Concentration Dependent Dielectric Constant Research Articles

Modeling of Electric Double Layer to Analyze Reaction Kinetics

Electric double layers (EDLs) play a fundamental role in various electrochemical processes such as colloid dispersion, surface charging, and charge-transfer reactions. Increasingly, the role of EDLs on reaction kinetics is being studied (1). Despite the extensive history of EDL modeling, there remain challenges in accurate prediction of its structure. The characteristic length of EDL (nanometer scale) exceeds the grasp of typical ab-initio molecular-dynamic simulations in order to capture a whole picture of the statistical equilibrium. While continuum models offer a means to estimate the thermodynamic equilibrium of EDLs with substantially lower computational cost than molecular dynamics, state–of-the-art continuum models require parameter fitting(2) due to the lack of appropriate expressions for microscopic interactions.Herein, we propose a predictive multiscale continuum model of EDL that eliminates the empirical fitting and relies more on physical relations. This model computes the distribution of the electrostatic potential, electron density, and species’ concentrations by minimizing the total grand potential of the system. The grand potential is calculated from entropic energy, electrostatic energy, electron energy, solute-solute interactions, and electrode-solute interactions. These energy expressions include not only the terms from previous work(2), but also the microscopic interactions that are newly introduced in this work: polarization of hydration shells, electrostatic interaction in parallel plane toward the electrode, and ion-size-dependent entropy. The parameters that identify the electrode and electrolyte materials are obtained from independent experiments such as vacuum work function, bulk modulus, X-ray diffraction, and concentration-dependent dielectric constant. The developed model reproduces the trends in the experimental differential capacitance with multiple electrode and electrolyte materials (Ag (111)-NaF, Ag (111)-NaClO4, and Hg-NaF), which verifies the accuracy and predictiveness of the model in a system without significant specific electrode-electrolyte interactions. The difference in the capacitance between mercury and silver electrode was attributed to the difference in the electron stability in the metal that is parameterized from the material’s bulk modulus. Finally, extension of the model to systems with specific electrode-electrolyte interactions (e.g., Pt (111)-NaClO4) and the analysis of reaction kinetics will be introduced. Overall, the model framework and findings will provide insights into the EDL structures and guidance on how to tailor electrode-electrolyte interface for improved reaction kinetics. Acknowledgements: This work was partially supported by the Hydrogen and Fuel Cell Technologies Office (HFTO), Office of Energy Efficiency and Renewable Energy, U.S. Department of Energy (DOE) through the Million Mile Fuel Cell Truck (M2FCT) consortia, technology managers G. Kleen and D. Papageorgopoulos, under contract number DE-AC02- 05CH11231, as well as by a CRADA with Toyota Central R&D Labs., Inc. and by the Center for Ionomer-based Water Electrolysis (CIWE), a DOE Energy Earthshot Research Center. Reference: S. J. Shin, D. H. Kim, G. Bae, S. Ringe, H. Choi, H. K. Lim, C. H. Choi and H. Kim, Nat Commun, 13, 174 (2022).J. Huang, J Chem Theory Comput, 19, 1003 (2023).

Resurrection of Hückel’s idea: Decoupling ion–ion and ion–water terms in activity coefficients via the state-dependent dielectric constant

Although Hückel proposed the basic idea of using a concentration dependent dielectric constant, ϵ(c), to compute the activity coefficients of ions in electrolytes in 1925 (Hückel, 1925), a large amount of modeling studies appeared in the literature only after 2010 when we published our II+IW theory (Vincze et al., 2010) that splits the excess chemical potential into two terms corresponding to interactions between ions (II) and interactions between ions and water (IW). In this approach, the two terms are decoupled, which means that they can be computed independently with ϵ(c) being the only link between them. Here, we review our theory and other works based on Hückel’s suggestion by discussing several issues that are partly cornerstones of the theory, partly make it possible to put the theory into a larger context. These issues include the role of ϵ(c) and the ionic radii used in the II and IW term, the statistical mechanical methods to estimate the II term, the existence and interpretation of individual activities, phenomena associated with strong ionic correlations in multivalent electrolytes, and explicit-water models used in molecular dynamics simulations.

EPC-SAFT advanced - Part I: Physical meaning of including a concentration-dependent dielectric constant in the born term and in the Debye-Hückel theory

The transition from aqueous electrolyte systems to non-aqueous electrolyte systems is highly demanded in industrial applications and especially challenging for physics-based thermodynamic models. Electrolyte thermodynamics is a complex matter, and still not all physico-chemical effects are accounted for in state-of-the-art equations of state. The dielectric constant of non-aqueous electrolyte systems changes drastically compared to aqueous systems. One main consequence is that ions are very differently solvated in non-aqueous medium compared to aqueous medium. The Born term represents a methodology to account for the influence of solvation energies of ions, which is based on influences of solvent and salt on the dielectric constant. Utilizing the Born term in electrolyte models is extensively debated, and it is often reasonably neglected in predominantly aqueous systems. Yet, it has a significant influence on transferability from aqueous to non-aqueous media i.e., systems with a large difference in polarity or permittivity compared to aqueous systems. In this work, a modified Born term was combined with electrolyte Perturbed-Chain Statistical Associating Fluid Theory (ePC-SAFT) by introducing additionally a salt concentration-dependent dielectric constant, henceforth called altered Born contribution. The new methodology was validated against infinite dilution properties for ion-solvent interactions: Gibbs energy of hydration and Gibbs energy of transfer of alkali halides from water to alcoholic solvents. Further, mean ionic activity coefficients (MIACs) of alkali halides in alcoholic solvents were quantitatively correct predicted with the advanced ePC-SAFT approach. Original ePC-SAFT parameters were applied for all predictions, and no further binary parameters were adjusted. Based on the success of the model predictions, the transferability of pure-ion ePC-SAFT parameters to organic solvents was verified and the incorporation of concentration-dependent dielectric constant into the altered Born contribution and Debye-Hückel theory was proven to be meaningful methods for the transfer of electrolyte thermodynamic models from aqueous to non-aqueous systems.

Activity Coefficients of Concentrated Salt Solutions: A Monte Carlo Investigation

Monte Carlo (MC) simulations were used to calculate single ion and mean ionic activity coefficients and water activity in concentrated electrolytes and at elevated temperatures. By using a concentration dependent dielectric constant, the applicability range of the MC method was extended to 3 mol·L−1 or beyond, depending on the salt. The calculated activity coefficients were fitted to experimental data by adjusting only one parameter, i.e., the cation radius. Fitted ionic radii obtained by such a procedure indicate the extent of cation–anion interaction in a salt solution. For example, the fitted radii of Li+ and Na+ in LiClO3 and NaClO3 indicate that Li+ is strongly hydrated and has a weak interaction with the ClO3− ion whereas Na+ forms ion pairs and loses its hydration. The single ion activity coefficients for protons and chloride ions in HCl were calculated by MC simulations and compared with experimental values obtained by ion selective electrodes. The calculated single ion activity coefficients for protons and chloride ions are much lower and higher, respectively, than the experimental values. However, the mean activity coefficients of HCl obtained by the MC simulations, ion selective electrodes and vapor pressure measurements are in good agreement. In the case of NaCl and KCl the calculated single ion activity coefficients of Na+, K+, and Cl− are much closer to the values obtained by ion selective electrodes. The results in HCl indicate that the hydrated proton is large and includes the chloride ion within the hydration shell, i.e., the apparent size of the chloride ion is negligible.

Incorporating a concentration-dependent dielectric constant into ePC-SAFT. An application to binary mixtures containing ionic liquids

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 [Cnmim]+, [Cnpy]+, [Cnmpy]+, [Cnmpyr]+, [C4m4py]+ and the IL-anions [BF4]-, [NTf2]-, [PF6]-, [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.

Microwave sensing technique based label-free and real-time planar glucose analyzer fabricated on FR4

In this paper, we have demonstrated a real-time label-free microwave analyzer fabricated on FR4 board for quantitative analysis of the glucose concentration. In this, sensing area configured of interdigital (IDT) structures is embedded in the centre of coplanar waveguide (CPW) transmission line. Here, centre of the IDT structures is acting as sensing area, where analyte under test (AUT) is confined by integrating a Poly-DiMethyl-Siloxane (PDMS) cavity on to the IDT structures. Analyte’s inherent electrical property that is dielectric constant is used as source of quantitative analysis when electromagnetic (EM) waves interact with AUT. The glucose concentration dependent dielectric constant causes shift in RF parameters which are investigated in the form of S-parameters, peak frequency, impedance, resistance and reactance. These parametric measurements are carried out using Vector Network Analyzer (VNA) and the variation in each parameter w.r.t the glucose concentration ranging from 0 g/mL to 1 g/mL, are closely observed. A rigorous study has been carried out for multidimensional characterization where measured sensitivity in terms of S11 magnitude is 15.30 dB/g/mL, S11 peak frequency is 235.32 MHz/g/mL, S12 magnitude is 5 dB/g/mL, S12 peak frequency is 168 MHz/g/mL, resistance is 10.91 Ω/g/mL, reactance is 23.82 Ω/g/mL, and impedance is 63.11 Ω/g/mL with linear regression coefficient always better that 0.87. Experimental results show that proposed planar analyzer has a great potential to determine glucose multidimensional quantitative analysis for high accuracy and with improved sensitivity.

Activity coefficients of individual ions in LaCl3 from the II+IW theory

ABSTRACTWe investigate the individual activity coefficients of ions in LaCl3 using our theory that is based on the competition of ion–ion (II) and ion–water (IW) interactions. The II term is computed from Grand Canonical Monte Carlo simulations on the basis of the implicit solvent model of electrolytes using hard sphere ions with Pauling radii. The IW term is computed on the basis of Born's treatment of solvation using experimental hydration-free energies. The results show good agreement with experimental data for La3+. This agreement is remarkable considering the facts that (i) the result is the balance of two terms that are large in absolute value (up to 20 kT) but opposite in sign, and (ii) that our model does not contain any adjustable parameter. All the parameters used in the model are taken from experiments: concentration-dependent dielectric constant, hydration free energies and Pauling radii.

Unraveling the behavior of the individual ionic activity coefficients on the basis of the balance of ion-ion and ion-water interactions.

We investigate the individual activity coefficients of pure 1:1 and 2:1 electrolytes using our theory that is based on the competition of ion-ion (II) and ion-water (IW) interactions (Vincze et al. J. Chem. Phys. 2010, 133, 154507). The II term is computed from grand canonical Monte Carlo simulations on the basis of the implicit solvent model of electrolytes using hard sphere ions with Pauling radii. The IW term is computed on the basis of Born's treatment of solvation using experimental hydration free energies. The two terms are coupled through the concentration-dependent dielectric constant of the electrolyte. We show that the theory can provide valuable insight into the nonmonotonic concentration dependence of individual activity coefficients. We compare our theoretical predictions against experimental data measured by electrochemical cells containing ion-specific electrodes. We find good agreement for 2:1 electrolytes, while the accuracy is worse for 1:1 systems. This deviation in accuracy is explained by the fact that the two competing terms (II and IW) are much larger in the 2:1 case, resulting in smaller relative errors. The difference of the excess chemical potentials of cations and anions is determined by asymmetries in the properties of the two ions: charge, radius, and hydration free energies.

Tunable defect mode in a semiconductor-dielectric photonic crystal containing extrinsic semiconductor defect

In this work, we theoretically investigate the tunable properties in the defect mode for a defective semiconductor-dielectric photonic crystal (SDPC). Here, the defect layer is an extrinsic semiconductor of n type InSb, n-InSb, which has a strong temperature- and concentration-dependent dielectric constant. With the use of n-InSb, the defect mode can be tuned as a function of temperature and concentration. The defect mode frequency is blue-shifted as the concentration or the temperature increases. In addition, this defect mode frequency shows a nearly linear dependence on the concentration but a nonlinear dependence on the temperature. Finally, by increasing the defect thickness leads to the presence of multiple defect modes which can be used to design a multichanneled filter.

Response to “Comment on ‘The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion–ion correlations’” [J. Chem. Phys. 134, 157101 (2011)

Fraenkel commented1 on our paper2 in which we offered a different view of the activity coefficient (or, equivalently, the excess chemical potential) of electrolytes. Fraenkel summarized the main points of our theory nicely. Also, he offered his smaller-ion-shell (SiS) model3, 4 as an alternative to our approach. In this response, we categorize our thoughts along a few main issues. (1) The most important issue for us is the question whether the (concentration-dependent) dielectric constant, (c), of the solution should be used in the model, or the dielectric constant of the pure solvent (that is equal to the dielectric constant of the solution at infinite dilution). We argue that is a macroscopic physical parameter whose value is well-defined and measurable for a given thermodynamic state point (different concentration corresponds to a different state point). In the implicit solvent model of electrolytes the ions are modeled on the molecular level, therefore, they are characterized by microscopic parameters such as the ionsize parameters (ISP). The solvent, on the other hand, is modeled on the basis of its average dielectric response, which is characterized by a macroscopic physical parameter: the dielectric constant. In our view, physical variables that enter the calculations as external parameters, such as the dielectric constant, should have the experimental values for the given state point. The structure of the solvent is influenced by the presence of ions and decreases with increasing concentration (dielectric saturation). Unfortunately, measurements of the static dielectric constant of an electrolyte have several technical difficulties. The extrapolation to zero frequency based on dielectric relaxation data over a frequency range requires special instrumentation to determine the dielectric loss. For example, newer experiments by Buchner et al.5 for NaCl gave (c) values that are larger than those reported by Barthel et al. previously.6 The absence of experimental data for (c) makes the applicability of our approach limited and, at the same time, from a practical point of view, necessarily requires the use of theories applying the dielectric constant of the solvent. The absence or availability of experimental data, however, does not influence the correctness of the physical picture behind our model. Our opinion is that solvation should be taken into account by a concentration-dependent dielectric constant rather than by an increased, solvated ionic radius. (2) Our paper might give the (false) impression that the question whether adjustable parameters are used or not is a central issue for us. It is not. It depends on the intention of the investigator: if one wants to adjust molecular parameters to fit with experiments, it is legitimate to do so. Actually, that is what we do in our ion channel studies, for example.7 There are parameters, however, whose values are well-defined so they should not be adjusted. Even if we adjust a parameter, the resulting value should have a sensible physical meaning. In general, we can shed a light on our standpoint regarding modeling if we cite Occam’s razor that “admonishes us to choose from a set of otherwise equivalent models of a given phenomenon the simplest one. In any given model, Occam’s razor helps us to ‘shave off’ those concepts, variables, or constructs that are not really needed to explain the phenomenon. By doing that, developing the model will become much easier, and there is less chance of introducing inconsistencies, ambiguities, and redundancies.”8 Paraphrasing this idea for the case of adjustable parameters, we can offer a less strong statement: “In a model, we should minimize the number of adjustable parameters.” (3) The third issue is the question of the statistical– mechanical method with which the results for the activity coefficient [and the ion–ion (II) term in our approach] are calculated. Our opinion is that the best method should be used even if it is computationally expensive (especially in the age of fast computers). Technically simpler methods might be justified by computational convenience, but when scientific knowledge is sought, the most accurate method is favorable. According to the complexity of the studied many-body systems, more complex methods usually provide more accurate results. (Alas, Occam’s razor does not help here.) For a well-defined molecular model of a many-particle system, computer simulations, if performed properly, provide exact results apart from statistical uncertainties and system-size effects. Theories necessarily use approximations. Their results, therefore, should be tested against simulation data. The simulation results contain only the error due to the imperfectness of the model, while the theoretical results also contain the error due to the imperfectness of the approximation to derive the theory. Separation of these two errors is important for the activity coefficient of electrolytes as demonstrated in our paper using grand canonical Monte Carlo (MC) simulations9 and the mean spherical

The capacitance of the electrical double layer of valence-asymmetric salts at low reduced temperatures

The low temperature anomalies in the double layer properties of electrolytes have been observed in recent simulations and reproduced by a number of theoretical methods. This anomaly means that the capacitance of the electrical double layer increases with increasing reduced temperature when the reduced temperature is small, while the behaviour is the reversed at high reduced temperatures. In this paper, we give a concise overview of Monte Carlo simulation results for a 1:1 electrolyte together with the results of various theories such as the Gouy–Chapman theory, various versions of the mean spherical approximation, density functional theories, the modified Poisson–Boltzmann theory, and a field theoretical approach. We report our simulation results for the reduced temperature dependence of the capacitance of 2:1 and 3:1 electrolytes. Finally, we investigate the effect of a concentration dependent dielectric constant by means of a Parsons–Zobel plot.

An equation of state for electrolyte solutions by a combination of low-density expansion of non-primitive mean spherical approximation and statistical associating fluid theory

A two-parameter equation of state (EOS) for electrolyte solutions is developed. The equation is in terms of Helmholtz free energy and incorporated with our previous results of the low-density expansion of non-primitive mean spherical approximation (MSA). The concentration dependent dielectric constant is thus inherently included in the model. The statistical associating fluid theory (SAFT) is introduced to represent the association interactions, including the solvent–solvent and ion–solvent. The EOS is tested for 15 aqueous alkali halide solutions at ambient condition. The equation can represent simultaneously the mean ionic activity coefficients, the osmotic coefficients and densities in a good accuracy up to saturated concentration. The comparisons with EOSs published earlier in the literature are carried out. The limitations of the model are also discussed.

The mean spherical approximation methodology applied to the acid–base equilibria of glycine in artificial seawater

We present results obtained by use of a statistical mechanics approximation based on integral equations, the mean spherical approximation, to predict equilibrium constants for glycine in artificial seawater and to illustrate the effect of the ionic medium. A relationship between the dissociation constants of glycine and the molar concentration of the seawater solution is derived in the range of salinities varying from 0.5 to 4%. The MSA is a simple analytical theory for electrolytes that is very useful in representing the thermodynamic properties of ionic solutions over a wide range of concentrations in terms of one simple parameter, Γ, and the charges, diameters, and concentrations of the ions. A concentration dependent dielectric constant rather than the pure solvent permittivity together with the concentration-dependent diameters of the cations are incorporated in the calculations. This is a way of including the effect of soft repulsions and attractions and the effect of solvation. In calculations, an optimized diameters for the protonated glycine ion and the proton, both of which are involved both in the equilibria of glycine, have been obtained.

Simple statistical mechanics of electrolytes with a concentration dependent dielectric constant. Part 1. The pressure equation

We have used an approximation of the Adelman's theory of solutions to take into account the non-pairwise additive effects on the interaction potentials for a real electrolyte solution. As a result, a solute concentration dependent dielectric constante ρ appears. The comparison with experimental data is done by means of the pressure equation. The influence ofe ρ on solution properties is analyzed in two molality ranges using as a reference simplified models but with the dielectric constant of the pure solvent ɛ instead ofe ρ.

Charged hard spheres in dipolar hard sphere solvents. A model for electrolyte solutions

The linearized and quadratic hypernetted-chain approximations are solved numerically for mixtures of charged hard spheres in dipolar hard sphere solvents. The pair correlation functions, concentration dependent dielectric constant, and thermodynamic properties are examined for several solvents and a range of ion concentrations. The ion–ion pair correlation functions contain many interesting structural features which are not present in the so-called primitive model. Solvent separated ion pairs are found to be a very important species in certain solutions.

Solution of the mean spherical approximation for hard ions and dipoles of arbitrary size

The general solution of the mean spherical approximation (MSA) for an arbitrary mixture of hard spherical ions and dipoles, in which the ions can be of different size, is found. This solution is given in terms of three parameters that are calculated by solving an algebraic equation. Two of these parameters are scaling parameters required to satisfy the general symmetry of the pair correlation functions, and are similar to the one introduced in the solution of the MSA for an ionic mixture in earlier work. For equal size and low ionic concentration, we get a rather explicit solution of the MSA, which is formally similar to the Waisman-Lebowitz solution of the restricted primitive model, but with a concentration-dependent dielectric constant.