Behavior Of Solutions Research Articles

Electrochemical Characterization of Recast Nafion® Film-Modified Electrodes in Acetonitrile with Various Electrolytes

Nafion® is a cation exchange polymer that is commonly used in aqueous energy applications such as fuel cells due to its ability to exclude anions and neutral molecules and increase apparent diffusion of cationic redox molecules. However, this behavior is not well studied in nonaqueous solutions. The behavior of platinum electrodes modified with recast Nafion® films in nonaqueous solutions was observed to be different from its well-studied behavior in aqueous solutions. The reversible redox couple tris(2,2′-bipyridine)ruthenium(II) hexafluorophosphate was studied in the nonaqueous, aprotic solvent acetonitrile with different electrolytes (tetrabutylammonium tetrafluoroborate, tetrabutylammonium trifluoromethanesulfonate, tetrabutylammonium hexafluorophosphate, and ammonium trifluoromethanesulfonate) using cyclic voltammetry and rotating disk voltammetry. An unmodified platinum electrode in the nonaqueous systems and a recast Nafion®-modified platinum electrode equilibrated in an aqueous solution of tris(bipyridine)ruthenium(II)chloride hexahydrate were used as controls. Results indicate that the polymer structure in acetonitrile conditions does not allow apparent (Dahms–Ruff) diffusion but does allow significant physical diffusion that would make Nafion a great immobilization option for modifying electrodes with catalysts in nonaqueous systems.

Pivotal Role of the Intracellular Microenvironment in the High Photodynamic Activity of Cationic Phthalocyanines.

To investigate the influence of phthalocyanine aggregation on their photodynamic activity, a series of six cationic water-soluble zinc(II) phthalocyanines bearing from four to sixteen 4-((diethylmethylammonium)methyl)phenoxy substituents was synthesized. Depending on their structure, the phthalocyanines have different aggregation behaviors in phosphate buffer solutions ranging from fully assembled to monomeric states. Remarkably, independent of aggregation in buffer, very high photodynamic efficiencies against the tumor cell lines MCF-7 and MDA-MB-231 in the nanomolar range were found for all investigated phthalocyanine, and the IC50(light) varied from 27 to 358 nM (3.5 J/cm2, 660 nm) with IC50(dark)/IC50(light) ratios up to ∼3700. This is due to the intracellular disassembly of aggregated phthalocyanines with the formation of monomeric photoactive forms, as demonstrated by fluorescence microscopy. Indeed, the interaction of aggregated phthalocyanines with serum proteins in a buffer resulted in the disassembly of nonluminescent aggregate species with the release of photoactive monomers bound to protein macromolecules.

Asymptotic expansions for harmonic functions at conical boundary points

We prove three theorems about the asymptotic behavior of solutions u to the homogeneous Dirichlet problem for the Laplace equation at boundary points with tangent cones. First, under very mild hypotheses, we show that the doubling index of u either has a unique finite limit, or goes to infinity; in other words, there is a well-defined order of vanishing. Second, under more quantitative hypotheses, we prove that if the order of vanishing of u is finite at a boundary point 0 , then locally u(x) = |x|^{m} \psi(x/|x|) + o(|x|^{m}) , where |x|^{m} \psi(x/|x|) is a homogeneous harmonic function on the tangent cone. Finally, we construct a convex domain in three dimensions where such an expansion fails at a boundary point, showing that some quantitative hypotheses are necessary in general. The assumptions in all of the results only involve regularity at a single point, and in particular are much weaker than what is necessary for unique continuation, monotonicity of Almgren’s frequency, Carleman estimates, or other related techniques.

Pirfenidone microcrystals for pulmonary delivery: Regulation of the precipitation behavior in the supercooled droplet.

Pirfenidone (PFD) is one of the first-line drugs for treating idiopathic pulmonary fibrosis, while directly delivering PFD to lung showed better efficiency. However, PFD is a non-glass former and easily precipitates into larger-sized crystals that are undesirable for pulmonary delivery. Hence, the fabrication of PFD particles with pulmonary delivery efficiency remains challenging. Herein, a series of particles were prepared by spray freeze drying a PFD and leucine mixed solution. The sub-ambient behavior of the mixed solution was evaluated via a differential scanning calorimeter. The effects of the PFD/leucine mass ratio and freezing temperature on the particle morphology, size, crystal polymorphism, molecular structure and in vitro aerosol performance were investigated. Shortening the lifetime of the droplet and adding proper amounts of leucine are the keys to decreasing the PFD crystal size and improving its dispersity. The optimal sample is SF-80D-P95L5-2, with high FPF and eFPF values of∼65.97% and∼27.86%, and owing to its high drug loading (95%), the FPD and eFPD are extremely high at∼6.27mg and∼2.65mg, respectively, equivalent to∼6.27mg and∼2.65mg PFD deposited in the lungs and alveoli, respectively, when 10mg dry powder is inhaled. This work provides a potential strategy for tuning the precipitation behavior of PFD microcrystals with high pulmonary drug delivery efficiency.

Normalized ground state solutions for critical growth Schrödinger equations with Hardy potential

In this article, we study the following Schrödinger equation \begin{align*} \begin{cases} -\Delta u -\frac{\mu}{|x|^2} u+\lambda u =f(u), &\text{in}~ \mathbb{R}^N\backslash\{0\},\\ \int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d} x=a, & u\in H^1(\mathbb{R}^{N}), \end{cases} \end{align*} where $N\geq 3$ , a > 0, and $\mu \lt \frac{(N-2)^2}{4}$ . Here $\frac{1}{|x|^2} $ represents the Hardy potential (or ‘inverse-square potential’), λ is a Lagrange multiplier, and the nonlinearity function f satisfies the general Sobolev critical growth condition. Our main goal is to demonstrate the existence of normalized ground state solutions for this equation when $0 \lt \mu \lt \frac{(N-2)^2}{4}$ . We also analyse the behaviour of solutions as $\mu\to0^+$ and derive the existence of normalized ground state solutions for the limiting case where µ = 0. Finally, we investigate the existence of normalized solutions when µ < 0 and analyse the asymptotic behaviour of solutions as $\mu\to 0^-$ .

On Large Amplitude Vibrations of the Softening Duffing Oscillator at Low Excitation Frequencies—Some Fundamental Considerations

The Duffing equation containing a cubic nonlinearity is probably the most popular example of a nonlinear oscillator. For its harmonically excited, slightly damped, and softening version, stationary large amplitude solutions at subcritical excitation frequencies are obtained when standard semi-analytical methods like Harmonic Balance or Perturbation Analysis are applied. These solutions have the shape of a nose in the amplitude-frequency diagram. In prior work, it has been observed that these solutions may contain large errors and that high ansatz orders may be necessary when applying the Harmonic Balance or other semi-analytical methods to make them converge. Some of these solutions are observed to be asymptotically stable, while in most cases, they are unstable. The current paper aims to give a descriptive explanation for this behavior of the nose solutions, which is mainly related to the exact solution of the free undamped vibrations. Based on this, approximations of the nose solutions are calculated with a procedure combining properties of Perturbation Analysis and Harmonic Balance. Therein, the exact solution of the free undamped vibrations is taken as the zeroth approximation, while higher-order solution parts are calculated by balancing the harmonics, and the phase shift of the zeroth approximation is calculated by a residuum minimization. This method just requires the solution of a system of linear algebraic equations, while systems of nonlinear algebraic equations have to be solved in the case of directly applying Harmonic Balance.

On the qualitative behavior of solutions for a quasilinear $$\Delta _p$$ equation in the Heisenberg group

On the qualitative behavior of solutions for a quasilinear $$\Delta _p$$ equation in the Heisenberg group

Unlocking the Potential of Redox‐Active Copper Complexes in Thin Films via Proton‐Coupled Electron Transfer for Enhanced Supercapacitor Performance

AbstractNanoscale redox‐active molecular films are promising candidates for next‐generation energy storage applications due to their ability to facilitate long‐range charge transport. However, establishing stable and efficient electrode‐molecule interfaces remains a critical challenge. In this study, the properties of redox‐active copper‐polypyridyl thin films covalently bonded to graphite rods are explored, investigating their potential as supercapacitors. Using an electrochemical grafting method, robust covalent interfaces are created, resulting in copper‐polypyridyl films prepared on graphite rods and indium tin oxide (ITO) electrodes, exhibiting both Cu(II) and Cu(I) redox states. These redox‐active mettalo‐oligomeric films demonstrate a structural transition between octahedral and tetrahedral geometries around the Cu(II), and Cu(I), respectively contributing to their charge storage capabilities. The combination of an electrical double‐layer capacitance and pseudocapacitance through Faradaic charge transfer is evaluated in different acidic electrolytes, showing significant capacitance enhancement. Notably, proton‐coupled electron transfer (PCET) at free pyridine‐N sites in Cu(I) polypyridyl complex is identified as a key factor in their distinct behavior in aqueous solutions, a finding supported by computational studies. This study shows the potential of binder‐free thin films for efficient supercapacitor applications, with a maximum areal capacitance of 6.8 mF cm⁻2 in aqueous media, representing an 1840% improvement over bare graphite rods.

Colloidal Dispersions of Sterically and Electrostatically Stabilized PbS Quantum Dots: Structure Factors, Second Virial Coefficients, and Film-Forming Properties.

Electrostatically stabilized nanocrystals (NCs) and, in particular, quantum dots (QDs) hold promise for forming strongly coupled superlattices due to their compact and electronically conductive surface ligands. However, studies of the colloidal dispersion and interparticle interactions of electrostatically stabilized sub-10 nm NCs have been limited, hindering the optimization of their colloidal stability and self-assembly. In this study, we employed small-angle X-ray scattering (SAXS) experiments to investigate the interparticle interactions and arrangement of PbS QDs with thiostannate ligands (PbS-Sn2S64-) in polar solvents. The study reveals significant deviations from the ideal solution behavior in electrostatically stabilized QD dispersions. Our results demonstrate that PbS-Sn2S64- QDs exhibit long-range interactions within the solvent, in contrast to the short-range steric repulsion characteristic of PbS QDs with oleate ligands (PbS-OA). Introducing highly charged multivalent electrolytes screens electrostatic interactions between charged QDs, reducing the length scale of the repulsive interactions. Furthermore, we calculated the second virial (B2) coefficients from SAXS data, providing insights into how surface chemistry, solvent, and size influence pair potentials. Finally, we explore the influence of long-range interparticle interactions of PbS-Sn2S64- QDs on the morphology of films produced by drying or spin-coating colloidal solutions. The long-range repulsive term of PbS-Sn2S64- QDs promotes the formation of amorphous films, and screening the electrostatic repulsion by the addition of an electrolyte enables the formation of crystalline domains. These findings highlight the critical role of NC-NC interactions in tailoring the properties of functional materials made of colloidal NCs.

The nature of the hydrophobic interaction varies as the solute size increases from methane's to C60's.

The hydrophobic interaction, often combined with the hydrophilic or ionic interactions, makes the behavior of aqueous solutions very rich and plays an important role in biological systems. Theoretical and computer simulation studies have shown that the water-mediated force depends strongly on the size and other chemical properties of the solute, but how it changes with these factors remains unclear. We report here a computer simulation study that illustrates how the hydrophobic pair interaction and the entropic and enthalpic terms change with the solute size when the solute-solvent weak attractive interaction is unchanged with the solute size. The nature of the hydrophobic interaction changes qualitatively as the solute size increases from that of methane to that of fullerene. The potential of mean force between small solutes has several well-defined extrema, including the third minimum, whereas the potential of mean force between large solutes has the deep contact minimum and the large free-energy barrier between the contact and the water-bilayer separated configurations. The difference in the potential of mean force is related to the differences in the water density, energy, and hydrogen bond number distributions in the vicinity of the pairs of hydrophobic solutes.

Darcy–Forchheimer gravity currents in porous media

We theoretically and experimentally study gravity currents of a Newtonian fluid advancing in a two-dimensional, infinite and saturated porous domain over a horizontal impermeable bed. The driving force is due to the density difference between the denser flowing fluid and the lighter, immobile ambient fluid. The current is taken to be in the Darcy–Forchheimer regime, where a term quadratic in the seepage velocity accounts for inertial contributions to the resistance. The volume of fluid of the current varies as a function of time as $\sim T^{\gamma }$ , where the exponent parameterizes the case of constant volume subject to dam break ( $\gamma =0$ ), of constant ( $\gamma =1$ ), waning ( $\gamma <1$ ) and waxing inflow rate ( $\gamma >1$ ). The nonlinear governing equations, developed within the lubrication theory, admit self-similar solutions for some combinations of the parameters involved and for two limiting conditions of low and high local Forchheimer number, a dimensionless quantity involving the local slope of the current profile. Another parameter $N$ expresses the relative importance of the nonlinear term in Darcy–Forchheimer's law; values of $N$ in practical applications may vary in a large interval around unity, e.g. $N\in [10^{-5},10^{2}]$ ; in our experiments, $N\in [2.8,64]$ . Sixteen experiments with three different grain sizes of the porous medium and different inflow rates corroborate the theory: the experimental nose speed and current profiles are in good agreement with the theory. Moreover, the asymptotic behaviour of the self-similar solutions is in excellent agreement with the numerical results of the direct integration of the full problem, confirming the validity of a relatively simple one-dimensional model.

On the existence and asymptotic behavior of weak solution of the kinetic Fokker-Planck equation in a bounded domain with absorbing boundary

We study here the kinetic Fokker-Planck equation in a bounded domain with absorbing boundary. We show the existence and the uniqueness of a weak solution to the initial-boundary problem and we associate it to a positive compact strongly continuous semigroup. Then, we establish that the weak solution satisfies the balanced exponential growth, meaning both exponential decay toward zero and a thermalization phenomenon.

Asymptotic behavior of complete conformal metric near singular boundary

The boundary behavior of the singular Yamabe problem has been extensively studied near sufficiently smooth boundaries, while less is known about the asymptotic behavior of solutions near singular boundaries. In this paper, we study the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive the optimal estimates for the background metric which is not necessarily conformally flat. In particular, we prove that the solutions are well approximated by the solutions in tangent cones at singular points on the boundaries.

Diffusion-induced stress in crystals: Implications for timescales of mountain building

Intracrystalline chemical diffusion offers valuable insights into the durations of metamorphic and igneous processes. However, it can yield timescale estimates for orogenic and subduction zone events that are considerably shorter than those obtained via isotopic geochronology. One potential explanation that has been offered for the discrepancy is that the interdiffusion of species with different atomic or ionic radii may generate intracrystalline, compositional stresses that alter or limit diffusional relaxation. In this study we test this idea by developing and applying the compositional stress theory of materials scientists F. Larché and J. Cahn to garnet from the Barrovian sillimanite zone, Scotland. Relaxed contacts from the garnet, independent diffusion chronometers, and thermal modeling all indicate a > 100 kyr duration for peak temperature metamorphism. Nonetheless, the garnet records sharp, μm-scale variations in calcium and iron contents that standard diffusion treatments predict should relax in 1–10 kyr at peak temperature conditions. Our results show that the development of compositional stress during diffusional relaxation can explain the preservation of the observed short wavelength compositional oscillations at a > 100 kyr timescale. Thus, it may be necessary to account for compositional stress when modeling diffusion in solid solutions with appreciable differences in their endmember molar volumes. This will be particularly relevant when considering sharp, μm-scale chemical gradients involving grossular, the garnet endmember with the largest molar volume relative to pyrope, almandine, and spessartine. Neglecting compositional stress in such cases could result in the underestimation of the timescales of lithospheric processes by potentially orders of magnitude. The effects of compositional stress in garnet are predicted to be the most pronounced under amphibolite and blueschist–eclogite facies conditions. At lower temperatures diffusion is limited, and at higher temperatures both plastic deformation and more ideal solid solution behavior will act to diminish the impact of stress.

Importance of Monitoring Frequency for Representation of Dissolved Organic Matter Dynamics in Urban Rivers

AbstractIn‐situ dissolved organic matter (DOM) monitoring frequencies have often been chosen for convenience or based on perceived wisdom, without fully assessing their impact on representation of DOM dynamics. To address this gap, we collected 5‐min fluorescence data in an urban headwater and resampled it at coarser intervals to investigate the impact of monitoring frequencies on the detectability of DOM dynamics during storms. Expecting hydrometeorological conditions to modify the impact of monitoring frequency, we categorized 85 storm events into groups: Group A (low intensity, short duration), Group B (high intensity, short duration), and Group C (low intensity, long duration). Surprisingly, our analysis indicated that monitoring frequency has minimal influence on commonly used biogeochemical indexes (e.g., maximum, hysteresis and flushing index), which are employed to characterize solute behavior, regardless of storm type. To facilitate a direct comparison between monitoring frequencies, we back‐interpolated coarser data into 5‐min intervals and calculated mean squared errors by comparing them with original high‐resolution data. Our findings indicated that in colder periods with predominately Type A and C storms, a coarser monitoring frequency (>30 min) can capture DOM dynamics. Conversely, in warmer periods when Type B storms dominate, a finer frequency (≤15 min) is necessary to capture key solute chemograph processes (e.g., first flush and dilution). Generally, we suggest a 15‐min monitoring frequency as optimal for similar urban headwater systems, and advocate an adaptive approach based on seasonal variations to improve efficiency, especially when power, data transfer, and storage are constraints.

Dynamics of classical solutions of a multi-strain diffusive epidemic model with mass-action transmission mechanism.

We study a diffusive epidemic model and examine the spatial spreading dynamics of a multi-strain infectious disease. In particular, we address the questions of competitive-exclusion or coexistence of the disease's strains. Our results indicate that if one strain has its local reproduction function spatially homogeneous, which either strictly minimizes or maximizes the basic reproduction numbers, then the phenomenon of competitive-exclusion occurs. However, if all the local reproduction functions are spatially heterogeneous, several strains may coexist. In this case, we provide complete information on the large time behavior of classical solutions for the two-strain model when the diffusion rate is uniform within the population and the ratio of the local transmission rates is constant. Particularly, we prove the existence of two critical superimposed functions that serve as threshold values for the ratio of the transmission rates and that of the recovery rates. Furthermore, when the populations' diffusion rates are small, our result on the asymptotic profiles of coexistence endemic equilibria indicate a spatial segregation of infected populations.

Electrified Nanogaps under an AC Field: A Molecular Dynamics Study.

The organization and dynamics of ions and water molecules at electrified solid-liquid interfaces are generally well understood under static fields, especially for macroscopic electrochemical systems. In contrast, studies involving alternating (AC) fields tend to be more challenging. In nanoscale systems, added complexity can arise from interfacial interactions and the need to consider ions and molecules explicitly. Here we use molecular dynamics (MD) simulations to investigate the behavior of NaCl aqueous solutions at different concentrations confined in nanogaps under AC fields ranging from 10 MHz to 10 GHz. We explore the impact of the gap size (2-60 nm) and of the solid material composing the electrode (silica, charged silica, or gold). Analysis of the transient and stable responses of the system shows that the total transverse dipole M z,total formed by the water molecules and the ions across the gap is always able to counter the applied field regardless of AC frequency, NaCl concentration, or electrode material. As expected, the ions lag at higher frequencies, leading to a capacitive behavior. This effect is fully compensated by water dipoles that lead the field, reaching a maximum lead at a specific frequency which depends on salt concentration and gap size. Changing the gap size affects the magnitude of M z,total. Finally, the electrode material is shown to affect the electrolyte behavior in the gap region. We anticipate these results to be useful for nanoscale dielectric spectroscopy, including scanning probes.

[formula omitted]-wave-like properties for entropy solutions to scalar parabolic–hyperbolic conservation laws

In this paper, we consider qualitative properties for entropy solutions to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equations have both properties of hyperbolic equations and those of parabolic equations, it is difficult to investigate the behavior of solutions to (CP). In our previous works, we focused on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with multiple discontinuity. Moreover, we constructed rarefaction wave type sub-, super-solutions to (CP) and investigated their properties.In the present paper, we investigate “N-wave-like properties” for entropy solutions to (CP) while we are not able to construct an analogue of N-waves. In particular, we derive generalized one-sided Lipschitz estimates (Oleinik type entropy estimates) and decay estimates for entropy solutions to (CP). Based on the decay estimates, we discuss the asymptotic profiles of entropy solutions to (CP) under some specific setting.

Approximate Solutions of the Coupled M-Truncated Fractional mKdV System Using the Adomian Decomposition Technique

This manuscript employs the Adomian decomposition technique (ADT) to develop solutions for the fractional space-time nonlinear mKdV system, incorporating an M-truncated fractional order and supposed initial conditions. The technique yields a power series expansion solution without the need for linearization, weak nonlinear assumptions, or perturbation theory. Software such as Maple or Mathematica was utilized to compute the Adomian formulas for the solution expansion. This technique can also be utilized for a range of nonlinear fractional-order models in mathematical physics. A graphical analysis is provided to demonstrate the behavior of Adomian solutions and how variations in non-integer order values influence the results. The technique is straightforward, clear, and widely applicable to other nonlinear fractional problems in both physics and mathematics. It is believed that these studies significantly advance our understanding of the nonlinear coupled fractional mKdV system and its potential applications in physics and engineering.

The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain

Abstract In this article, we investigate the behavior of weak solutions for the three-dimensional Navier-Stokes-Voigt-Brinkman-Forchheimer fluid model with memory and Tresca friction law within a thin domain. We analyze the asymptotic behavior as one dimension of the fluid domain approaches zero. We derive the limit problem and obtain the specific Reynolds equation, while also establishing the uniqueness of the limit velocity and pressure distributions.