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This study presents a new equation of state (EOS) for charged hard sphere fluids that incorporates ion-ion association. The EOS is developed using the Debye–Hückel (DH) theory, reference cavity approximation, and Wertheim's theory. Predictive accuracy is evaluated by comparing the model's predictions with Monte Carlo simulations for various charged hard-sphere fluids. The assessment focuses on mean ionic activity coefficient, individual ionic activity coefficient, and osmotic coefficients. The results demonstrate good agreement between the model and simulations, indicating its success for different electrolyte systems. Incorporating ion-ion association improves accuracy compared to the DH theory. The importance of the cavity function and ion-dipole interactions is emphasized in accurately representing structural properties. Overall, the developed EOS shows promising predictive capabilities for charged hard sphere fluids, providing validation and highlighting the significance of ion-ion association in thermodynamic predictions of electrolyte solutions. [ABSTRACT FROM AUTHOR]

Ionic liquid (IL) assisted liquid–liquid-extraction (LLE) has been emerged as fascinating pathway to extract gallium (Ga(III)) from electronic waste. The computational investigation of LLE invokes the UNIQUAC model with extended Pitzer–Debye–Hückel (PDH*) framework to accommodate long range interactions due to the presence of ions. The combination of quantum chemical calculation and global optimization technique have been incorporated to estimate the model parameters. Herein, phosphonium and ammonium-based ILs having 26 tie-lines are used for sole Ga(III) extraction and 14 tie-lines for co-extraction. The sole Ga(III) extraction exhibits deviation between 0.5 and 0.8%, and co-extraction ranges from 0.85 to 1.5% and thus highlights the goodness of estimation. [ABSTRACT FROM AUTHOR]

The problem of single-ion activity coefficients is addressed, primarily, in view of application of ion-selective electrodes (ISEs). A brief historical survey is followed by the consideration of the information delivered by measurements of the EMF of galvanic cells with and without liquid junction. It is shown that obtaining practically relevant data, the content of a particular ion in mixed solution requires measurements with cells with liquid junction. These measurements, in turn, require the use of individual ion activities. Several conventions aimed at accessing these values are discussed. The above-mentioned issues can be useful for teaching electrochemistry. An empirical modification of the Davies' equation is proposed: l o g γ i = - A z i 2 I 1 + a Kiell B I + D i z i I . The values of individual, ion-specific parameters a Kiell and D i have been found based on the best agreement with literature data on the mean electrolyte activity coefficients. Activity coefficients of cation, γ + , and anion, γ - , are calculated separately using the equation proposed, and the respective mean electrolyte activity coefficient is calculated according to its definition using the obtained single-ion activity coefficients: γ ± υ = γ + υ + γ - υ - . The calculated mean electrolyte activity coefficients are compared with literature data on mean electrolyte activities of 53 electrolytes (at 25 °C). The consistency of the proposed values of the individual ion parameters is achieved: for a particular ion, they remain the same in any electrolyte. For KCl and CaCl2, it is also checked whether individual ion parameter values are suitable over the temperature range from 15 to 35 °C (relevant in view of ISE application), and whether these values allow for a consistent interpretation of the ISE response in pure and mixed solutions. [ABSTRACT FROM AUTHOR]

By using a recently formulated Legendre transform approach to the thermodynamics of charged systems, we explore the general form of the screening length in the Voorn–Overbeek-type theories, which remains valid also in the cases where the entropy of the charged component(s) is not given by the ideal gas form as in the Debye–Hückel theory. The screening length consistent with the non-electrostatic terms in the free energy ansatz for the Flory–Huggins and Voorn–Overbeek type theories, derived from the local curvature properties of the Legendre transform, has distinctly different behavior than the often invoked standard Debye screening length, though it reduces to it in some special cases. [ABSTRACT FROM AUTHOR]

Achievement of carbon neutrality requires the development of electrochemical technologies suitable for practical energy storage and conversion. In any electrochemical system, electrode potential is the central variable that regulates the driving force of redox reactions. However, quantitative understanding of the electrolyte dependence has been limited to the classic Debye-Hückel theory that approximates the Coulombic interactions in the electrolyte under the dilute limit conditions. Therefore, accurate expression of electrode potential for practical electrochemical systems has been a holy grail of electrochemistry research for over a century. Here we show that the 'liquid Madelung potential' based on the conventional explicit treatment of solid-state Coulombic interactions enables quantitatively accurate expression of the electrode potential, with the Madelung shift obtained from molecular dynamics reproducing a hitherto-unexplained huge experimental shift for the lithium metal electrode. Thus, a long-awaited method for the description of the electrode potential in any electrochemical system is now available. Electrode potential in any electrochemical systems has long been discussed by classical Debye-Hückel theory which holds only under extremely dilute concentrations. Here, the authors establish the concept 'liquid Madelung potential' to comprehensively describe the potential shift at practical concentrations. [ABSTRACT FROM AUTHOR]

The effect of a permanent magnetic field on the crystallization of calcium carbonate from carbonate aqueous solutions was studied. Based on the equations for the first- and second-order dissociation constants of carbonic acid, the stability constants of NaCO3- and NaHCO30 complexes, the mass balance and electroneutrality equations, the values of the system components and the ratio between the components of the carbonate subsystem were determined, and it was established that the main species in experimental conditions are СО32-, NaСО3- and HСО3- and the main components of the carbonate subsystem are СО32- (86.7%) and HСО3- (13.19%). Studies have shown that with a significant excess of CO32- over HCO3- ions in the solution at temperatures of 18-20˚C, calcite and vaterite are most likely formed. During the crystallization of calcium carbonate from an aqueous carbonate solution in a magnetic field of 125-250 mT, the size of the crystallites increases significantly compared to the absence of a magnetic field.

The electrostatic screening length predicted by Debye–Hückel theory decreases with increasing ionic strength, but recent experiments have found that the screening length can instead increase in concentrated electrolytes. This phenomenon, referred to as underscreening, is believed to result from ion–ion correlations and short-range forces such as excluded volume interactions among ions. We use Brownian Dynamics to simulate a version of the Restrictive Primitive Model for electrolytes over a wide range of ion concentrations, ionic strengths, and ion excluded volume radii for binary electrolytes. We measure the decay of the charge–charge correlation among ions in the bulk and compare it against scaling trends found experimentally and determined in certain weak coupling theories of ion–ion correlation. Moreover, we find that additional large scale ion structures emerge at high concentrations. In this regime, the frequency of oscillations computed from the charge–charge correlation function is not dominated by electrostatic interactions but rather by excluded volume interactions and with oscillation periods on the order of the ion diameter. We also find the nearest neighbor correlation of ions sharing the same charge transitions from negative at small concentrations to positive at high concentrations, representing the formation of small, like-charge ion clusters. We conclude that the increase in local charge density due to the formation of these clusters and the topological constraints of macroscopic charged surfaces can help explain the degree of underscreening observed experimentally. [ABSTRACT FROM AUTHOR]

We consider a classical two‐component plasma analog of the restricted primitive model of electrolyte, where the hard‐core interaction is replaced by a soft differentiable potential. Within the Born–Green–Yvon hierarchy for the equilibrium distribution functions, we shed light on an infection mechanism where the charge correlations are polluted by the density correlations at the critical point of the liquid–gas transition. This should imply an algebraic decay of critical charge correlations, consistently with the violation of the Stillinger–Lovett sum rule observed in Monte Carlo simulations. In connection with this effect of density correlations on the charge correlations, we also provide a discussion on an extended version of the Debye–Hückel theory that calculates charge correlations in the restricted primitive model for an externally imposed spatial fluctuation of charges. PACS Numbers: 05.30.‐d, 05.70.Ce, 52.25.Kn. [ABSTRACT FROM AUTHOR]

Being an expensive metal, the large-scale requirement of gold in the ever-growing market of electronics increases the overall cost of manufacturing. Hence, recycling gold from electronic waste mitigates the demand and subsequently bounds the price of electronic goods. However, the extraction of gold from electronic waste poses a severe hazard to the environment. Because of tuneable and fascinating properties, ionic liquids (ILs) have emerged as state-of-the-art solvents to extract gold from electronic waste. Herein, the modeling of IL-assisted liquid-liquid-extraction incorporating UNIQUAC extended Pitzer–Debye–Hückel* (PDH*) thermodynamic model has been used for the extraction process of gold from an aqueous acidic solution. The long-range, weak electrostatic interactions present in ionic liquids are adequately represented by the extended PDH* model. The model parameters are obtained from the polarizable continuum medium (PCM) calculation, conductor-like screening model for real solvent (COSMO-RS), and genetic algorithm formulation. Altogether, 11 hydrophobic ILs consisting of 9 imidazolium, 1 pyrrolidinium, and 1 ammonium class with 51 tie line data points are considered for the modeling. The estimated root mean square deviation of 0.002–0.99% over 51 tie lines reveals the goodness of the fitting. The projected percentage extraction efficiency lies in the range of 98.5% to 99.3%. [ABSTRACT FROM AUTHOR]

A minimum-modified Debye-Hückel (DH) theory for electrolytes with size asymmetry is developed. Compared with the conventional DH theory, the minimum-modified DH theory only introduces an extra surface charge density to capture the electrostatic effect of the size asymmetry of the electrolytes and hence facilitates a boundary element method for electrostatic potential calculation. This theory can distinguish the electrostatic energies and excess chemical potentials of ions with the same sizes but opposite charges, and is applied to a binary primitive electrolyte solution with moderate electrostatic coupling. Compared with the hyper-netted chain theory, the validity of this modified DH theory demonstrates significant improvement over the conventional DH theory. [ABSTRACT FROM AUTHOR]

Understanding the performance of graphene devices in contact with highly concentrated aqueous electrolytes is key to integrating graphene into next-generation devices operating in sea water environments, biosensors, and high-density energy production/storage units. Despite significant efforts toward interpreting the structure of the electrochemical double layer at high concentrations, the interface between graphene-based materials and concentrated aqueous solutions has remained vaguely described. In this study, we demonstrate the use of graphene-based chemiresistors as a technique to indirectly quantify the experimental screening length of concentrated electrolytes that could clarify the interpretation of electrochemical measurements conducted at low ionic strength. We report a breakdown of the Debye–Hückel theory in the proximity of graphene surfaces at lower concentrations (10–50 mM) than previously reported for other systems, depending on cation size, dissolved oxygen concentration, and degree of graphene defectivity. [ABSTRACT FROM AUTHOR]

The role of Debye charging method in developing extended Debye–Hückel theories with concentration dependent solution permittivity is examined. It is shown that there are different charging methods giving accordingly different equations for the free energy of solution and activity coefficients. They are named regular and extended Debye charging methods respectively. In the case of the extended Debye charging the ion–ion interaction term in the ionic activity coefficient takes the form of the original Debye–Hückel expression but with permittivity of pure solvent substituted by the concentration dependent permittivity of solution and the ion–water interaction (solvation) term reduces to the Born expression. The regular Debye charging leads to more cumbersome expressions for mean ionic activity coefficient. The capabilities of the two extended Debye–Hückel models to predict activity coefficients in aqueous solutions of alkali metal chlorides are compared. [ABSTRACT FROM AUTHOR]

The main focus of this article is to mathematically formulate the microfluidics-based mechanical system for nanofluids. A 50:50 mixture of propylene glycol (PG) and water is used as a heat transfer fluid because of its tremendous anti-freezing properties, and nontoxicity and it is safe to be utilized at the domestic level. Titanium dioxide (titania) nanoparticles are suspended in the working fluid to enhance its heat transfer ability. The fluid flow is induced by electroosmosis in a microtube, which is further assisted by cilia beating. The impacts of Joule heating and non-linear thermal radiation are also considered. The simplification of the dimensionless system is done under lubrication theory and the Debye-Hückel linearization principle. The nonlinear system of equations is executed for a numerical solution by adopting the symbolic mathematical software Maple 17 using the command "dsolve" along with the additional command "numeric" to get the numerical solution. This command utilizes a low-ordered method along with accuracy-enhancing schemes such as the deferred correction technique and Richardson extrapolation to get a numerical answer of desired accuracy, where we can choose the accuracy level and mesh points according to our requirements. The detailed analysis of results obtained from the numerical treatment of the considered problem indicates that the efficiency of the PG + water enhances due to the suspension of the nanoparticles and heat is rapidly removed from the system. Further, the velocity of the fluid is augmented by decreasing the thickness of the electric double layer and raising the strength of the electric field in the forwarding direction. [ABSTRACT FROM AUTHOR]

Charge correlations in dense ionic fluids give rise to novel effects such as long-range screening and colloidal stabilization which are not predicted by the classic Debye–Hückel theory. We show that a Coulomb or charge-frustrated Ising model, which accounts for both long-range Coulomb and short-range molecular interactions, simply describes some of these ionic correlations. In particular, we obtain, at a mean field level and in simulations, a non-monotonic dependence of the screening length on the temperature. Using a combination of simulations and mean field theories, we study how the correlations in the various regimes are affected by the strength of the short ranged interactions. [ABSTRACT FROM AUTHOR]

Reversible ion binding equilibria in polyelectrolyte solutions are strongly affected by interactions between dissociated ionic species. We examine how the structural correlations between ionic groups on polyelectrolytes impact the counterion binding. Treating the electrostatic correlation free energy using the classical Debye-Hückel expression leads to complete counterion dissociation in the concentrated regime. This unphysical behavior is shown to stem from improper regularization of the self-energy of dissociated ions and polyions and is mitigated by smearing point-like charges across a finite width. The influence of the self-energy on counterion binding is elaborated on by generalizing the Debye-Hückel free energy to polyelectrolytes with variable fractal dimension and stiffness. In the dilute regime, a greater propensity for binding is found for chains with more compact architectures, which in turn reduces the harsh self-repulsions of tightly packed arrangements of charge. In the concentrated regime, the effects of electrostatic correlations weaken due to screening and the extent of binding is governed by a balance of short-ranged interactions and the translational entropy of ions. [ABSTRACT FROM AUTHOR]

Random polyampholytes (PAs) contain positively and negatively charged monomers that are distributed randomly along the polymer chain. The interaction between charges is assumed to be given by the Debye-Huckel potential. We show that the size of the PA is determined by an interplay between electrostatic interactions, giving rise to the polyelectrolyte effect due to net charge per monomer (σ) and an effective attractive PA interaction due to charge fluctuations, δσ. The interplay between these terms gives rise to non-monotonic dependence of the radius of gyration, Rg, on the inverse Debye length, κ, when PA effects are important ( δ σ σ > 1). In the opposite limit, Rg decreases monotonically with increasing κ. Simulations of PA chains, using a charged bead-spring model, further corroborate our theoretical predictions. The simulations unambiguously show that conformational heterogeneity manifests itself among sequences that have identical PA parameters. A clear implication is that the phases of PA sequences, and by inference intrinsically disordered proteins (IDPs), cannot be determined using only the bare PA parameters (σ and δσ). The theory is used to calculate the changes in Rg on N, the number of residues for a set of IDPs. For a certain class of IDPs, with N between 24 and 441, the size grows as Rg ∼ N0.6, which agrees with data from small angle X-ray scattering experiments. [ABSTRACT FROM AUTHOR]

The screening, the configurational correlations and the interaction correlations in the theory of binary electrolytes are examined, by exploiting the charge conservation and the properties of the equilibrium state. Validity conditions for the linearised Debye–Huckel screened potential are provided. A non-zero ionic radius is employed in order to account for solvation effects, and limitations on the contribution of the ionic radius are derived. A possible stabilisation of the ions in equilibrium positions, arising from counterion configurational correlations, is analysed for associated electrolytes, and the corresponding correlation energy is derived. A van der Waals-type equation is presented for strong electrolytes. Some well-known technical difficulties of the theory of electrolytes are revisited, with emphasis on statistical-dynamical correlations, excluded volume and multiple-boundary conditions. [ABSTRACT FROM AUTHOR]

The Debye–Hückel Limiting Law (DHLL) correctly predicts the thermodynamic behavior of dilute electrolyte solutions. Most articles and books explain this law using Peter Debye and Erich Hückel's original formalism of linearizing the Poisson–Boltzmann equation for a simple electrolyte model. Brilliant in its own right, this approach does not fully explain which microstates contribute in the range of the Debye–Hückel theory. Notably, the original formalism does not establish the Energy Multiplicity Distribution (EMD), which is the energy distribution of a system's microstates. This work establishes an analytical expression for the EMD that satisfies the DHLL. Specifically, an EMD that is proportional to exp (a U el 3 ) satisfies the DHLL for a monovalent electrolyte solution. Here, Uel is the effective electrostatic energy due to ion–ion interactions. The proposed proportionality shows quantitative agreement with the simulated EMDs of a Coulomb lattice gas that corresponds to an aqueous sodium chloride solution at a concentration of 3.559 × 10−4 M. The lattice gas that is used does not incorporate solvent molecules, but the Coulomb interactions are scaled through a permittivity that emulates the solvent—similar to the Debye–Hückel theory. Moreover, this work explains the proportionality by partitioning Uel into a set of energy contributions using minimal spanning graphs. This discussion on the EMD is new in the field. It widens the scope of the Debye–Hückel theory and could lead to a new parameterization option for developing equations of state. [ABSTRACT FROM AUTHOR]

The Debye–Huckel and Bronsted-Bjerrum theory has been applied to supported liquid membrane process to understand the effect of ionic strength on separation factor. The effect of ionic strength due to different salts and their concentration on the separation factor of Cu(II) and Zn(II) by supported liquid membrane using Di-2 ethyl-hexyl phosphoric acid (D2EHPA) as a mobile carrier has been studied. For those studies, different salts, i.e., NaCl, NaNO3, NaClO3, Na2SO4, and CH3COONa, have been used. A mathematical model for the dependency of separation factor of Cu(II) and Zn(II) on ionic strength considering Debye–Huckel theory and Bronsted–Bjerrum theory has been developed. Experiments were carried out using a batch-type permeator of micro-porous polypropylene thin sheet as the solid support for the liquid membrane in the presence of different ionic strength of anions. The calculated separation factor is well agreed with the obtained results within 95% confidence interval. [ABSTRACT FROM AUTHOR]

Long-range ion-induced correlations between water molecules have been observed by secondharmonic or hyper-Rayleigh scattering experiments with conflicting results. The most recent work observed a large difference between the results for H2O and D2O, and large discrepancies with the previously proposed theory. However, the present observations are in quantitative agreement with the model where the ion electric field induces second harmonic generation by thewater molecules, and ionion correlations given by the Debye-Huckel theory account for intensity saturation at high ion concentration. Thiswork compares experimental results with theory and addresses the apparent discrepancies with previous experiments. [ABSTRACT FROM AUTHOR]

A dielectric response theory of solvation beyond the conventional Born model for polar fluids is presented. The dielectric response of a polar fluid is described by a Born response mode and a linear combination of Debye-Hückel-like response modes that capture the nonlocal response of polar fluids. The Born mode is characterized by a bulk dielectric constant, while a Debye-Hückel mode is characterized by its corresponding Debye screening length. Both the bulk dielectric constant and the Debye screening lengths are determined from the bulk dielectric function of the polar fluid. The linear combination coefficients of the response modes are evaluated in a self-consistent way and can be used to evaluate the electrostatic contribution to the thermodynamic properties of a polar fluid. Our theory is applied to a dipolar hard sphere fluid as well as interaction site models of polar fluids such as water, where the electrostatic contribution to their thermodynamic properties can be obtained accurately. [ABSTRACT FROM AUTHOR]

Electrolyte thermodynamics is a complex and broad subject of immense importance in very diverse applications. The models proposed for electrolyte solutions have some similarities to those for non-electrolytes but also significant differences. Not just due to additional contributions but also because of the way the models are developed for electrolytes vs. non-electrolytes. Moreover, there are still fundamental issues unresolved in electrolyte thermodynamics. It is still today the activity coefficient models, often extensions of local-composition models that are used in engineering practice for electrolytes e.g., the Pitzer, electrolyte NRTL and extended UNIQUAC. In this review, however, we investigate the area of equations of state (EoS) for electrolytes, a very rich field which essentially started with the Fürst and Renon model in 1993. Since then numerous electrolyte EoS (e-EoS) have been proposed; the literature is both rich and confusing. We have decided in this work to review mostly electrolyte versions of cubic and CPA (Cubic-Plus-Association) EoS, although some of the observations made may be applicable to e-EoS of the SAFT type as well, and some of them are also briefly discussed. Reviewing e-EoS is not an easy task due to especially the diversity of modeling and parameter estimation approaches which are followed as well as the way the models have been validated. Almost none of the e-EoS proposed in literature can be compared "on equal terms" with another e-EoS. Thus, a critical comparative analysis is proposed here, including some recent developments of the e-CPA approach. When possible, different modeling—parameter estimation—validation approaches are compared. It is hoped that this review can provide an insight on the current state-of-the-art of some e-EoS proposed in literature and point out areas where further research is needed. [ABSTRACT FROM AUTHOR]

An inverse patchy colloid (IPC) refers to a particle having patches that are mutually repulsive while the bare particle–patch interactions are attractive. We model a system of charged IPCs with a single patch using Debye-Hückel theory. We derive the full analytical expression for the effective interaction energy for a pair of IPCs as a function of inter-particle separation and particle orientations. The model is applicable for various patch coverage, ionic strengths and dielectric permittivity of the medium. Approximate solutions are derived under high screening limit and the results are compared with the full solution. The model can be used in molecular simulations such as Monte Carlo simulations to accurately describe the interaction between a pair of IPCs for a given experimental conditions. [ABSTRACT FROM AUTHOR]

A method for determining the activity of the solvent in electrolyte solutions by molecular dynamics simulations is presented. The electrolyte solution is simulated in contact with the pure solvent. Between the two phases, there is a virtual membrane, which is permeable only for the solvent. In the simulation, this is realized by an external field which acts only on the solutes and confines them to a part of the simulation volume. The osmotic pressure, i.e., the pressure difference between both phases, is obtained with high accuracy from the force on the membrane, so that reliable data on the solvent activity can be determined. The acronym of the new method is therefore OPAS (osmotic pressure for activity of solvents). The OPAS method is verified using tests of varying complexity. This includes a comparison of results from the OPAS method for aqueous NaCl solutions to results from the literature which were obtained with other molecular simulation methods. Favorable agreement is observed not only for the solvent activity but also for the activity coefficient of NaCl, which is obtained by application of the Gibbs-Duhem equation. [ABSTRACT FROM AUTHOR]

Electrostatic interactions among colloidal particles are often described using the venerable (two-particle) Derjaguin–Landau–Verwey–Overbeek (DLVO) approximation and its various modifications. However, until the recent development of a many-body theory exact at the Debye–Hückel level (Yu in Phys Rev E 102:052404, 2020), it was difficult to assess the errors of such approximations and impossible to assess the role of many-body effects. By applying the exact Debye–Hückel level theory, we quantify the errors inherent to DLVO and the additional errors associated with replacing many-particle interactions by the sum of pairwise interactions (even when the latter are calculated exactly). In particular, we show that: (1) the DLVO approximation does not provide sufficient accuracy at shorter distances, especially when there is an asymmetry in charges and/or sizes of interacting dielectric spheres; (2) the pairwise approximation leads to significant errors at shorter distances and at large and moderate Debye lengths and also gets worse with increasing asymmetry in the size of the spheres or magnitude or placement of the charges. We also demonstrate that asymmetric dielectric screening, i.e., the enhanced repulsion between charged dielectric bodies immersed in media with high dielectric constant, is preserved in the presence of free ions in the medium. [ABSTRACT FROM AUTHOR]

We generalize the inverse patchy colloid model that was originally developed for heterogeneously charged particles with two identical polar patches and an oppositely charged equator to a model that can have a considerably richer surface pattern. Based on a Debye-Hückel framework, we propose a coarse-grained description of the effective pair interactions that is applicable to particles with an arbitrary patch decoration. We demonstrate the versatility of this approach by applying it to models with (i) two differently charged and/or sized patches, and (ii) three, possibly different patches. [ABSTRACT FROM AUTHOR]

Brownian dynamics simulations of a coarse-grained bead-spring chain model, with Debye-Hückel electrostatic interactions between the beads, are used to determine the root-mean-square end-to-end vector, the radius of gyration, and various shape functions (defined in terms of eigenvalues of the radius of gyration tensor) of a weakly charged polyelectrolyte chain in solution, in the limit of low polymer concentration. The long-time diffusivity is calculated from the mean square displacement of the centre of mass of the chain, with hydrodynamic interactions taken into account through the incorporation of the Rotne-Prager-Yamakawa tensor. Simulation results are interpreted in the light of the Odjik, Skolnick, Fixman, Khokhlov, and Khachaturian blob scaling theory (Everaers et al., Eur. Phys. J. E 8, 3 (2002)) which predicts that all solution properties are determined by just two scaling variables--the number of electrostatic blobs X and the reduced Debye screening length, Y. We identify three broad regimes, the ideal chain regime at small values of Y, the blob-pole regime at large values of Y, and the crossover regime at intermediate values of Y, within which the mean size, shape, and diffusivity exhibit characteristic behaviours. In particular, when simulation results are recast in terms of blob scaling variables, universal behaviour independent of the choice of bead-spring chain parameters, and the number of blobs X, is observed in the ideal chain regime and in much of the crossover regime, while the existence of logarithmic corrections to scaling in the blob-pole regime leads to non-universal behaviour. [ABSTRACT FROM AUTHOR]

In a recent article [J. Solution Chem. 50, 105 (2021)], Le and Tran discussed the nonlinear decrement of static permittivity of electrolyte solutions described by the Langevin function and its role in predicting activity coefficients using the extended Debye-Hückel (EDH) theory [Shilov I.Yu., Lyashchenko A.K. J. Phys. Chem. B 119, 10087 (2015)]. In this comment, we consider the mathematical description of permittivity concentration dependence, rebut some statements made by Le and Tran and provide an example of employing the Langevin function in the EDH theory. [ABSTRACT FROM AUTHOR]

Zeta potential (ZP) is a parameter that expresses the electrochemical equilibrium between particles and liquids like in nanoparticle (NP) colloidal solutions with applications in medicine, pharmaceuticals, chemical production, mineral processing, and water and soil purification. Smoluchowski's theory applies to the ZP particles that are larger the interfacial layer but neglects surface conductivity. The Debye‐Hückel theory correctly approximates the concentration of ions in a double layer but fails to account for the dependence of ZP on the concentration of counterions. Determining ZP of NPs is essential to proper NP characterization. For instance, developing well‐defined therapeutic‐relevant nanoformulations needs information on NPs size, surface charge, stability and agglomeration behaviour. This approach has many practical challenges, from inadequate knowledge of operating standards to sampling, data interpretation and good laboratory practice for the experiments replicability. However, in drug delivery research, very little literature can provide a clear, succinct explanation of these techniques. Looking for specific guidelines to overcome frequently encountered problems during ZP measurements. This article explores factors influencing colloidal particle stability. Measurement criteria such as applied voltage, number of measurements, electrophoretic mobility (EPM), size distribution, surface shape, temperature, viscosity, particle concentration, zeta potential, nanoparticles, colloidal suspension, electrophoretic mobility, and pH. [ABSTRACT FROM AUTHOR]

If the molecules of a given solvent possess significant quadrupolar moment, the macroscopic Maxwell equations must involve the contribution of the density of the quadrupolar moment to the electric displacement field. This modifies the Poisson-Boltzmann equation and all consequences from it. In this work, the structure of the diffuse atmosphere around an ion dissolved in quadrupolarizable medium is analyzed by solving the quadrupolar variant of the Coulomb-Ampere's law of electrostatics. The results are compared to the classical Debye-Hückel theory. The quadrupolar version of the Debye-Hückel potential of a point charge is finite even in r = 0. The ion-quadrupole interaction yields a significant expansion of the diffuse atmosphere of the ion and, thus, it decreases the Debye-Hückel energy. In addition, since the dielectric permittivity of the electrolyte solutions depends strongly on concentration, the Born energy of the dissolved ions alters with concentration, which has a considerable contribution to the activity coefficient γ ± known as the self-salting-out effect. The quadrupolarizability of the medium damps strongly the self-salting-out of the electrolyte, and thus it affects additionally γ ±. Comparison with experimental data for γ ± for various electrolytes allows for the estimation of the quadrupolar length of water: LQ ≈ 2 Å, in good agreement with previous assessments. The effect of quadrupolarizability is especially important in non-aqueous solutions. Data for the activity of NaBr in methanol is used to determine the quadrupolarizability of methanol with good accuracy. [ABSTRACT FROM AUTHOR]

The shrinkage of microbubbles that are less than about 50 µm in diameter is a well-known phenomenon that results from the surface tension. It has also been shown recently that hydroxyl ions have an extremely strong affinity for gas–water interfaces including bubble surfaces. A theoretical model is proposed that predicts bulk nanobubble stability in water, based on a force balance that results from the shrinkage of microbubbles. This model was designed to test the hypothesis that the surface tension of a shrinking microbubble can ultimately be balanced by the repulsion of the hydroxyl ions that initially adsorb onto the microbubble surface prior to shrinking. The present model considers the forces due to ionic repulsion as microbubbles shrink under the surface tension. No special assumptions are required in the present model other than the recently reported strong affinity hydroxyl ions have to gas/water interfaces. The Debye-Hückel theory was used to determine the number of ions on the microbubble surface. The results of this model predict a stable balance between the surface tension and the electrostatic repulsion of hydroxyl ions for nanobubble diameters less than 1100 nm. This predicted maximum in nanobubble size is shown to be consistent with experimental findings. [ABSTRACT FROM AUTHOR]

We examine in detail the activity coefficient of higher-charge electrolytes, which, in dilute solutions, can display negative deviations from the Debye–Hückel limiting law instead of the usual positive deviations typical of lower-charge electrolytes. This fact is of considerable relevance for scientists concerned with extrapolation to infinite dilution of thermodynamic and kinetic quantities. It is shown that this "strange" behavior originates merely from the electrostatic interactions between each ion and all other ions, with no necessity of hypothesizing the presence of chemical association; these negative deviations, indeed, are predicted even at the level of the "primitive model" (ions assumed as charged, unpolarizable, rigid spheres inside an unstructured, isotropic, dielectric fluid). Three different approximations for the behavior of the primitive model of low-charge and high-charge electrolytes are tested, in addition to the Debye–Hückel theory; i.e. IPBE (a numerical accurate integration of the Poisson–Boltzmann equation), the Mayer theory of the electrolytes in the so-called DHLL + B2 approximation, and the Bjerrum theory. In the Supporting Information, the fundamentals of the respective algorithms are reported, and the effects produced by the differences of size between cations and anions, are also examined. [ABSTRACT FROM AUTHOR]

A Monte Carlo (MC) study is made of the persistence length q and the binary cluster integral β (or the excluded-volume strength B) for polyelectrolytes by the use of the discrete Kratky-Porod wormlike chain with hard-core-effective Debye-Hückel electrostatic pair potentials. The quantity q is determined from the initial decay rate of the bond correlation function after preliminary confirmation of the validity of this procedure using the chain with Lennard-Jones pair potentials. The quantity B is determined from the mean-square radius of gyration along with q by the use of the quasi-two-parameter (QTP) excluded-volume theory. They are evaluated for two model cases of polyelectrolytes, sodium hyaluronate as an example of semiflexible polymers and poly(sodium 4-styrenesulfonate) as a typical example of flexible polymers, both in aqueous sodium chloride. The behavior of MC data so obtained for q and B as functions of added salt concentration c is examined in detail, comparing them with the Odijk-Skolnick-Fixman theory of q and the Fixman-Skolnick (FS) theory of B and also with literature experimental data. In particular, the MC values of B are in almost complete agreement with the FS theory for large c, although the latter still overestimates B somewhat for small c. The values of B themselves and also the validity of the QTP theory in general are discussed in comparison with the case of nonionic polymers. [ABSTRACT FROM AUTHOR]

This corresponds to the EDH model in the work by Lei et al.1 Equations (13) and (14) can be derived from the expression for the excess Gibbs energy of solution: 17 HT ht 18 HT ht 19 HT ht It follows from the equivalence between Equations (5) and (13) and between Equations (6) and (14) that there are only two extended DH models for I G i SP E sp that should be considered. [Extracted from the article]

In order to investigate the impacts of a concentration dependent static permittivity in the Debye–Hückel theory, two electrostatic Helmholtz free energy models and four activity coefficient models, with the ion‐solvent interactions naturally included, are derived under different assumptions. The effects of static permittivity and model parameters are analyzed by predicting the mean ionic activity coefficients. It is found out that it is reasonable to assume a constant static permittivity in deriving the electrostatic Helmholtz free energy model but it is highly recommended to take the concentration dependence of static permittivity into account in subsequent calculations of thermodynamic properties. The activity coefficient model derived in this way accurately predicts the mean ionic activity coefficients of the investigated systems up to 0.1 mol/kg H2O, which indicates that the size parameters in the Debye–Hückel theory might be determined in advance before it is integrated into a more complete thermodynamic model. [ABSTRACT FROM AUTHOR]

So far, it has rarely been acknowledged that the initial version of the Debye-Hückel theory provided for different mean distances of closest approach to different ions, but the final formulation of the theory was presented with a single geometric parameter for all the ions in a solution. Here the initial design is recalled by deriving the Debye-Hückel equation for the mean activity coefficient in binary solutions with two geometric parameters for the two ion species present and, in the course of this, looking critically at details of the derivations. The equation is compared with the SiS model of electrolyte solutions (Mol. Phys. 108, 1435 (2010)) both in theoretical aspects and as regards their abilities to reproduce experimental data. It is found that the SiS model, as distinct from the original theory, is thermodynamically inconsistent in that it derives the mean activity coefficient from chemical potentials of ions, which do not represent partial derivatives of a single free-energy function of the solution. In addition, arguments are set forth that the geometric parameters in both equations cannot be interpreted in terms of a realistic structural chemistry. The results of comparing the two equations with experiment are found to bear out this view. [ABSTRACT FROM AUTHOR]

The article presents a study on active particles as mobile microelectrodes for selective bacteria electroporation and transport. Topics discussed include self-propelling micromotors are emerging as a promising micro- and nanoscale tool for single-cell analysis necessary to manipulate matter via dielectrophoresis can be induced at the surface of a polarizable active metallodielectric Janus particle; and important and novel experimental tool for single-cell analysis and targeted delivery.

A quantitative model of ion-stabilized gas bubbles is suggested. Charging the bubbles by the ions, which are capable of adsorption, and the screening by a cloud of counter-ions, which are less absorptable, is modeled. It is shown that, subject to the charge of bubble, two regimes of such screening can be realized. For low-charged bubbles, the screening is described in the framework of the known linearized Debye-Huckel approach, when the sign of the counter-ion cloud is preserved everywhere in the liquid, whereas at large charge this sign is changed at some distance from the bubble surface. This effect provides the mechanism for the emergence of two types of compound particles having the opposite polarity, which leads to the aggregation of such compound particles into fractal clusters. Based on experimental data, arguments in favor of the existence of the clusters composed of the ion-stabilized bubbles in aqueous electrolyte solutions are advanced. This paper provides theoretical grounds for the experimental results presented in the previous paper (part I) published in this journal. [ABSTRACT FROM AUTHOR]

Starting from the Debye model for frequency-dependent specific heat and the Vogel-Fulcher-Tammann (VFT) model for its relaxation time, an analytic expression is presented for the heat capacity versus temperature trace for differential scanning calorimetry (DSC) of glass transitions, suggesting a novel definition of the glass transition temperature based on a dimensionless criterion. An explicit expression is presented for the transition temperature as a function of the VFT parameters and the cooling rate, and for the slope as a function of fragility. Also a generalization of the results to non-VFT and non-Debye relaxation is given. Two unique ways are proposed to tackle the inverse problem, i.e., to extract the fragility from an experimental DSC trace. Good agreement is found between theoretically predicted DSC traces and experimental DSC traces for glycerol for different cooling rates. [ABSTRACT FROM AUTHOR]

Using self-consistent field theory (SCFT), we study the conformational properties of polyelectrolyte dendrimers. We compare results for three different models of charge distributions on the polyelectrolytes: (1) a smeared, quenched charge distribution characteristic of strong polyelectrolytes; (2) a smeared, annealed charge distribution characteristic of weak polyelectrolytes; and (3) an implicit counterion model with Debye-Huckel interactions between the charged groups. Our results indicate that an explicit treatment of counterions is crucial for the accurate characterization of the conformations of polyelectrolyte dendrimers. In comparing the quenched and annealed models of charge distributions, annealed dendrimers were observed to modulate their charges in response to the density of polymer monomers, counterions, and salt ions. Such phenomena is not accommodated within the quenched model of dendrimers and is shown to lead to significant differences between the predictions of quenched and annealed model of dendrimers. In this regard, our results indicate that the average dissociated charge [formula] inside the dendrimer serves as a useful parameter to map the effects of different parametric conditions and models onto each other. We also present comparisons to the scaling results proposed to explain the behavior of polyelectrolyte dendrimers. Inspired by the trends indicated by our results, we develop a strong segregation theory model whose predictions are shown to be in very good agreement with the numerical SCFT calculations. [ABSTRACT FROM AUTHOR]

Correlation between the elastic modulus (B) and the Raman shift (Δω) of TiO2 and their responses to the variation of crystal size, applied pressure, and measuring temperature have been established as a function depending on the order, length, and energy of a representative bond for the entire specimen. In addition to the derived fundamental information of the atomic cohesive energy, binding energy density, Debye temperature and nonlinear compressibility, theoretical reproduction of the observations clarified that (i) the size effect arises from the under-coordination induced cohesive energy loss and the energy density gain in the surface up to skin depth; (ii) the thermally softened B and Δω results from bond expansion and bond weakening due to vibration; and, (iii) the mechanically stiffened B and Δω results from bond compression and bond strengthening due to mechanical work hardening. With the developed premise, one can predict the changing trends of the concerned properties with derivatives of quantitative information as such from any single measurement alone. [ABSTRACT FROM AUTHOR]

Characteristic non-Debye behaviors of heat capacity data, Cp(T), which are mainly manifested in the form of non-monotonic dependences (maxima) of the respective Cp(T)/T3 curves in the cryogenic region, are described by means of an unprecedented interpolation formula. Least mean square fittings of duly comprehensive Cp(T) data sets available for GaN and ZnO show excellent agreement, from the liquid-helium-hydrogen region up to high temperatures. We show that, in contrast to misleading suggestions made in some recent thermo-physical papers, Debye's heat capacity formula used under the presumption of a fixed Debye temperature is clearly incapable of providing reasonable numerical simulations of measured Cp(T) dependences. [ABSTRACT FROM AUTHOR]

The effects of non-equilibrium charge screening in mixtures of oppositely charged interacting molecules on surfaces are analyzed in a closed system. The dynamics of charge screening and the strong deviation from the standard Debye-Hückel theory are demonstrated via a new formalism based on computing radial distribution functions suited for analyzing both short-range and long-range spacial ordering effects. At long distances the inhomogeneous molecular distribution is limited by diffusion, whereas at short distances (of the order of several coordination spheres) by a balance of short-range (Lennard-Jones) and long-range (Coulomb) interactions. The non-equilibrium charge screening effects in transient pattern formation are further quantified. It is demonstrated that the use of screened potentials, in the spirit of the Debye-Hückel theory, leads to qualitatively incorrect results. [ABSTRACT FROM AUTHOR]

We propose a Debye-Hückel-like screening model (DESMO) that generalizes the familiar conductor-like screening model (COSMO) to solvents with non-zero ionic strength and furthermore provides a numerical generalization of the Debye-Hückel model that is applicable to non-spherical solute cavities. The numerical implementation of DESMO is based upon the switching/Gaussian (SWIG) method for smooth cavity discretization, which we have recently introduced in the context of polarizable continuum models (PCMs). This approach guarantees that the potential energy is a smooth function of the solute geometry and analytic gradients for DESMO are reported here. The SWIG formalism also facilitates analytic implementation of two other PCMs that are based on a screened Coulomb potential: the 'integral equation formalism' (IEF-PCM) and the 'surface and simulation of volume polarization for electrostatics' [SS(V)PE] method. Fully analytic implementations of these screened PCMs are reported here for the first time. Numerical results, for model systems where an exact solution of the linearized Poisson-Boltzmann equation is available, demonstrate that these screened PCMs are highly accurate. In realistic test cases, they are as accurate as the best available three-dimensional finite-difference methods. In polar solvents, DESMO is nearly as accurate as more sophisticated screened PCMs, but is significantly simpler and more efficient. [ABSTRACT FROM AUTHOR]

The nanoparticles in solution represent a model system, where the well-established colloidal theories such as the Debye–Hückel theory and/or Derjaguin–Landau–Verwey–Overbeek theory can be implemented to predict the nanoparticle phase behavior. Recently, reentrant phase transitions in a wide range of colloids (e.g., inorganic and organic nanoparticles, polymers, and biomolecules) have been observed, which are not consistent with these theories. The colloids in the reentrant phase behavior undergo a phase change and return back to the original phase with respect to a specific physiochemical parameter (e.g., ionic strength, concentration of different additives, temperature, and so on). The nanoparticle–polymer/multivalent ion systems, demonstrating such phase transition and the corresponding phase behavior in terms of interparticle interactions, have been probed by small-angle scattering. It has been shown how the tuning in interparticle interactions using external parameters can lead to reentrant phase behavior and use the nanoparticle aggregation for building nanohybrids. The deviations of the present observations from those of the standard colloidal theories and the anticipated challenges are also discussed. Image 1 [ABSTRACT FROM AUTHOR]

Primitive thermodynamic models for electrolyte solutions require the dielectric constant ε. This property strongly depends on the concentration of the electrolytes in the mixture. Neglecting this dependency might be reasonable for modeling solutions at low electrolyte concentrations. However, in solutions containing ionic liquids (ILs) and especially for the calculation of liquid-liquid equilibria (LLE) of systems with ILs, liquid phases often contain high IL concentrations. At such conditions, neglecting the influence of concentration on ε is an oversimplification. In this work, an approach to account for the concentration-dependent dielectric constant within the Debye-Hückel theory was implemented into electrolyte Perturbed-Chain Statistical Associating Fluid Theory (original ePC-SAFT). This new approach was then applied to model LLE of binary mixtures containing water and commonly used hydrophobic ILs. These common ILs are comprised of the IL-cations [C n mim]+, [C n py]+, [C n mpy]+, [C n mpyr]+, [C 4 m 4 py]+ and the IL-anions [BF 4 ]-, [NTf 2 ]-, [PF 6 ]-, [TFO]-. The LLE of binary mixtures water + IL were modeled at ambient pressure and different temperatures with the new ePC-SAFT and with the original ePC-SAFT [Ji et al. DOI: 10.1016/j.fluid.2012.05.029] without the concentration-dependent ε. Overall, the new approach within ePC-SAFT shows superior modeling as well as correlation capability compared to original ePC-SAFT, which was concluded by comparing both models with LLE data from literature. [ABSTRACT FROM AUTHOR]

This paper discusses various consequences of the fact that the physical nature of process of generation, propagation and measurement of biopotentials is inherently ionic. Source of macroscopic biopotential is the temporary deficit of charge captured by ionic channels, which polarizes the organ border. Electric properties of the living tissue seem to be dominated by the electrolyte which polarizes easily and fast. Current theory of biopotential which treats the body as a volume conductor of certain conductivity is an idealization, which does not consider polarization of tissue. Polarization in quasistatic approximation may be reliably described using the well-known Debye–Hückel theory for electrolytes, derived directly from Boltzmann equation and Poisson equation. The polarization theory for biopotential generation, propagation and measurement is conceptually simple and enables to model wider range of experimental conditions, such as dependence of the potential on ionic strength. Modification of the basic numerical assumption for biopotential modeling: the electroneutrality condition is proposed. The accuracy of the theory is verified by comparison with existing experimental results, which shows very good accuracy. This paper shows the importance of a proper physical description to interpretation of biopotentials. The topic is of special importance to electrophysiology and neuroscience, but affects all biopotential measurements, theories and numerical studies. • Biopotentials come from ion diffusion polarizing the organ border. • Language of excess charge and polarization nicely complements the theory of membrane potential. • Living tissue resembles an electrically isolated conductor, filled with electrolyte. • Debye–Hückel theory for electrolytes well describes propagation of biopotentials. • Polarization theory may help to improve simulation in neuroscience. • Many macroscopic experimental cases may be described using single theory. [ABSTRACT FROM AUTHOR]

Abstract Because the weight of interactions in liquid desiccants varies with solute type and concentration, it is difficult for existing typical models to accurately quantify the interactions in mixed liquid desiccants near saturated solubility. A statistical thermodynamic model is proposed to predict the vapor pressure of mixed liquid desiccant solutions near saturated solubility by re-quantifying the weight of interactions between all species in solution. The model considers that charge interaction and non-charge interaction are still dominant, and charge-dipole interaction between ions and solvent molecules increases with the increase of solute concentration. Rigorous expressions are given to calculate the interactions: ion-ion charge interaction is described by modifying the Fowler-Guggenheim theory, charge-dipole interaction is expressed by combining the McMillan-Mayer theory with the Debye-Hückel theory, and non-charge interaction is calculated based on the extended UNIQUAC equation. The prediction accuracy comparison between the new model and two typical models shows that the new model has better accuracy, and the superiority of the model increases with the increase of solute concentration and temperature. The model can be recommended for the liquid desiccant solutions consisting of Li+, Ca2+, Cl− or Br− with concentration ranging from infinite dilution to near saturation and temperature ranging from 283.15 to 353.15 K. Highlights • A new statistical model re-quantifies the weight of interactions in liquid desiccants. • Vapor pressures of single and mixed desiccants near saturated solubility are predicted. • The developed model improves accuracy significantly in high temperature and concentration. [ABSTRACT FROM AUTHOR]