Concentration-independent Diffusion Coefficients Research Articles

Improving hydrogen permeability and sustainability of Nb30Ti35Co35 eutectic alloy membrane by substituting Co using Fe

The microstructure and hydrogen permeation performance of Nb30Ti35Co35-xFex (x = 0, 5, 10, 15, 20) alloys have been investigated. With Fe less than 15 at%, the as-cast Nb30Ti35Co35-xFex ingots exhibit fully eutectic structure. When the Fe content is higher than 15 at%, primary bcc-(Nb, Ti) phase appears in combination with eutectic structure. Substituting Co using Fe leads to slightly increased hydrogen solubility but highly enhanced hydrogen permeability, which comes mainly from the increased hydrogen concentration-independent diffusion coefficient D∗. With Fe content up to 10 at%, Nb30Ti35Co35-xFex membranes exhibit stable and higher hydrogen permeation flux than Nb30Ti35Co35 at 673 K during hydrogen permeation test up to 72 h. Nb30Ti35Co35-xFex alloys of fully eutectic structures exhibit no hydrogen-induced failure when cooled down to room temperature under hydrogen atmosphere, indicating the potential application of the membrane at lower temperature range. An optimal combination of hydrogen permeability and hydrogen embrittlement resistance is achieved at 10 at% Fe, since further increase of Fe leads to comparable D∗ as that of Nb30Ti35Co25Fe10 but higher hydrogen solubility.

Water diffusion in poly(vinyl alcohol) membranes: A rigorous analysis of the pervaporation process

Abstract In the design of crosslinked polymer elements for separation and fuel cell technology, controlled drug delivery devices or micro-scale sensor-actuator systems, profound knowledge of mass transport in the polymer network is of substantial interest for a systematic ab initio evaluation of separation efficiency, drug release rates and response characteristics. The study of solvent transport in polymeric membranes is frequently based on the pervaporation process, analysing the flux of one or more permeants under well-defined boundary conditions. Yet, for quantitative analysis, assumptions such as concentration-independent diffusion coefficients and neglect of gas-phase boundary layer resistances and local mass transfer effects are common, but remain largely unverified as information on the solvent distribution in the membrane is not usually available. The present study therefore introduces an advanced experimental methodology and combines the routine permeation flux measurement with the determination of local concentration profiles in the membrane, allowing diffusion coefficients to be specified directly as a function of solvent content. For physically crosslinked poly(vinyl alcohol) membranes, it could thus be demonstrated that water transport is successfully described by pure Fickian diffusion with an exponential expression accounting for the concentration-dependent solvent diffusion coefficient.

Pervaporation of methyl–ethyl ketone and water mixture: Determination of concentration profile

Experimental studies were conducted to separate methyl–ethyl ketone (MEK) from water through pervaporation using poly di-methyl siloxane (PDMS) and poly (ether) block amide (PEBA) membranes. Experiments were carried out at different feed MEK concentrations (10, 15, and 20 wt.%), temperatures (40–60 °C), and downstream pressures (3–7.5 mm Hg), and their influences on flux and selectivity were observed. The selectivity of MEK–PDMS was found to be around 100 for 10 wt.% MEK–water solution at 40 °C with 3 mm Hg downstream pressure, whereas it was 13.5 for MEK-PEBA under the same conditions. However, PEBA membranes offered much higher fluxes. For the PEBA membrane, a flux of around 0.12 kg m − 2 h − 1 was typically observed under the same conditions, whereas this value was 0.06 kg m − 2 h − 1 for the PDMS membrane. The selectivity of MEK–PDMS was 7.4 times higher than that of MEK-PEBA, but in terms of flux, the PEBA membrane offered almost double the flux of the PDMS membrane. Concentration profiles across the PDMS membrane were determined based on an earlier published work (Mulder and Smolders, 1984). Concentration profiles were developed under the following two conditions: (i) apparent diffusion coefficients independent of concentration and (ii) doubling the apparent diffusion coefficients. Analysis of transport across membranes for an MEK-water mixture by pervaporation required efforts beyond the usual consideration of concentration-independent diffusion coefficients. Accordingly, concentration profiles across the membrane were obtained using the determined values of polymer–liquid and liquid–liquid interaction parameters.

A model kinetics for nucleation and diffusion-controlled growth of immiscible alloys

A system of mean field rate equations is employed for describing the kinetics of solid-solid phase separation within the immiscibility gap of binary alloys. The system allows us to study the time evolution of both supersaturation and diffusion length of the components in the metastable phase. It is shown that in the case of simultaneous nucleation the system of differential equations leads to a simple formula for the characteristic time of the transformation in terms of material parameters and initial supersaturation. The nucleation rate is computed on the basis of the classical nucleation theory and the alloy is assumed to behave as a regular solution. It turns out that for low values of the initial supersaturation the nucleation process can be considered as simultaneous. It is also found that thermally activated nucleation takes place for supersaturation values lower than about 0.21. The assumption of a concentration-independent diffusion coefficient and the effect of nucleus curvature on interface composition have been analyzed and discussed.

Studying diffusion of partially miscible and systems near their consolute point by laser line deflection

We propose that miscible systems can be divided into three types: miscible in all proportions, partially miscible but not near a consulate point, and a system with a consulate point. We studied two types of miscible systems by laser line deflection (LLD) in order to understand behavior observed in a spinning drop tensiometer; namely, that some miscible drops became diffuse over time while others maintained a sharp concentration profile as the drop dissolved. We previously studied a miscible system, dodecyl acrylate–poly(dodecyl acrylate) using LLD and found that the system relaxed as expected with a concentration-independent diffusion coefficient. We used LLD to study two other miscible systems: 1-butanol–water, and isobutyric acid–water above the latter's upper critical solution temperature (UCST). The transition zone boundaries in the 1-butanol and water system propagated into the component that was not saturated with the other component. The boundaries propagated with a t 0.09. When an equilibrated IBA–water system was heated above its UCST (26 °C), the transition zone propagated into the IBA-rich region. The propagation exhibited t 0.06 behavior. The propagation of transition zone boundaries showed the same qualitative behavior under 1 g acceleration in the LLD experiments as under the higher acceleration in the tensiometer so it is not necessary to invoke barodiffusion as the cause of the behavior although it may still affect the rate of diffusion.

Investigations on sorption, diffusion and permeation of chloro-alkanes and -alkenes through fluoroelastomeric membranes

Sorption, diffusion and permeation of halogenated liquids such as chloroform, dichloromethane, carbon tetrachloride, 1,2-dichloroethane, 1,1,2,2-tetrachloroethane, trichloroethylene and tetrachloroethylene into commercial fluoroelastomer (FC-2177 D) were studied at 30, 40 and 50°C using the sorption gravimetric method. The concentration-independent diffusion coefficients were calculated using Fick's equation from the initial linear portion of the sorption curves. The liquid concentration profiles were also computed using the analytical solution of Fick's equation with the appropriate initial and boundary conditions, and these were presented as a function of penetration depth of molecular migration and time of immersion. These results are discussed in terms of the molecular size of liquid penetrants and temperature. The transport phenomenon was found to follow anomalous behavior. From a temperature dependence of diffusion and permeation coefficients, the Arrhenius activation parameters were estimated. The activation energies for diffusion and permeation did not exhibit any systematic dependence on size of the penetrating molecules, but these were influenced by the nature of the liquids. None of the liquids showed any degradative effects on the membrane.

Transient Signal Analysis Using Complementary Metal Oxide Semiconductor Capacitive Chemical Microsensors

This work explores the possibility to discriminate analytes based on their nonequilibrium signals in polymer-coated capacitive chemical microsensors. The analyte uptake in the chemically sensitive polymer layers of 3-7-microm thickness has been analyzed using a diffusion model and the dynamic sensor response data. The shapes of the response profiles have been calculated analytically. Despite the simplifications in the model, the observed transient signal profiles could be described accurately. Comparison of the measured diffusion coefficients (on the order of 10(-12) m2/s) with literature values measured at similar concentration levels showed good agreement. Concentration-independent diffusion coefficients for several analyte/polymer combinations (poly(etherurethane)/all analytes; poly(epichlorohydrin)/alcohols) as well as slightly concentration-dependent diffusion coefficients (poly(epichlorohydrin)/toluene or ethyl cellulose/toluene) have been found in the investigated concentration range of tens to hundreds of pascals gas-phase partial pressure. The diffusion times of water and the first aliphatic monohydric alcohols in the polymers are strongly correlated to their molecular size. The discrimination of these substances based on dynamic sensor data of a single sensor could be demonstrated. In particular, the analysis of mixtures of analytes with similar chemical behavior (water/ethanol or methanol/ethanol) by means of analyzing the response profile of single-exposure steps or by applying a series of decreasingly long alternating target gas exposure and carrier gas exposure steps has been performed.

Molecular transport of ketones and nitriles into commercial fluoroelastomeric membranes

The molecular transport of ketones such as acetone, ethyl methyl ketone, methyl isobutyl ketone, cyclohexanone and acetophenone, and nitriles such as acrylonitrile and acetonitrile, through Fluorel 3M membranes (FC-2177 D) was carried out in the temperature range of 30–50°C using the sorption gravimetric method. With the sorption data, the concentration-independent diffusion coefficients were calculated from Fick's diffusion equation. The liquid concentration profiles were also calculated from the analytical solution of Fick's diffusion equation with the appropriate initial and boundary conditions, and these were presented as a function of penetration depth of molecular migration and time of immersion. Because of the linearly increasing trend of the diffusion coefficients with temperature, efforts were made to estimate the Arrhenius parameters. Experimental values and the computed quantities were used to determine the membrane solvent interactions and to propose suitable applications in various situations. None of the solvents showed any degradative effects on the polymer membrane.

Measuring the Mutual Diffusion Coefficient for Dodecyl Acrylate in Low Molecular Weight Poly(dodecyl acrylate) with Laser Line Deflection (Wiener's Method) and the Fluorescence of Pyrene

Diffusion of small molecules into glassy polymers is quite complicated and almost always non-Fickian. Little work has been done with the diffusion of low molecular weight polymers that are liquids at room temperature (such as poly(dodecyl acrylate)) into their miscible monomers. We have studied three molecular weights under 20 000 to determine if poly(dodecyl acrylate) diffusion into dodecyl acrylate could be treated with Fick's law and if so to determine the values of the diffusion coefficients. We compare two methods for measuring the diffusion of dodecyl acrylate into poly(dodecyl acrylate): We used laser line deflection (Wiener's method) and improved upon the method from published reports. We also used the dependence of pyrene's fluorescence on the viscosity to measure the concentration distribution, and thus to extract the diffusion coefficient. After an initial relaxation period, diffusion in all cases followed Fick's law with a single concentration-independent diffusion coefficient. Comparison of the diffusion coefficients obtained by both methods yielded the same order of magnitude for the diffusion coefficients (10(-7) cm2/s) and showed the same trend in the dependence on the average molecular weight of the polymer (a decrease in the diffusion coefficient with an increase in the molecular weight).

Diffusion of vanadium in GaAs

The diffusion of Vanadium has been studied in V-doped GaAs layers (GaAs:V) grown by Metal-Organic Chemical Vapour Deposition (MOCVD) using secondary ion mass spectroscopy (SIMS). The vanadium (V) concentration profiles of sandwiched structures made of alternatively undoped and V doped GaAs layers have shown a concentration independent diffusion coefficient (DV) for varying V doping levels from 1018 to 1019cm−3. Measurements of DV at 550, 615 and 680°C indicate that the temperature dependence of DV can be represented by the Arrhenius equation: Dv=2.4×10-6exp(-1.51eV/kBT)cm2s−1. It is suggested that V diffuses via interstitial sites.

Sorption, diffusion, and swelling characteristics of sodium alginate and its blend membranes with poly(vinyl alcohol) in water-acetic acid mixtures

Sorption, diffusion, and swelling characteristics of sodium alginate and its blend membranes with poly(vinyl alcohol) were investigated for water–acetic acid mixtures by using a gravimetric method at 30, 40, and 50°C. The membranes were characterized by X-ray diffraction and Fourier transform infrared techniques. Concentration-independent diffusion coefficients were obtained by applying Fick's relationship before completion of equilibrium sorption. Permeation coefficients were calculated from sorption and diffusion coefficients. Concentration profiles of liquids were computed considering the sheet geometry for the membrane by solving Fick's equation under suitable boundary conditions. Arrhenius activation parameters were computed for the transport processes. Experimental results and calculated quantities were discussed to understand membrane–solvent interactions. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 94: 1139–1150, 2004

Fickian Diffusion Equation—an Unsolved Problem

Two forms of diffusion equation are discussed: (1) the conventional Fick’s law with concentration-independent diffusion coefficient and (2) the flux as proportional to the chemical potential gradient with the proportional constant independent of concentration (which is, in most cases, equivalent to the application of Fick’s law with a diffusion coefficient inversely proportional to concentration). It is shown that appropriate empirical form of the resistant forces could validate both forms thermodynamically. It is concluded that the ideal form of diffusion equation is absent. It is also shown that the second form of the flux equation might describe the self-diffusion flux better.

Effect of electro-chemical properties of chloride salts on their diffusional parameters in symmetrical cellulose acetate membranes

A relation was obtained between electro-chemical properties of alkali, alkali earth and aluminium chlorides (LiCl, NaCl, KCl, RbCl, CsCl, MgCl 2, CaCl 2, AlCl 3) and the distribution coefficient K and the overall diffusion coefficients D in symmetrical cellulose acetate membranes. Symmetrical cellulose acetate membranes were cast to have a wider range of water content, 15–30%. K and D were measured by the unsteady- and steady-state dialysis method. The K values increased with the increase of water content and in the increasing order of CaCl 2, MgCl 2, AlCl 3, LiCl, KCl, NaCl, RbCl and CsCl, and are in the range of 10 −2 for alkali chlorides, 10 −5-10 −4 for alkali earth chlorides, and 10 −4 for AlCl 3. D was found to increase with the increase of the solute concentration, and the extrapolated values of D to zero concentration D (0) were obtained as 0.01-0.1 μm 2/s for membranes cast from a dense dope solution and 0.1–0.4 μm 2/s for thin membranes. D (0) increases with the increase of water content in the membrane and in the order of NaCl, LiCl, MgCl 2, RbCl, CaCl 2, CsCl, KCl and AlCl 3 for dense membranes and NaCl, RbCl, CsCl, KCl and LiCl for thin membranes. D can be divided into the concentration independent diffusion coefficients in the dense part ( D d ), and in the porous ( D p ), applying a two-part (perfect or dense and imperfect or porous) model of the membrane. D d does not depend on the water content in the membrane ( W W ) but on dope composition. Contrary to D d , D p increases with the increase of W W and can be correlated as D p,c = D d exp (γ × W W ). It is shown that the averaged D d , D d increased almost linearly with the increase of the product of valence times ionic mobility u o of the solutes at infinite dilution, and that the parameter γ increased with the increase of Stokes radius of ion r Stokes. The value of K increased slightly with the increase of water content and decreased with the increase of the Flory—Huggics parameter χ, which is calculated from the measured values of distribution coefficients and data from the literature. It was also found that the gradient of linear decrease of χ (λ cation) and of linear increase of χ (λ anion) depend on equivalent ionic conductivity of anion and cation of salt, λ an and λ cat, respectively. It seems possible to predict permeability for some specific solute through a membrane from measured diffusional values such as D (0) and K for NaCl or some standard solute.

Effect of electro-chemical properties of chloride salts on their diffusional parameters in symmetrical cellulose acetate membranes

A relation was obtained between electro-chemical properties of alkali, alkali-earth and aluminium-chlorides (LiCl, NaCl, KCl, RbCl, CsCl, MgCl 2, CaCl 2, AlCl 3) and the distribution coefficient K and the overall diffusion coefficients D in symmetrical cellulose acetate membranes. Symmetrical cellulose acetate membranes was cast to have a wider range of water content, 15–30%. K and D were measured by the unsteady- and steady-state dialysis method. The K values increase with the increase of water content and in the increasing order of CaCl 2, MgCl 2, AlCl 3, LiCl, KCl, NaCl, RbCl and CsCl, and are in the range of 10 −2 for alkali chlorides, 10 −5-10 −4 for alkali earth chlorides, and 10 −4 for AlCl 3. D was found to increase with the increase of the solute concentration and the extrapolated values of D to zero concentration D (0) were obtained as 0.01–0.1 μm 2/s for membranes cast from a dense dope solution and 0.1–0.4 μm 2/s for membranes from a thin one. D (0) increases with the increase of water content in the membrane and in the order of NaCl, LiCl, MgCl 2, RbCl, CaCl 2, CsCl, KCl and AlCl 3 for membranes from dense and NaCl, RbCl, CsCl, KCl and LiCl for membranes from thin. D can be divided into the concentration independent diffusion coefficients in the dense part D d and in the porous D p , applying a two-part (perfect or dense and imperfect or porous) model of the membrane. D d does not depend on the water content in the membrane W w , but on dope composition. Contrary to D d , D p increases with the increase of W w and can be correlated as D p,c = D̄ind exp (γ × W w ). It is shown that the averaged D d , D̄ind increases almost linearly with the increase of the product of valence times ionic mobility u° of the solutes at infinite dilution, and that the parameter γ increases with the increase of Stokes radius of ion r Stokes. The value of K increases slightly with the increase of water content and decreases with the increase of the Flory—Huggins parameter χ. The Flory—Huggins parameter χ is calculated from the measured values of distribution coefficients and data offered from literatures. It was found that the gradient of linear decrease of χ (λ cat) and of linear increase of χ (λ an) depends on equivalent ionic conductivity of anion and cation of salt, λ an and λ cat, respectively. It seems possible to predict permeability for some specific solute through a membrane from measured diffusional values such as D (0) and K for NaCl or some standard solute.

Effect of electro-chemical properties of sodium salts of various anions on their diffusional parameters in symmetrical cellulose acetate membranes

A relation was obtained between electro-chemical properties of sodium salts (NaCl, NaBr, and Na 2SO 4), and the thermodynamic property of permeability in symmetrical cellulose acetate membranes, the distribution coefficient K and the kinetic property, the overall diffusion coefficients D. K and D were obtained by the method we proposed using measured unsteady- and steady-state dialysis data. The K values increase with the increase of water content and are in the range of 10 −2 for sodium halides and 10 −3 for Na 2SO 4. D is found to increase with the increase of the solute concentration, and the extrapolated values of D to zero concentration D (0) are obtained as 0.015–0.03 μm 2/s and increase with the increase of water content in the membrane. D can be divided into the concentration independent diffusion coefficients in the dense part of the membrane D d and in the porous D p , applying a two-part (perfect or dense and imperfect or porous) model of the membrane. Contrary to D d , D p increases with the increase of W w and can be correlated as D p,c = D d exp (γ × W w ). It is shown that the averaged D d , D D increases with the increase of the quantity of the ionic mobility u∘ of the solutes at infinite dilution divided by valence, and that the parameter γ increases with the increase of the ionic mobility u∘. The value of K increases slightly with the increase of water content and decreases with the increase of the Flory—Huggins parameter χ. The Flory—Huggins parameter χ is calculated from the measured values of distribution coefficients and data obtained from the literature. And it was found that the gradient of linear decrease of χ (λ cation) depends on equivalent ionic conductivity of anion of salt, λ an.

Trace element diffusion in andesitic melts: an application of synchrotron X-ray fluorescence analysis

—We have investigated the diffusivity of trace elements in hydrous iron-free andesitic melts containing 4.5 to 5.2 wt.% water at a pressure of 500 MPa and at temperatures between 1100 and 1400°C using the diffusion couple technique. The studied elements can be combined in several groups of particular geochemical interest: low field strength elements (LFSE: Rb, Sr, Ba), transition elements (Cr, Fe, Ni, Zn), rare earth elements (REE: La, Nd, Sm, Eu, Gd, Er, Yb, Y), and high field strength elements (HFSE: Zr, Nb, Hf). The diffusion profiles of the trace elements were measured using the synchrotron X-ray fluorescence (SYXRF) microprobe. H 2O distribution in the samples was analyzed by IR microspectroscopy. Diffusion profiles are excellently reproduced, assuming concentration-independent diffusion coefficients. For all trace elements, the temperature dependence of diffusion in the hydrous melt can be described by a simple Arrhenius law. In general, the diffusivity decreases from the LFSE, to transition elements, to REE, and to the HFSE, a trend that can be correlated to the increase of charge in the same order. The activation energy shows a similar trend, increasing from 129 kJ/mol for Rb to 189 kJ/mol for Zr. For the transition elements Cr and Fe, the activation energy is relatively high (228 and 193 kJ/mol, respectively), which can be explained by increasing contributions of divalent cations to the diffusion flux with increasing temperature. Higher diffusivity of Eu compared to its neighbor elements also is attributed to contributions of divalent cations. Modeling Eu-diffusivity using data of Sr as representative for Eu 2+, and of Sm and Gd as representative for Eu 3+, shows that at all temperatures Eu 3+ is clearly dominating in the hydrous melt. To quantify the effect of water, an additional experiment was performed at 1400°C using a nominally anhydrous melt. The obtained diffusion coefficients are (for most of the elements) by one and a half orders of magnitude lower than for a melt containing 4.5 wt.% H 2O. Chemical diffusion coefficients D η, which were calculated from the viscosity data of Richet et al. (1996) using the Eyring equation, and which assumed a jump distance of 3 Å, are in excellent agreement with the diffusivity of the HFSE for both dry and hydrous melt. Most of the investigated elements show a linear relation between log diffusivity and log viscosity, enabling the prediction of diffusivities in hydrous andesite systems at various conditions. Provided viscosity data are available, we suggest that this relation can be a useful tool to estimate trace element diffusivities for silicate melts with different compositions. The new diffusion data show that water strongly enhances diffusivity of trace elements in andesitic melts. After 10,000 yr at 1200°C, diffusion produces in the dry melt relatively short profiles with lengths (defined as x=( Dt) 1/2) between 0.8 and 0.07 m (for Sr and Zr, respectively), whereas in hydrous melts (5 wt.% water), profiles are much longer with lengths between 3.9 and 0.92 m (for Sr and Zr, respectively).

Diffusional characteristics of sodium chloride in symmetrical cellulose acetate membranes measured by an unsteady-state dialysis method

Unsteady-state dialysis and steady-state reverse osmosis (RO) experiments for aqueous NaCl solutions were carried out with cellulose acetate (CA) symmetrical membranes. The distribution and overall diffusion coefficients of NaCl, K and D were obtained from the unsteady-state dialysis experimental data, taking the effect of osmosis into consideration. D was found to increase with the increase of the solute concentration. The concentration dependency of D can be explained by applying a two-part (perfect or dense and imperfect or porous) model of the membrane. The concentration independent diffusion coefficients of NaCl in the dense part D d and in the porous D p are obtained from the assumption of the two-part model for the membrane. D d does not depend on the water content in the membrane W w , but on dope composition. Contrary to D d , D p increases with the increase of W w and can be correlated only with W w and D p ( W w =0) is found almost equal to the averaged D d for each type of membrane: D p,c = D d exp ( γ × W w ). It seems from these facts that the structure of membrane becomes tight with the decrease of the ratio of formamide to CA and loose with the decrease the concentration of CA in dope solutions. D c(c = 0) at the zero concentration can be calculated as follows: D c (0) ≈ W w D p,c + (1−W w) D d . Solute permeability ω obtained by RO experiments agrees well with ω c = [K · D c (0)] (ℓ · β) , those calculated from D c (0) and distribution coefficient K.

Praseodymium-doped Ti:LiNbO 3 waveguides

by diffusion doping is investigated by means of secondary neutral mass spectrometry and secondary ion mass spectrometry. The diffusion of praseodymium in LiNbO3 can be described by Fick’s laws of diffusion with a concentration-independent diffusion coefficient and a limited solubility of praseodymium in LiNbO3 increasing exponentially with rising temperature. The diffusion depends on the Li2O content of the LiNbO3 crystal. For LiNbO3 crystals with a nominal slight difference in the congruent composition, the diffusion constants and activation energies for Z-cut LiNbO3 are 3.28×10-5 cm2/s and 2.27 eV, and 1.39×10-5 cm2/s and 2.24 eV, respectively. Titanium-doped waveguides are formed in Pr:LiNbO3 and characterised in relation to waveguide loss and absorption in the visible and near infrared.

Non-metal diffusion coefficients for the Ta–C and Ta–N systems

The diffusivity of carbon in tantalum and tantalum carbides was investigated in the temperature range 1700°C–2200°C, and that of nitrogen in tantalum and tantalum nitrides between 1700°C and 1950°C. The concentration-independent diffusion coefficients were obtained in all phases by investigating the enhanced layer growth in wedge-shaped specimens. In the nonmetal-rich phases having a broad homogeneity range, i.e. in δ-TaC 1− x , β-Ta 2C 1− x and β-Ta 2N 1− x , also the concentration-dependent diffusion coefficients were calculated and compared with the concentration-independent diffusion coefficients. The calculation of the concentration-dependent diffusion coefficients was performed by fitting a modified error function on the measured concentration profiles assuming the nonmetal diffusivity being an exponential function of the nonmetal concentration. Strong dependence of the diffusion coefficient on the nonmetal concentration was found for the δ-TaC 1− x phase, whereas the nonmetal diffusion coefficients were nearly concentration-independent for β-Ta 2C 1− x and β-Ta 2N 1− x .

Combined refinement of diffusion coefficients applied on the Nb-C and Nb-N systems

A novel technique was used for the calculation of diffusion coefficients in the niobium carbides and nitrides prepared by reaction diffusion. The temperature ranges investigated were 1500 °C to 2100 °C for the Nb-C system and 1400 °C to 1800 °C for the Nb-N system. Three independent theoretical approaches were applied and their results are compared. In the metalloid-rich phases, the concentration-dependent diffusion coefficients were calculated from the concentration profiles; two models of layer growth were used to obtain the concentration-independent diffusion coefficients in all phases. It was found that the diffusion coefficient of carbon in δ-NbC1−x shows a decrease with increasing metalloid concentration, whereas the diffusivity of nitrogen in δ-NbN1−x is nearly independent of the nonmetal concentration. The concentration dependence of the carbon diffusion coefficients in δ-NbC1−x is a result of a lower activation energy of carbon diffusion in the substoichiometric δ-NbC1−x than in the δ-NbC. On the contrary, the activation energy of nitrogen in δ-NbN1−x does not change with the nitrogen concentration. This behavior could be explained by the different occupancies of metal sublattices, which remain constant in δ-NbC1−x but decrease with increasing nonmetal concentration in δ-NbN1−x.