Solution Theory Research Articles

Weak Solutions of Mullins–Sekerka Flow as a Hilbert Space Gradient Flow

We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and 3, primarily motivated from a gradient flow perspective. Previous existence results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3, 1995) or Röger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both a sharp energy dissipation principle and a weak formulation of the contact angle at the intersection of the interface and the domain boundary. To incorporate these, we introduce a functional framework encoding a weak solution concept for Mullins–Sekerka flow essentially relying only on (i) a single sharp energy dissipation inequality in the spirit of De Giorgi, and (ii) a weak formulation for an arbitrary fixed contact angle through a distributional representation of the first variation of the underlying capillary energy. Both ingredients are intrinsic to the interface of the evolving phase indicator and an explicit distributional PDE formulation with potentials can be derived from them. The existence of weak solutions is established via subsequential limit points of the naturally associated minimizing movements scheme. Smooth solutions are consistent with the classical Mullins–Sekerka flow, and even further, we expect our solution concept to be amenable, at least in principle, to the recently developed relative entropy approach for curvature driven interface evolution.

Carbon dioxide capture using diaminoalkane-grafted MIL-101(Cr)s: Critical role of geometric configuration of loaded amines

Preventing the increase of CO2 content in the atmosphere is crucial to mitigating climate change. In this study, we investigated the adsorption of CO2 using isostructural Cr-based MOFs, MIL-101 and MIL-100, after grafting diaminoalkanes with various chain lengths to determine the effect of the geometric configuration of the amines and to develop a competitive adsorbent. The performance of CO2 adsorption (e.g., adsorption capacity, selectivity, and adsorption heat) over MIL-101 increased as the size of diaminoalkanes increased, topped with 1,4-butanediamine, and declined with longer diaminoalkanes. The favorable contribution of grafted 1,4-butanediamine in increasing the performances in CO2 adsorption could be explained by the adequate size of the amines for the optimized configuration in the pore window of the MIL-101. However, interestingly, none of the MIL-100s grafted with diaminoalkanes showed a similar positive effect probably because of the smaller pore window of MIL-100 than that of MIL-101. To the best of our knowledge, this is the first study to demonstrate the importance of geometric configuration (or, the chain length of amines) in amine-grafted MOFs for the CO2 adsorption that follows the general mechanism that requires interactions between two uncoordinated amines for the formation of ammonium carbamates. Moreover, one of the modified MIL-101 grafted with 1,4-butanediamine exhibited remarkable performance in CO2 adsorption with 4.0 times the adsorption capacity of CO2 at 15 kPa and 20.6 times CO2/N2 selectivity (based on ideal adsorbed solution theory, at 100 kPa) compared to the virgin MIL-101.

Viscosity Anomalies of Magnetic Fluid Stabilized by a Double Layer of Surfactant in Water

Purpose to study the temperature dependence of the viscosity of a water-based magnetic fluid.Method. Consists of measuring the dynamic viscosity of a magnetic fluid based on cobalt ferrite particles stabilized in water by a double layer of surfactant. Lauric acid (first layer) and a mixture of lauric acid and sodium dodecyl sulfate (second layer) were used as a stabilizer. Measurements were performed using a Brookfield DV-II+Pro rotational viscometer equipped with a coaxial cylinder system. The measuring system of the viscometer was thermostated using a KRIO-VT-12-1 thermostat.Results. The temperature dependences of dynamic viscosity were measured for three samples of magnetic fluid based on cobalt ferrite particles of various concentrations in the temperature range 0–90C°. The obtained temperature dependences of viscosity are radically different from the temperature dependences both predicted by known theories and experimentally observed for kerosene-based magnetic fluids. According to well-known theoretical models, the ratio of the viscosity of the magnetic fluid to the viscosity of the base medium is a certain universal function of particle concentration. Different models offer different types of this feature. But it clearly follows from them that the viscosity ratio should not depend on temperature. For magnetic fluids based on kerosene, it has been experimentally established that its relative viscosity decreases with increasing temperature. However, according to the results obtained, the relative viscosity of a water-based magnetic fluid does not decrease with increasing temperature, but increases significantly. That is, the viscosity of a water-based magnetic fluid decreases more slowly with increasing temperature than the viscosity of water.Conclusion. The observed dependencies completely contradict the known patterns, both theoretical and experimentally established for kerosene-based magnetic fluids. The results obtained may be useful for the further development of the theory of aqueous colloidal solutions with particle stabilization by a double layer of surfactants.

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.

Local composition-regular solution theory for analysis of pharmaceutical solubility in mixed-solvents

Local composition regular solution theory (LC-RST) model was developed by introducing non-equal i–j and j–i parameters into regular solution theory (RST) to account for specific molecular interactions. The LC-RST model was applied to the correlation of solubilities of active pharmaceutical ingredients (API) in hydrogen bond donor (HBD) – hydrogen bond acceptor (HBA) mixed-solvent systems. In the assessment of 32 API-mixed-solvent systems that contained 11 different APIs and 1947 data, it was determined that the LC-RST model had a logarithmic average relative deviation (ARDln) in solubility of 0.028, the Wilson model had an ARDln value of 0.24 and the purely predictive RST model had an ARDln value of 1.29. Partial molar excess enthalpies and entropies of the API in the mixed-solvents determined from the LC-RST model allowed estimation of API solubility regions where specific interactions occur between API and HBD-HBA complex molecules. In the correlation scheme, component numbers were designated as API molecule (1) – HBD molecule (2) - HBA molecule (3) and systems were grouped according to HBD molecule interactions with the API (pro-solvent, anti-solvent, synergistic with HBA molecule). Phenomenological parameter sensitivity analysis of the LC-RST model showed that dominant specific interactions tended to be 1–2 or 2–1 molecules for pro-solvent groupings, 2–3 or 2–3 molecules for anti-solvent groupings and all i–j and j-–i molecular pairs for synergistic HBD-HBA molecule groupings. Based on parameters determined from a given API-mixed-solvent system, the LC-RST model allows qualitative prediction of solubilities for other systems. The LC-RST model is simple to apply and it retains inherent predictive characteristics of regular solution theory.

Efficient modeling and simulation of gas separations applying Maxwell-Stefan approach and Ideal Adsorbed Solution Theory

We present an efficient way to incorporate the Maxwell-Stefan approach (Krishna, 1990) for modeling gas separation processes, i.e., adsorbent particles and membranes. Ideal Adsorbed Solution Theory (Myers and Prausnitz, 1965) is applied for the description of competitive equilibria. The proposed calculation method introduces analytical expressions for the partial derivatives of the loadings w.r.t. fluid-phase concentrations, developed by Rubiera Landa et al. (2013), in the calculation of the thermodynamic correction factors, as required by Maxwell-Stefan flux equations. These expressions are solved simultaneously with transient mass balances that describe the gas separation processes in a differential-algebraic equation formulation. Robustness and advantages of the proposed solution approach are demonstrated with several examples adapted from the literature.

Construction of the mosaic aluminophosphate zeotypes based on nonclassical crystallization and in situ selective etching strategies

Owing to short diffusion lengths and more accessible active sites, hierarchical zeolitic materials can substantially improve catalytic activities and lifetimes compared to the conventional counterparts. Here we develop a straightforward and mesopore-template-free strategy to prepare aluminophosphate-based LTA zeotypes with mosaic architectures and hierarchical porosities. The influence of synthesis parameters and time-dependent study on the crystallization process were studied comprehensively using multiple techniques. The results demonstrate that the mosaic architectures were formed based on nonclassical crystallization, i.e., phase transformation of a lamellar phase to aligned crystalline AlPO4-42 nanoparticles followed by crystal fusion, and in situ selective etching of grain boundaries and defects. Meanwhile, heteroatom-substituted AlPO4-42 zeotypes with the mosaic morphologies can also be synthesized, which show enhanced ideal adsorbed solution theory (IAST) selectivities and efficient dynamic separation performances in the CO2/N2 mixture. This work not only provides a novel mesoporogen-free strategy for the preparation of hierarchical zeotypes, but also sheds new insight into the crystallization pathway.

Temperature dependent extensions of the Cahn–Hilliard equation

The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In this paper we focus on the dynamics of these binary media, when the underlying temperature is not constant.The aim of this paper is twofold. We first derive two distinct models that extend the classical Cahn-Hilliard equation with an evolutionary equation for the absolute temperature. Secondly, we analyse the local well-posedness of classical solution for one of these systems.Our modelling introduces the systems of PDEs by means of a general and unified formalism. This formalism couples standard principles of mechanics together with the main laws of thermodynamics. Our work highlights how certain assumptions on the transport of the temperature effect the overall physics of the systems.The variety of these thermodynamically consistent models opens the question of which one should be more appropriate. Our analysis shows that one of the derived models might be more desirable to the well-posedness theory of classical solutions, a property that might be natural as a selection criteria.We conclude our paper with an overview and comparison of our modelling formalism with some equations, which were previously derived in literature.

The impact of trace amounts of CO2 on the high-pressure adsorption of CH4 on 5A zeolite

The impact of trace amounts of CO2 on CH4 adsorption in a zeolite at elevated pressure was examined using experimental, binary adsorption isotherms, and molecular simulations. CH4 upgrading is important for a variety of applications, and the use of zeolites for CH4 upgrading has been considered previously but, to the best of our knowledge, not at pressures between 60 and 70 bar. The non-ideality of the adsorption was examined by comparing the experimental loadings to predicted loadings from the Ideal Adsorbed Solution Theory (IAST). When adsorption occurs at a total pressure of 58–66 bar, and a CO2 fraction below 0.42 mol %, there was less than 5 % error in the CH4 IAST predicted loading compared to the experimental value, but the deviation in CO2 loading between IAST and experiments, even at these low CO2 mole percentages, varied nominally from 4 to 13 %. Above 0.42 mol % deviations from IAST were more severe. Grand Canonical Monte Carlo simulations captured the trends in the adsorption data and illustrated how CO2 molecules align in the pore when adsorbed such that oxygens on both ends of the molecule can interact with cations in the zeolite. The results highlight the applicability of adsorbent materials to separate CO2 from process streams at elevated pressure and provide molecular insights that can aid the design of adsorbent structures.

Interchangeable effect of polyols-based zeolite on the separation of CO2, CH4, and N2 gases

The adsorptive separation process for CO2/CH4 and CH4/N2 gas separation has commercial potential, in biogas, Coalbed Methane process etc. although it remains to strive due to the absence of effective adsorbents. Zeolite NaY was used in this study due to its uniform pore structures, surface properties, and potential gas separation adsorbent. Zeolite NaY was modified by polyether polyols (PEPO) to get an adsorbent for selective separation of CO2/CH4 and CH4/N2 gas mixtures. The PXRD, FT-IR, GPC, 1H NMR, TEM, and surface area analysis were used to characterize the PEPO and PEPO-modified zeolite. The single component equilibrium adsorption desorption isotherms with PEPO modified NaY zeolites and NaY zeolite for CO2, CH4 and N2 were measured at 298, 313, and 328 K and up to 5 bar pressure. The isotherm data of CO2, CH4 and N2 were fitted in Dual Site Langmuir equations, and adsorption selectivity of CO2/CH4 and CH4/N2 gas mixtures was predicted using Ideal Adsorbed Solution Theory (IAST) method for 90:10, 50:50 (vol/vol) CH4/N2 and 80:20, 50:50 (vol/vol) CO2/CH4 feed gas mixtures representing coal bed methane gas composition and pressure conditions. The high surface area and hydroxyl groups in PEPO-modified NaY enhanced selectivity for CO2/CH4 and CH4/N2 gas separation. The selectivity predicted by the IAST method were 106.12 and 95.34 for 0.5 % PEPO-NaY (80:20 v/v and 50:50 v/v) for CO2/CH4 and 3.95 and 3.52 for 2 % PEPO-NaY (90:10 v/v and 50:50 v/v) for CH4/N2 at 298 K up to 5 bar respectively.

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

Efficient results on unbounded solutions of fractional Bagley-Torvik system on the half-line

<abstract><p>The fractional Bagley-Torvik system (FBTS) is initially created by utilizing fractional calculus to study the demeanor of real materials. It can be described as the dynamics of an inflexible plate dipped in a Newtonian fluid. In the present article, we aim for the first time to discuss the existence and uniqueness (E&U) theories of an unbounded solution for the proposed generalized FBTS involving Riemann-Liouville fractional derivatives in the half-line $ (0, \infty) $, by using fixed point theorems (FPTs). Moreover, the Hyers-Ulam stability (HUS), Hyers-Ulam-Rassias stability (HURS), and semi-Hyers-Ulam-Rassias stability (sHURS) are proved. Finally, two numerical examples are given for checking the validity of major findings. By investigating unbounded solutions for the FBTS, engineers gain a deeper understanding of the underlying physics, optimize performance, improve system design, and ensure the stability of the motion of real materials in a Newtonian fluid.</p></abstract>

Linear constraints on variables in influence diagrams for causal models

The paper considers some ways of representing probabilistic causal models using Bayesian network theory (hereafter referred to as BN). These models describe well problems with different types of uncertainty. The theory of BN extended with some additional types of nodes called influence diagrams or IDs. Influence diagrams make it possible to consider a number of solution options, to evaluate them quantitatively, and to select the best of the considered options. However, it is practically impossible to find an optimal solution in an ID. It is not even possible to create a system of linear constraints on some variables in an ID, although there is a large class of practical problems with such constraints. The paper describes the idea of extending ID to describe linear constraints on some variables of the BN. In the future, it will help to use the ideas of linear programming in ID to find an optimal solution in the sense of LP for problems with different types of uncertainties and causal relationships between some variables. This work has been done under grant AP19679142 "Search for optimal solutions in Bayesian networks in models with linear constraints and linear functionals. Development of algorithms and programs " (2023-2025) of MES RK. This project will develop the theory for finding optimal solutions in Bayesian networks. Optimality will be understood in the sense of linear programming - a system of linear constraints, extremum of a linear functional. The theory will be implemented in a software product.

Mathematical models and algorithms for determining the modes of an intelligent stage control sensor

In article discusses models of unlimited track circuits with a receiver, adapting to different working conditions, applied double receiver with one power supply in the middle, which allows you to extend the length of the continuous rail circuit. For this, a goal is set, in which is the development of models of jointless track circuits with an adaptive receiver, in which the rail line must take into account the total stress distribution along the rail line and disturbing factors. The article presents the equivalent circuits of continuous welded rail circuits with an adaptive receiver, algorithmic expressions for determining currents and voltages at the ends of the rail circuit, the coefficients of rail lines with an adaptive receiver are developed, calculation algorithms are presented. In conclusion, a graph is given, according to which it is shown that a continuous rail circuit with an adaptive receiver can operate with an insulation resistance three times lower than the threshold value in any modes of the rail circuit, taking into account the fact that, minimum allowable insulation resistance for tone-on-tone continuous rail circuits is taken as Ri=0,7ohm*km. Derived for the theory and synthesis of solutions, and also proposed new approaches to ensure the operating modes of the rail circuit, taking into account the real and weather conditions of operation.

Trace SO2 capture within the engineered pore space using a highly stable SnF62--pillared MOF.

Developing reliable solid sorbents for efficient capture and removal of trace sulfur dioxide (SO2) under ambient conditions is critical for industrial desulfurization operations, but poses a great challenge. Herein, we focus on SNFSIX-Cu-TPA, a highly stable fluorinated MOF that utilizes SnF62- as pillars, for effectively capturing SO2 at extremely low pressures. The exceptional affinity of SNFSIX-Cu-TPA towards SO2 over CO2 and N2 was demonstrated through single-component isotherms and corroborated by computational simulations. At 298 K and 0.002 bar, this material displays a remarkable gas uptake of 2.22 mmol g-1. Among various anion fluorinated MOFs, SNFSIX-Cu-TPA shows the highest SO2/MF62- of 1.39 mmol mmol-1 and exhibits a low Qst of 58.81 kJ mol-1. Additionally, SNFSIX-Cu-TPA displays excellent potential for SO2/CO2 separation, as evidenced by its ideal adsorbed solution theory (IAST) selectivity of 148 at a molar fraction of SO2 of 0.01. Dynamic breakthrough curves were obtained to reveal the effective removal of trace SO2 from simulated flue gas (SO2/CO2/N2; v/v/v 0.2/10/89.8) with a high dynamic capacity of up to 1.52 mmol g-1. Furthermore, in situ TGA demonstrated the efficient and reversible capture of 500 ppm SO2 over 20 adsorption-desorption tests. This durable material presents a rare combination of exceptional SO2 capturing performance, good adsorption selectivity, and mild regeneration, thus making it a good candidate for a realistic desulfurization process.

Synergistic effects between a non-ionic and an anionic surfactant on the micellization process and the adsorption at liquid/air surfaces.

Predicting the behaviour of solutions with surfactants of significantly different critical micelle concentration (CMC) values remains a challenge. The study of the molecular interactions within micelles and interfaces in surfactant combinations used in everyday products is essential to understand these complex systems. In this work, the equilibrium and dynamic surface tension in the presence of mixed non-ionic (tristyrylphenol ethoxylates) and anionic (sodium benzene sulfonate with alkyl chain lengths of C10-C13) surfactants, commonly encountered as delivery systems in agrochemicals, were studied and their CMC values were determined. For the surfactant mixtures, four molar ratios were examined: nEOT/nNaDDBS = 0.01, 0.1, 1, 4 and two different cases were analysed, the premixed and the add one by one surfactant. The surface tension for single surfactants stabilised quickly, while the mixtures needed a long time to reach equilibrium; up to 15 h for the premixed mixtures and 40 min when surfactants were added one by one. The CMC values for the nEOT/nNaDDBS = 0.01, 0.1 premixed surfactant mixtures were found to be in between the CMC values of the single surfactants, but those for the nEOT/nNaDDBS = 1 and 4 mixtures were lower than the CMCs of both single surfactants. Calculations based on the regular solution theory suggested that there are attractive forces in the mixed micelles and at the interface layers, while the supramolecular assemblies in the bulk (i.e., micelles) and at interfaces (surfactant films) are preferentially enriched in EOT.

Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes

For a semi-linear Schrödinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude, obeying different scaling limits. The corresponding nets of approximate solutions represent actual generalised solutions for the singular-perturbed Schrödinger equation. The behaviour of such nets is investigated, comparing the distinct scaling regimes that yield, respectively, the Hartree equation with point interaction Hamiltonian vs the ordinary Hartree equation with the free Laplacian. In the second case, the distinguished regime admitting a generalised solution in the Colombeau algebra is studied, and for such a solution compatibility with the classical Hartree equation is established, in the sense of the Colombeau generalised solution theory.

Influence of Surfactant Structure on Polydisperse Formulations of Alkyl Ether Sulfates and Alkyl Amidopropyl Betaines.

Surfactants provide detergency, foaming, and texture in personal care formulations, yet the micellization of typical industrial primary and cosurfactants is not well understood, particularly in light of the polydisperse nature of commercial surfactants. Synergistic interactions are hypothesized to drive the formation of elongated wormlike self-assemblies in these mixed surfactant systems. Small-angle neutron scattering, rheology, and pendant drop tensiometry are used to examine surface adsorption, viscoelasticity, and self-assembly structure for wormlike micellar formulations comprising cocoamidopropyl betaine, and its two major components laurylamidopropyl betaine and oleylamidopropyl betaine, with sodium alkyl ethoxy sulfates. The tail length of sodium alkyl ethoxy sulfates was related to their ability to form wormlike micelles in electrolyte solutions, indicating that a tail length greater than 10 carbons is required to form wormlike micelles in NaCl solutions, with the decyl homologue unable to form elongated micelles and maintaining a low viscosity even at 20 wt % surfactant loading with 4 wt % NaCl present. For these systems, the incorporation of a disperse ethoxylate linker does not enable shorter chain surfactants to elongate into wormlike micelles for single-component systems; however, it could increase the interactions between surfactants in mixed surfactant systems. For synergy in surfactant mixing, the nonideal regular solution theory is used to study the sulfate/betaine mixtures. Tail mismatch appears to drive lower critical micelle concentrations, although tail matching improves synergy with larger relative reductions in critical micelle concentrations and greater micelle elongation, as seen by both tensiometric and scattering measurements.

Computing Bayes–Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging

Determining equilibria in auction games is computationally hard in general, and no exact solution theory is known. We introduce an algorithmic framework in which we discretize type and action space and then learn distributional strategies via online optimization algorithms. We show that the equilibrium of the discretized auction game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the approach approximates the analytical (pure) Bayes–Nash equilibrium closely. In standard models in which agents are symmetric, we find equilibrium in seconds. The method allows for interdependent valuations and different types of utility functions, and it can be used to find equilibrium in auction markets and beyond.

Fine Tuning Metal-Organic Frameworks with Halogen Functional Groups for Ethylene Purification.

One-step C2H4 purification from a mixture of C2H2/C2H4/C2H6 could be achieved by metal-organic framework (MOF) NTUniv-70 with an F-functional group. The selectivities of C2H4/C2H6 and C2H4/C2H2 of NTUnvi-70 based on ideal adsorbed solution theory were at least twice that of the original MOF platform, which was in line with the enthalpy of adsorption (Qst) and breakthrough testing. Grand canonical Monte Carlo simulations indicated that the C-H···F interactions played an important role in enhanced C2H4/C2H6 and C2H4/C2H2 adsorption selectivities.