Correlation effects in tracer diffusion and ionic conductivity. II

In this paper it is shown that the physical or conductivity correlation factor can be simply and directly expressed in terms of the well-known tracer correlation factor and a new two particle correlation factor. A Monte Carlo calculation of these correlation factors was performed to illustrate the use of the expression. The physics of the two-particle correlation factor are discussed.     

Correlation effects in diffusion in a two sublattice structure

With a Monte Carlo method we have investigated tracer and conductivity (physical) correlation factors in the simple cubic lattice gas with two f.c.c. inequivalent sublattices and self exclusion by the atoms. Except at very low temperatures, excellent agreement was found between our results for the conductivity correlation factor and Richard's approximate treatment. At an occupation of 0.5 the Haven Ratio exhibits the characteristic discontinuity found recently in other lattice gases which give ordered arrangements. On the concentration axis at low temperature we found another discontinuity in the Haven Ratio. All of these discontinuities are associated with changes in mechanism.     

Ionic conductivity and tracer diffusion in a lattice containing random traps

A model where the lattice contained a number of randomly distributed traps equal to the number of diffusing particles was investigated rigorously. Monte Carlo methods were used to determine the relevant correlation factors in the tracer diffusion and ionic conductivity. It was found that the temperature dependence of the correlation factors contributed a relatively small component to the activation energy of the overall transport process. The relevance of the findings to atomic transport in fluorite-related oxides containing divalent dopant ions is commented on.     

A Monte Carlo study of ionic transport in a simple cubic random alloy via the interstitialcy mechanism: effects of non-collinear and direct interstitial jumps

Self-diffusion and ionic conduction via the interstitialcy mechanism in a simple cubic, binary random alloy AB were investigated as a function of composition using Monte Carlo simulation. It was found that allowance for non-collinear jumps (partly) replacing concurrent collinear site exchanges leads to a reduction in diffusion correlation effects. This goes along with a shift of the diffusion percolation threshold to lower concentrations of the (more) mobile component B. Even stronger changes of mass and charge transport compared to an exclusively collinear interstitialcy scheme are observed for additional contributions of direct interstitial jumps. It is remarkable that for both extensions of interstitialcy mediated diffusion, the Haven ratio appears to be greater than unity in certain composition ranges poor in B. All results rely on the calculation of tracer and interstitialcy correlation factors in the simplest possible three-dimensional lattice structure. Yet they may have more general relevance to the interpretation of tracer self-diffusion data and ionic conductivity measurements on crystalline materials.     

Relating tracer and chemical diffusion coefficients in intermetallic compounds taking the DO3 and A15 structures

In this article we employ computer simulation to explore the validity of the Darken/Manning relation between the chemical diffusion coefficient and the tracer diffusion coefficients of the components in stoichiometric intermetallic compounds A₃B taking the DO₃ and A15 structures at vanishingly small vacancy contents. The analysis centres on the validity of Manning’s random alloy expression for the vacancy wind factor. The models for both the DO₃ and A15 structures use eight atom-vacancy exchange frequencies. For the DO₃ structure it is found that the actual vacancy wind factor is usually somewhat larger than that predicted by Manning but overall the agreement is good. At worst the use of Manning’s expression would underestimate this factor by about 30 or 40%. For the A15 structure a similar result is found except when diffusion along the chains in the structure is rapid. Then Manning’s expression fails badly when a constant geometrical tracer correlation factor is employed. In both the structures if the geometrical correlation factor is varied to reflect the structure actually explored by the atoms (mainly the majority atoms A) the agreement is improved very dramatically.     

Diffusion kinetics of isolated and paired species: a Monte Carlo study

In this paper, we address a four-frequency diffusion model that allows diffusion of paired and unpaired atoms (or small molecular species) in an otherwise empty lattice. The model is useful for describing tracer and collective diffusion of interstitials in dilute interstitial solid solutions and of adsorbed species on surfaces. Diffusion kinetics of this model have been treated analytically by McKee, Le Claire and Okamura and Allnatt. We use Monte Carlo computer simulation to calculate, for the first time, the correlation factors and the linear solute diffusion enhancement factor. Absolute values of the tracer correlation factors and the linear enhancement factors are in excellent agreement with the results of the theory of Okamura and Allnatt, but only in semi-quantitative agreement with results of the other theory. A general argument is also presented to show that the chemical diffusion coefficient does not exhibit correlation effects. This is verified by computer simulation.     

Diffusion of 22Na and 45Ca and ionic conduction in two standard soda-lime glasses

The tracer diffusivities of ²²Na and ⁴⁵Ca in two high-quality silica glasses produced by the Deutsche Glastechnische Gesellschaft as standard glasses I and II have been measured in the temperature range between 473 K and 783 K. The temperature dependences of the tracer diffusion coefficients in both glasses follow Arrhenius laws. The diffusion of ²²Na is more than six orders of magnitude faster than the diffusion of ⁴⁵Ca. The ionic conductivity was determined by frequency-dependent impedance spectroscopy and the conductivity diffusion coefficient D_σ was deduced from the dc conductivity via the Nernst–Einstein relation. The temperature dependences of D_σ for both glasses follow also Arrhenius functions. The activation parameters and pre-exponential factors for tracer diffusion and for conductivity diffusion were determined. The activation enthalpy of ²²Na and the activation enthalpy of the dc conductivity are equal, showing that the conductivity of standard glasses is due to the motion of Na ions. The viscosity diffusivities D_η were determined from available viscosity data using the Stokes–Einstein relation. They are considerably slower than both tracer diffusivities. The Haven ratios H_R are temperature independent for both glasses. The diffusivities of ²²Na and ⁴⁵Ca in soda-lime glasses increase with increasing Na₂O content.     

Drift mobility correlation in a simple cubic lattice

A Monte Carlo method is described for the calculation of the drift mobility of interacting ions on a simple cubic lattice in an electric field. This formalism provides a useful operational model for electromigration in interstitial solid solutions. The results, which show a correlation in the drift of the ions, can be interpreted either in terms of a deviation from the Nernst-Einstein relation in the sense of vacancy wind effects or as an intrinsic correlation effect in the diffusion coefficient of the charge carriers. It is also shown that the usual tracer correlation factor can be calculated upon subtracting the square of the average drifts from the average squared displacements.     

Electrode materials, interface processes and transport properties of yttria-doped zirconia

The kinetics of the oxygen exchange reactions at the electrodes of a galvanic cell using yttria-doped zirconia single crystals (9.5 mole-% Y₂O₃) as solid electrolyte and Pt or Ag as electrode materials was studied by complex impedance spectroscopy. The electrode resistance when using silver was found to have negligible values over the temperature range 180 – 900 °C. In agreement with these results, oxygen sensors were tested successfully at temperatures as low as 200 °C. According to the performance of silver as electrode material, an electrochemical method was developed to determine the oxygen diffusion coefficient in doped zirconia. The results obtained, compared to those of conductivity and oxygen tracer diffusion measurements, have allowed us to obtain information both on the structure of the defects in yttria-doped zirconia and on the correlation factor. Paper presented at the 4th Euroconference on Solid State Ionics, Renvyle, Galway, Ireland, Sept. 13–19, 1997     

The Manning factor for direct exchange and ring diffusion mechanisms

Abstract

In this paper, we consider lattice-based diffusion kinetics for the direct exchange and ring mechanisms as possible proxy diffusion mechanisms for diffusion in liquid alloys. For these mechanisms, we assessed the Manning factor that arises from the Darken–Manning relation relating the interdiffusion coefficient and tracer diffusion coefficients and which can be obtained experimentally. The maximum values of the Manning factor for these two mechanisms occur when the exchange only takes place between the atoms of different type but not between the atoms of the same type. These values have strong composition dependence and reach a value of 2 (ignoring tracer correlation factors) for the direct exchange mechanism at equal compositions of the two components in binary alloys. But for the three atom ring mechanism, these values as a function of composition have a much more complicated form that sits below the direct exchange mechanism for compositions between 10 and 90%. When all exchanges (allowed by a mechanism) occur with approximately the same probability, then the Manning factor is about unity for all compositions.     

Correlation effects for cation diffusion via vacancy-pairs in fluorite-related oxides

Recent research has suggested that diffusion via vacancy-pairs could be an important contribution to cation diffusion in fluorite-related oxides, such as yttria-stabilized zirconia. In this paper, a combination of analytical development and Monte Carlo computer simulation is used to analyze various diffusion correlation effects for cation and oxygen ion diffusion via tightly bound vacancy-pairs in the fluorite structure. It is shown that the application of sum-rule relations provides exact expressions for the collective correlation functions. It is also shown that a formalism inspired by Manning's diffusion kinetics formalism gives accurate expressions for tracer correlation factors when tested against Monte Carlo simulation results. It is also shown that the tracer correlation factors follow the impurity-form, thereby simplifying an interpretation of the diffusion isotope effect.     

Ionic conductivity and anion diffusion in single crystals of lead chloride

The ionic conductivity and self-diffusion of chlorine ions in undoped single crystals of zone-refined lead chloride have been measured over the temperature range 625–370K. These measurements suggest that the conductivity and self-diffusion are due to simple vacancy migration as confirmed by the observed correlation factor. The activation energy of formation of vacancies is found to be 1.55 eV and that for migration of an anion is 0.38 eV. In the extrinsic region, the two measurements show marked discrepancies which are explained by assuming the presence of oxygen impurity ions in the lattice, and the mechanisms for their contribution to the observed excess conductivity are discussed.     

Development and evaluation of an improved correlation based PTV method

An improved correlation based Particle Tracking Velocimetry (PTV) algorithm was proposed in the present paper. The path tracking of the tracer particles was achieved through a correlation operation of the small interrogation window around the studied tracer particles at two-time steps. The central positions of the tracer particles were determined by the correlation operation of the tracer particle image with a Gaussian particle mask in order to improve the accuracy to identify the central positions of particles up to sub-pixel level. The performance of the present improved correlation based Particle Tracking Velocimetry (PTV) algorithm was evaluated by using both synthetic VSJ standard PIV images and actual PIV images of a self-induced sloshing. Compared with other conventional PTV methods, the present improved correlation based PTV algorithm was found to be able to provide better solution and more robust for suppression the effect of background noise in the PIV images.     

Correlated tracer diffusion in an ordered interface structure

In this paper, we postulate a simple two-dimensional structure that attempts to capture the character of many of the qualitative findings from computer simulations of diffusion paths in grain boundaries. We postulate two types of mechanism: those where single atom jumps occur and those where multiple atom jumps occur. We derive analytical expressions for the tracer correlation factors, including those correlation factors that appear in the analysis of the diffusion isotope effect. We also carry out Monte Carlo simulations of these correlation factors. We find very good agreement between the derived expressions for the correlation factors and the simulation results. We are able to show that, in the absence of knowledge about the kinetic energy factor ΔK, isotope effect experiments cannot differentiate between a simple atom-jump mechanism and multiple-atom-jump mechanism.     

Silicon diffusion in molybdenum disilicide: correlation effects

Correlation factors for silicon diffusion by a vacancy mechanism in the silicon sublattice of the tetragonal MoSi₂ structure have been calculated by combining an analytical and a Monte Carlo approach. The ratio of the silicon diffusivity perpendicular to the tetragonal axis to that parallel to the tetragonal axis is also deduced. An effect of forward correlation of tracer atom jumps in the silicon sublattice with the corresponding partial correlation factor of 1.5 appears at small frequencies of silicon atom jumps along the tetragonal axis with respect to the jump frequencies in the silicon layer perpendicular to the tetragonal axis of the MoSi₂ structure. The anisotropy of silicon diffusion in MoSi₂ measured by Salamon et al. is explained in terms of correlation effects of silicon diffusion on its own sublattice.     

Diffusion and ionic conductivity in sodium-β-alumina below room temperature

Sodium tracer diffusivity and ionic conductivity have been measured from room temperature down to ?79°C on the same single crystals of sodium-β-alumina of composition 1.23Na₂O·11Al₂O₃. The Haven ratio decreases from 0.35 at room temperature to 0.18 at ?79°C. The low value of H_R and its temperature dependence can be explained in terms of high correlation effects for ionic diffusion due to large associated defects present at low temperatures.     

Computer simulation of solute-enhanced diffusion kinetics in dilute fcc alloys

This paper describes a Monte Carlo study of the linear enhancement factor for the solvent tracer diffusivity in dilute fcc alloys. The model used is the well-known five-frequency model with isolated solute atoms. It is shown that the analytical treatments by Howard and Manning and by Ishioka and Koiwa have major shortcomings in their handling of correlation effects for certain combinations of the atom-vacancy exchange frequencies. Many of the values of the exchange frequency ratios that have been determined from experiment by way of these treatments are probably in some error.     

Study of cation tracer diffusion in NaF

Cation tracer diffusion coefficients were measured in pure NaF crystals in the intrinsic ionic conductivity range (876–970 °C). The results can be rationalized satisfactorily in terms of contributions to the observed Na tracer diffusivities arising from both free vacancies and neutral vacancy pairs, the latter contribution amounting to about 53 per cent of the total Na diffusion at the highest measuring temperature. The best-fit defect parameters derived in an earlier conductivity study [21] from this laboratory on similar NaF crystals give for the free vacancy contribution D_v^*(Na) = 4·25 exp (?2·21 eV/kT) cm²s^?1. A combination of these D_v^*(Na) values with the present diffusion data yields for the vacancy-pair contribution D_p^*(Na) = 1·15 × 10⁸exp (?4·04 eV/kT) cm²s^?1. Comparison of the present findings with published values of the anion tracer diffusion coefficient in NaF showed that D_p^* (F) is 2·3 to 4·4 times larger than D_p^*(Na) over the temperature range of our observations, the difference between the two contributions increasing with decreasing temperature. When approximate account is taken of the temperature-dependence of the two pair correlation factors, this last result indicates that the anion jumps into the vacancy pair occur with a higher frequency, and increasingly so at lower temperatures, than do those involving the cations.     

Strong correlation between membrane effective fixed charge and proton conductivity in the sulfonated polysulfone cation-exchange membranes

The effectiveness of fixed charge in sulfonated polysulfone membranes and its correlation with proton conductivity and physicochemical properties have been investigated in this work. The membranes were prepared with various concentrations of sulfonating agent (6% to 10% v/v) and followed by the characterizations that include membrane potential measurements, proton conductivity, and physicochemical properties (contact angle, water uptake, and ion-exchange capacity). Here, the effective fixed-charge concentrations of the membranes were obtained based on the data of membrane potential measurements using the Teorell–Meyer–Siever equation. The analysis results exhibit that a strong correlation between effective charge concentration and proton conductivity, which is expressed by the linear increase of proton conductivity with QX. This correlation is also supported by the membranes physicochemical data, such as water uptake, ionic exchange capacity, surface contact angle against water and functional analysis using FTIR. Finally, it was also developed an ionic conductivity equation that describes the correlation between proton conductivity and QX values.     

About ionic conductivity/diffusion relationships in yttria doped zirconia

The values of the oxygen self-diffusion coefficients (measured using¹⁸O tracer) are compared to the ionic conductivity (measured by impedance spectroscopy) of 9.5 mol% yttria doped zirconia single crystals in the temperature range 240–800 °C in air. Electrical conductivity measurements in polycrystals, exhibiting a grain boundary contribution to the ionic conductivity, are furthermore discussed in the frame of a “brick/boundary model”. Paper presented at the 8th EuroConference on Ionics, Carvoeiro, Algarve, Portugal, Sept. 16–22, 2001.