosion in flue-gas heat exchangers for improving thermal storage efficiency in geothermal power plants

Geothermal energy is a renewable and stable energy source that plays a crucial role in providing heat. However, underground sulfur fumes have been restricting the heat exchange and storage rate for years. Accurately predicting the condensation of acidic substances of flue gas heat exchangers is crucial in geothermal heat storage. In this paper, Acid and water vapor have been firstly utilized to forecast the deposition of acidic substances on the heat exchanger’s surface. Utilizing gas–liquid equilibrium data of H2SO4-H2O solution and multi-component transport theory, a numerical model has been developed to analyze the impact of acidic deposition and corrosion characteristics. The results shown a correlation between the distribution of acidic substances deposition within heat exchanger and temperature distribution. When water vapor volume fraction increased from 3 % to 18 %, there was a corresponding increase of acid condensation rate on surface. Specifically, the condensation rate increased by 9.6 %, 16.1 %, and 21.8 % respectively under the flow rates of 5 m/s, 8 m/s, and 10 m/s, while the acid mass fraction decreased by 22.4 %, 20.7 %, and 20.1 % respectively. Acidic substances predominantly deposit in specific areas of the heat exchanger. Moreover, Water vapor content increasing can lead to reduction deposition of acidic substances, this may be a potential method to reduce loss of heat exchanger and increase heat storage. These findings offer a significant step forward heat exchanger technology and geothermal energy application.

Existence and Hyers–Ulam Stability of Stochastic Delay Systems Governed by the Rosenblatt Process

Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided.

The role of topological photon spheres in constraining the parameters of black holes

In this paper, we investigate the topological photon sphere from two distinct perspectives. In the first view, we examine the existence and characteristics of topological photon (anti-photon) spheres for black holes with different structures, such as Einstein–Young–Mills non-minimal, AdS black holes surrounded by Chaplygin-like dark fluid, and Bardeen-like black holes in Einstein–Gauss–Bonnet gravity. Furthermore, we delve into the deeper perspective of the necessity of photon spheres for super-compact gravitational structures such as black holes. By leveraging this necessity, we propose a classification of the parameter space of black hole models based on the existence and positioning of photon spheres. This approach enables the determination of parameter ranges that delineate whether a solution represents a black hole or a naked singularity. In essence, the paper illustrates the utility of the photon sphere as a notable test for establishing the permissible and non-permissible parameter ranges within specific theories of black hole solutions.

Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality:(⋆)∫|Stu0−Stv0|φ0dx≤∫|u0−v0|Gtφ0dx,∀φ0≥0,∀u0,∀v0, where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying (⋆). This amounts to finding an optimal weighted L1 contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation∂tφ=supξ⁡{F′(ξ)⋅Dφ+tr(A(ξ)D2φ)}. Since weighted L1 contraction results are mainly used for possibly nonintegrable L∞ solutions u, the natural spaces behind this duality are L∞ for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions φ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called Lint∞. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].