Nernst-Einstein Equation Research Articles

Experimental investigation of the impact of additives in low clinker cementitious materials on multi-ion transference numbers and diffusion coefficients

In this paper, multi-ion transference numbers were determined throughout the chloride migration testing of low clinker cement-based materials. For this purpose, five cement pastes were studied: pure Portland paste (as a reference) and four other pastes, based on limestone filler, fly ash, slag or silica fume. The transference numbers were calculated from the concentration evolution in the three zones of the migration cell (catholyte, anolyte and sample), considering that: (i) ions moved in the catholyte and anolyte; (ii) ions leached from the sample; (iii) ions were generated from the electrode processes, and (iv) the pore solution of the sample evolved during the test. The chloride transference numbers of pastes with pure Portland cement, limestone filler, fly ash, slag or silica fume are almost zero at the beginning of the migration test but 0.23; 0.18; 0.06; 0.05 and 0.20 at the end of the test, respectively. The ion transference numbers obtained were used for the calculation of diffusion coefficients of chlorides, sodium and potassium using the Nernst-Einstein equation.

Simplified Universal Equations for Ionic Conductivity and Transference Number

Nernst-Einstein equation can provide a reasonable estimate of the ionic conductivity of dilute solutions. For concentrated solutions, alternate methods such as Green–Kubo relations and Einstein relations are more suitable to account for ion-ion interactions. Such computations can be expensive for multicomponent systems. Simplified mathematical expressions like the Nernst-Einstein equation do not exist for concentrated multicomponent mixtures. Newman’s treatment of multicomponent concentrated solutions yields a conductivity relation in terms of species concentration and Onsager phenomenological coefficients. However, the estimation of these phenomenological coefficients is not straightforward. Here, mathematical formulations that relate the phenomenological coefficients with the friction coefficients are developed, leading to simplified, ready-to-use expressions of conductivity and transference numbers that can be used for a wide range of ionic mixtures. This approach involves spectral decomposition of the matrix of Onsager phenomenological coefficients. The general analytical expressions for conductivity and transference number are simplified for binary electrolytes, and numerical solutions are provided for ternary and quaternary mixtures with ion dissociation.

Evaluating the contributions to conductivity in room temperature ionic liquids.

The conductivity of room temperature ionic liquids is not described adequately by the Nernst-Einstein equation, which accounts only for Brownian motion of the ions. We report on the conductivity of the ionic liquid 1-butyl-3-methylimidazolum bis(trifluoromethylsulfonyl) imide (BMIM TFSI), comparing the known conductivity of this RTIL to the diffusion constants of the cationic and anionic species over a range of length scales, using time-resolved fluorescence depolarization and fluorescence recovery after photobleaching (FRAP) measurements of chromophores in the RTIL. Our data demonstrate that the diffusional contribution to molar conductivity is ca. 50%. Another mechanism for the transmission of charged species in RTILs is responsible for the "excess" molar conductivity, and we consider possible contributions.

Kinetics in low-temperature-isothermal oxidation of tungsten carbide under different humidity conditions

The isothermal oxidation experiments of tungsten carbides (WC) were carried out at levels of inequality in relative humidity (RH) levels and below 200 °C. Evaluated resistivity fluctuations and structural transformations during the low-temperature corrosion process were detected. In this paper, we present a detailed investigation into the resistivity behavior of WC under varying humidity conditions. Our innovative approach enabled us to witness a rapid decrease in resistivity under conditions of 85 % RH and 85 °C, from 0.00374 Ω-cm to 0.00177 Ω-cm, followed by a stable linear increase after 0.7 h at a constant rate of 0.000924 Ω-cm/h. Notably, a clear correlation between humidity levels and resistance change rates was found, where higher humidity led to faster resistivity changes. In sharp contrast, a resistivity change rate of less than 5 % under the conditions of water and oxygen independently was observed. The low-temperature oxidation mechanism of WC was described by XRD and XPS spectra. Moreover, the Nernst-Einstein equation and Wagner's model are used to establish the relationship between resistivity and oxidation degree for analyzing and predicting the reaction kinetics and oxidation behavior, indicating that the oxidation degree of WC increased logarithmically with oxidation time.

Computation of Electrical Conductivities of Aqueous Electrolyte Solutions: Two Surfaces, One Property.

In this work, we computed electrical conductivities under ambient conditions of aqueous NaCl and KCl solutions by using the Einstein-Helfand equation. Common force fields (charge q = ±1 e) do not reproduce the experimental values of electrical conductivities, viscosities, and diffusion coefficients. Recently, we proposed the idea of using different charges to describe the potential energy surface (PES) and the dipole moment surface (DMS). In this work, we implement this concept. The equilibrium trajectories required to evaluate electrical conductivities (within linear response theory) were obtained by using scaled charges (with the value q = ±0.75 e) to describe the PES. The potential parameters were those of the Madrid-Transport force field, which accurately describe viscosities and diffusion coefficients of these ionic solutions. However, integer charges were used to compute the conductivities (thus describing the DMS). The basic idea is that although the scaled charge describes the ion-water interaction better, the integer charge reflects the value of the charge that is transported due to the electric field. The agreement obtained with experiments is excellent, as for the first time electrical conductivities (and the other transport properties) of NaCl and KCl electrolyte solutions are described with high accuracy for the whole concentration range up to their solubility limit. Finally, we propose an easy way to obtain a rough estimate of the actual electrical conductivity of the potential model under consideration using the approximate Nernst-Einstein equation, which neglects correlations between different ions.

Hydration and Mobility of Alkaline Metal Cations in Sulfonic Cation Exchange Membranes.

The interconnection of ionogenic channel structure, cation hydration, water and ionic translational mobility was revealed in Nafion and MSC membranes based on polyethylene and grafted sulfonated polystyrene. A local mobility of Li+, Na+ and Cs+ cations and water molecules was estimated via the 1H, 7Li, 23Na and 133Cs spin relaxation technique. The calculated cation and water molecule self-diffusion coefficients were compared with experimental values measured using pulsed field gradient NMR. It was shown that macroscopic mass transfer is controlled by molecule and ion motion near sulfonate groups. Lithium and sodium cations whose hydrated energy is higher than water hydrogen bond energy move together with water molecules. Cesium cations in possession of low hydrated energy are directly jumping between neighboring sulfonate groups. Cation Li+, Na+ and Cs+ hydration numbers (h) in membranes were calculated from 1H chemical shift water molecule temperature dependences. The values calculated from the Nernst-Einstein equation and the experimental conductivity values were close to each other in Nafion membranes. In MSC membranes, calculated conductivities were one order of magnitude more compared to the experimental ones, which is explained by the heterogeneity of the membrane pore and channel system.

Insight into the structural and transport properties of methyl and benzyl triphenyl phosphonium based deep eutectic solvents using molecular dynamics simulations

Molecular dynamics simulation of twelve phosphonium based DESs based on methyl triphenyl phosphonium bromide and benzyl triphenyl phosphonium chloride as hydrogen bond acceptors (HBA) with glycerol, ethylene glycol, triethylene glycol and trifluoroacetamide as hydrogen bond donors (HBDs) are carried out with the Generalized Amber Force Field between 25 °C and 95 °C at 1 atm pressure. Isobaric-isothermal ensemble is used for predicting the structural properties, namely, radial distribution function and coordination number. Further, time resolved trajectory data is correlated to predict the transport properties such as, self-diffusivity, binary-diffusivity, ionic conductivity and viscosity. The temperature dependence of predicted density is observed with 4.96 × 10−2 absolute average deviations from experimental dataset. Further, the self-diffusivities are predicted using Einstein's approach and ionic conductivity is calculated from Nernst-Einstein equation as well as Green-Kubo formulation. Further, the viscosity is also predicted using the Green-Kubo formulation. The predicted ionic conductivities and viscosities are fitted in the Vogel-Fulcher-Tammann (VFT) equation to calculate the VFT parameters for the respective DESs.

Using LAMMPS to shed light on Haven’s ratio: Calculation of Haven’s ratio in alkali silicate glasses using molecular dynamics

Haven and Verkerk studied the diffusion of ions in ionic conductive glasses with and without an external electric field to better understand the mechanisms behind ionic conductivity. In their work, they introduced the concept now known as Haven’s ratio (HR), which is defined as the ratio of the tracer diffusion coefficient (Dself) of ions to the diffusion coefficient from steady-state ionic conductivity (Dσ), calculated by the Nernst–Einstein equation. Dσ can be challenging to obtain experimentally because the number of charge carriers has to be implied, a subject still under discussion in the literature. Molecular dynamics (MD) allows for direct measurement of the mean squared displacement (r2) of diffusing cations, which can be used to calculate D, avoiding the definition of a charge carrier. Using MD, the authors have calculated the r2 of three alkali ions (Li, Na, and K) at different temperatures and concentrations in silicate glass, with and without the influence of an electric field. Results found for HR generally fell close to 0.6 at lower concentrations (x = 0.1) and close to 0.3 at higher concentrations (x = 0.2 and 0.3), comparable to the literature, implying that the electric field introduces new mechanisms for the diffusion of ions and that MD can be a powerful tool to study ionic diffusion in glasses under external electric fields.

Experimental and computational studies on the ion dissociation states of 1-butyl-3-methyimidazolium tetrafluoroborate in water and alcohols

Herein, we investigated the dissociation states of ion pairs in the prototype ionic liquid (IL), 1-butyl-3-methylimidazolium tetrafluoroborate ([BMIm]BF4), when mixed with water or alcohol, a homologous substituent of water, during experimentation and computation. Specifically, the degree of dissociation of an ion pair was determined through the density, electrical conductivity and self-diffusion coefficient using the Nernst–Einstein equation, and microscopic information regarding the solution was obtained through molecular dynamics simulations. A schematic model was proposed based on the Experimental and computational results. In the region of low IL concentration, [BMIm] and BF4 existed in a loosely aggregated or free ionic state in the [BMIm]BF4/water mixture. However, [BMIm] and BF4 mainly formed ion pairs, and the ion pairs did not aggregate much with each other in the [BMIm]BF4/alcohol mixture.

TEM analysis and molecular dynamics simulation of graphene coated Al-Cu micro joints

This study compares friction-stir spot welds (FSSW) of pure Al to Cu, with and without graphene interlayer (GL), for tensile load and electrical conductivity (σ). The weld interface of Al-Cu fabricated without a GL is found with brittle intermetallic compounds (IMC) like Al2Cu. The presence of brittle IMCs significantly affects the tensile load and σ. In contrast, the sample with GL suppresses the brittle IMCs and enhances the formation of Al4C3 IMC. The presence of Al4C3 strengthens the weld joint by 26.94% concerning the without GL samples. Further, it was observed that thinner and high-density twins are formed in the samples with GL. The formation of thinner deformation twins is also possible for increased tensile load and σ. The thicker twins in the samples without GL inhibit the electron flow and increase electrical resistivity. The molecular dynamics (MD) simulation was performed to study the in-situ formation of deformed twins. In addition, the MD simulation provides insight into the influence of graphene during the formation of IMCs based on diffusion coefficients of individual atoms. The σ of the Al-Cu joint can be estimated using a cluster Nernst-Einstein equation, which is dependent on the diffusion coefficient obtained from MD simulation.

The electrochemical measurement of salt diffusion coefficient, apparent cation transference number and ionic conductivity for a thin-film electrolyte.

Further development of the electrochemical measurement procedure, namely the Symmetric Polarization Procedure (SPP), is described. The SPP allows for an additional verification of measurement assumptions through a quick and simple inspection of a symmetry between the procedure outcomes. It also improves the precision of estimated transport properties. A considerable emphasis was also put on detailed outcomes analysis. In particular, a newly developed approach to the analysis of restricted diffusion data is proposed. This approach is based on a power law which is a common form for the rate equation, therefore it allows for a precise estimation of the salt diffusion coefficient and, innovatively, the diffusion domain. Importantly, as a result of this approach, the subdiffusive-like response is recognized in every electrolyte examined herein. The viscoelastic relaxation of polymer matrix is given as most probable cause of such response. Additionally, four approaches that lead to a quantification of the apparent cation transference number are also extensively discussed. It has been demonstrated, how a knowledge about the salt diffusion coefficient, the apparent cation transference number and the ionic conductivity can be used to evaluate the electrochemical performance of an electrolyte. It is also shown, how these properties can be utilized to approximate the diffusion-limited current and the deviation from Nernst-Einstein equation.

Hydration and ion interactions of zwitterionic homopolymers with varying carbon spacer lengths

We use molecular dynamics simulations to examine the role of polymerization and changing monomer carbon spacer length on the hydration and ion association properties of carboxybetaine and sulfobetaine zwitterionic homopolymers with LiCl and KCl salts in aqueous solution. Our simulation results suggest that polymerization leads to increased water and ion effective residence time compared to previous studies while varying carbon spacer length causes a change in the partial charge of zwitterionic functional groups. This change in partial charge affects the distribution and quality of water and ion interactions. Additionally, investigating the ionic conductivity by use of the Nernst-Einstein equation, we found that KCl had higher conductivity than LiCl for both polymer types, which is explained by the shorter effective residence time of K+ ions to the polymer than that of Li+.

Determination of Diffusion Coefficients of Heavy Metal Ions (Ni3+, Zn2+, Ba2+, and Mn2+) at Infinite Dilution through Electrolytic Conductivity Measurements

One important parameter to examine the behavior and mass transfer properties of heavy metal ions is the diffusion coefficient. Due to the costly methods of its determination, a simple process correlating the molar conductivity data to diffusion coefficient was utilized. Molar conductivity data were determined for five (5) different dilute concentrations of the chlorides of the heavy metal ions (Ni3+, Zn2+, Ba2+, and Mn2+) and at temperatures ranging from 303.15 to 323.15 K. The infinite dilution diffusion coefficients of the heavy metals were estimated using the Nernst-Haskell equation and Nernst-Einstein equation. The molar conductivity and the diffusion coefficients values of the ions were in the order of Ba2+ > Mn2+ > Zn2+ > Ni3+ with the Ba2+ having the highest molar conductivity with a correlated infinite dilution diffusion coefficient of 1.6565 × 10-9 m2/s at 303.15 K. This study was able to predict the values of the infinite dilution diffusion coefficient of heavy metal ions and could contribute to a better understanding of the mobility of heavy metal ions in a water environment

Effect of Concentration and Temperature on the Structure and Ion Transport in Diglyme-Based Sodium-Ion Electrolyte.

Glyme-based sodium electrolytes show excellent electrochemical properties and good chemical and thermal stability compared with existing carbonate-based battery electrolytes. In this investigation, we perform classical molecular dynamics (MD) simulations to examine the effect of concentration and temperature on ion-ion interactions and ion-solvent interactions via radial distribution functions (RDFs), mean residence time, ion cluster analysis, diffusion coefficients, and ionic conductivity in sodium hexafluorophosphate (NaPF6) salt in diglyme mixtures. The results from MD simulations show the following trends with concentration and temperature: The Na+---O(diglyme) interactions increase with concentration and decrease with temperature, while the Na+---F(PF6-) interactions increase with concentration and temperature. The mean residence time suggests that Na+---O(diglyme) are significantly longer lived compared with that of Na+---F(PF6-) and H (diglyme)---F(PF6-), which shows the affinity of diglyme to the Na+ ions. The ion cluster analysis suggests that the Na+ ions largely exist as solvated ions (coordinated to diglyme molecules), whereas some fractions exist as contact-ion pairs, and negligible fractions as aggregated ion pairs, with the latter two increasing slightly with temperature and more with ion concentration. The magnitude of the diffusion coefficients of Na+ and PF6- ions decreases with concentration and increases with temperature, where the Na+ ion has slightly lower mobility compared with the PF6- anion. The simulated total ionic conductivities show qualitative trends comparable to experimental data and highlight the need for the inclusion of ion-ion correlations in the Nernst-Einstein equation, especially at higher concentrations and lower temperatures.

The relationship between the apparent diffusion coefficient and surface electrical resistivity of fly ash concrete

Electrical resistivity shows promise in predicting the mass transport of ions in concrete. This could be of great value because of the low cost, speed, and convenience of performing electrical resistivity measurements. This study uses the Nernst-Einstein equation to correlate the apparent iodide diffusion coefficient (Dic) of fly ash–cement paste and the surface electrical resistivity (ρsr) of fly ash concrete of the same age. This relationship is investigated at three different ages for 19 different fly ash sources at 20% and 40% fly ash replacement to the cement. A factor K which shows the correlation between ρsr and Dic is calculated using the Nernst-Einstein equation, and the relationship is evaluated with a regression analysis. While different K factors were found to relate ρsr and Dic, it was not possible to only use one K factor to relate Dic and ρsr for 20% or 40% replacement of fly ash for the materials and mixtures investigated. Despite there not being a single relationship, this work suggests a practical approach to use the ρsr in a specification to obtain a Dic of the desired value or lower.

Electrophoretic NMR of Concentrated Electrolytes for Li-Ion Batteries: Understanding the Role of Solvent Motion

In this work cation, anion, and solvent velocities are reported in concentrated Li-ion battery electrolytes using an electrophoretic NMR (eNMR) methodology. Recent computational studies suggest that in concentrated electrolytes, ion aggregation may lead to transport of Li+ in negatively-charged clusters, resulting in negative cationic transference numbers.1,2 Additionally, increasing ion–solvent coordination may lead to a significant non-zero solvent velocity. These unusual transport properties have important implications for fast-charging performance, but as of yet only limited experimental evidence exists for these phenomena.3 Pulsed-field-gradient (PFG) NMR experiments on their own give self-diffusion coefficients but do not report on the drift of charged species, or induced solvent motion; by running PFG under synchronized application of an electric field (i.e., eNMR), the sign and magnitude of electrophoretic mobilities can be determined with spectroscopic specificity. Through 7Li, 19F and 1H eNMR measurements on carbonate- and glyme-based concentrated electrolytes, we show average species velocities in good agreement with conductivity data, and also directly measure transference numbers in the solvent reference frame. eNMR may provide rapid screening of potential electrolytes at various concentrations to discern which systems (both liquid- and polymer-based) possess efficacious transport properties.(1) France-Lanord, A.; Grossman, J. C. Correlations from Ion Pairing and the Nernst-Einstein Equation. Phys. Rev. Lett. 2019, 122 (13), 136001. https://doi.org/10.1103/PhysRevLett.122.136001.(2) Molinari, N.; Mailoa, J. P.; Kozinsky, B. General Trend of a Negative Li Effective Charge in Ionic Liquid Electrolytes. The Journal of Physical Chemistry Letters 2019, 10 (10), 2313–2319. https://doi.org/10.1021/acs.jpclett.9b00798.(3) Gouverneur, M.; Schmidt, F.; Schönhoff, M. Negative Effective Li Transference Numbers in Li Salt/Ionic Liquid Mixtures: Does Li Drift in the “Wrong” Direction? Physical Chemistry Chemical Physics 2018, 20 (11), 7470–7478. https://doi.org/10.1039/C7CP08580J.

DC conductivity of Li2O and SrO doped borophosphate glasses

Borophosphate glasses with dopants SrO and Li2O in the composition of 20(B2O3) + 30(P2O5) + (50-x) (SrO) + x (Li2O) where x = 10, 20, 30, 40 and 50 were synthesized by standard melt quenching technique. All the glass samples were annealed to extract the thermal strains presence in it if any, its amorphous nature was evidenced by XRD characterization, density of the as synthesized glass series has been calculated at the ambient temperature using Archimedes’ principle and also its molar volume was estimated. With increase in mole fraction of Li2O, density of the glass samples found deceasing and the molar volume observed to be increasing as per the trends. Using Nernst-Einstein equation, high temperature dc activation energy has been calculated.

Hydration, self-diffusion and ionic conductivity of Li+, Na+ and Cs+ cations in Nafion membrane studied by NMR

Hydration of Nafion 117 perfluorinated sulfonic cation-exchange membrane in alkaline ion forms was investigated by high resolution 1H NMR. Hydration numbers of Li+, Na+ and Cs+ cations were 5 ± 1, 6 ± 1 and 1 ± 0.2, correspondingly for membrane equilibrated with water vapor at 98% RH. As opposed to Li+ and Na+, which form separate ion pair, Cs+ cation directly contacts with membrane sulfonate group. Cation self-diffusion coefficients were measured by pulsed field gradient NMR technique on 7Li, 23Na and 133Cs nuclei for the first time. Self-diffusion coefficients are changed in the next rows Li+ ≤ Na+ > Cs+. Self-diffusion activation energies of Li+ and Na+ cations are about 20 kJ/mol which is close to water self-diffusion activation energy in these membranes, but Cs+ self-diffusion activation energy is distinctly more (25 kJ/mol). Ionic conductivities calculated on the basis of Nernst–Einstein equation from cation self-diffusion coefficients 1.6∙10−2, 2∙10−2, 6∙10−3 S/cm for Li+, Na+, Cs+ cations, correspondingly, are closely approximating to conductivities measured by impedance spectroscopy: 1.3∙10−2, 1.1∙10−2, 2.3∙10−3 S/cm for Li+, Na+, Cs+ cations, correspondingly, but calculated values are appreciably more compared with experimental meanings.

Defect Structure, Transport Properties, and Chemical Expansion in Ba0.95La0.05FeO3– δ

This work systematically investigates the oxygen nonstoichiometry, defect structure and mass/charge transport properties of Ba0.95La0.05FeO3−δ (BLF). Thermogravimetric measurement and coulometric titration were performed to determine change in oxygen nonstoichiometry (δ) with oxygen partial pressure () in 10−17 ≤ (/atm) ≤ 0.21 and 850 ≤ (T/°C)≤900 range. The δ−−T plot showed “S-type” shape with a crossover plateau around δ ≈ 0.475. Defect modelling was used to explain the nonstoichiometry data and partial molar thermodynamic quantities for oxygen were determined. The DC 4-probe conductivity measurement was performed which showed a typical semiconductor-to-metal transition with temperature around the conductivity maxima at 450 °C. Electrical conductivity relaxation (ECR) measurement was performed to extract partial ionic conductivity, chemical diffusivity and surface exchange kinetics of oxygen ion by using Nernst-Einstein equation and Fick’s law. Isothermal chemical expansion was measured by dilatometry measurement in the range of 10−2 ≤ (/atm) ≤ 0.21 and 850 ≤ (T/°C) ≤ 950.

High Formability and Fast Lithium Diffusivity in Metastable Spinel Chloride for Rechargeable All‐Solid‐State Lithium‐Ion Batteries

Materials with high formability and Li‐ion diffusivity are desired to realize safe bulk‐type all‐solid‐state Li‐ion batteries with high energy density. Spinel‐type Li2FeCl4, which is expected as a Li‐ion‐conductive electrode, adopts the high‐temperature cubic phase (space group: Fd‐3m) by a mechanochemical synthesis method. The powder‐compressed pellet shows a high relative density of 92% and large neck formation between the particles. An analysis of the distribution of relaxation times from the AC impedance results indicates that the contribution of the grain boundary impedance is almost negligible (≈3%) in the powder‐compressed pellet. Even the Li diffusion coefficient, which is underestimated by the Nernst–Einstein equation, is several orders of magnitude higher than those of conventional oxide and sulfide electrode materials. Analyzing the diffusion pathways using a classical force field calculation suggests that stabilizing the high‐temperature phase at room temperature delocalizes Li ions in the diffusion pathway, thus realizing high Li‐ion diffusivity. The Li2FeCl4 green compact containing 10 wt% carbon as an electron‐conductive additive shows a one‐electron charge reaction with a capacity of 126 mAh g−1, with no large overpotential at a high operating voltage of 3.6 V versus Li/Li+ at 30 °C.