Darken Model Research Articles

On determining the surface segregation kinetics of Sb in binary and ternary Fe alloys during continuous annealing

In the present research, Sb surface segregation kinetics for Fe-(0.01/0.03)Sb and Fe-2Mn-(0.01/0.03)Sb at.% alloys were determined based on the modified Darken model and linear heating followed by isothermal annealing. The modified Darken model was used to evaluate the measured Sb segregation profile during continuous galvanizing heat treatments. Based on the kinetic analysis, increasing temperatures and isothermal holding times led to an increase in Sb surface segregation. Sb segregation was observed at the surface for both the Fe-xSb and Fe-2Mn-xSb alloys after annealing. It was found that Sb surface diffusion in the Fe-Sb alloys was faster than in the Fe-2Mn-Sb alloys, which was attributed to the crystal structure and the defect density in the matrix. The segregation rate was determined from the Darken curves and a higher segregation rate was observed in Fe-xSb alloys. The activation energy of Sb diffusion for the Fe-Sb and Fe-2Mn-xSb alloys was determined to be ∼ 193±18 kJ/mol, which is close to the activation energy of bulk Sb diffusion in α-Fe.

Mutual diffusion coefficients of three alkanes (n-heptane, n-octane and isooctane) in cyclohexane from 288.15 K to 308.15 K

Mutual diffusion coefficients of three alkanes (n-heptane, n-octane and isooctane) in cyclohexane were measured by digital holographic interferometry under atmospheric pressure. The measured temperature points were 288.15 K, 298.15 K and 308.15 K, respectively and the mass fraction of the measured solution range was from 0.05 to 0.95. The extended relative uncertainty of the measurement system was estimated to be less than 2.4%. The dependence of mutual diffusion coefficient on temperature and solution concentration was studied. The mutual diffusion coefficients of these three binary mixtures increase with the increase of temperature and have a nearly linear relationship with the solution concentration. And the image processing process of holographic interference image is introduced in detail. Zhang et al.’s model, Darken model and Vignes model were used to predict the mutual diffusion coefficients of the three binary mixtures, the absolute average of relative deviations (AARDs) are 2.97%, 3.89% and 6.18%, respectively and Zhang et al.’s model is the one with the best agreement among the three models.

Fe equilibrium and kinetic segregations of surface/subsurface layers upon annealing Fe on Pt(111) in vacuum/oxygen/hydrogen environment

Fe equilibrium and kinetic segregations of surface and subsurface layers upon annealing 1ML (monolayer)-Fe/Pt(111) under vacuum/oxygen/hydrogen environment are simulated by introducing oxygen and hydrogen adsorption enthalpy into the modified Darken model. The simulation results show that (1) Fe subsurface dissolution and Pt surface segregation take place upon annealing Fe/Pt(111) in vacuum; (2) Fe surface segregation takes place upon annealing Pt/Fe/Pt(111) in oxygen environment; (3) Fe/Pt(111) and Pt/Fe/Pt(111) layered structures are formed respectively at low and elevated temperatures upon annealing FeO/Pt(111) in hydrogen environment. The experimental observations are well interpreted that the reversible modulation of Fe structures on Pt(111) surface upon annealing 1ML-Fe/Pt(111) could be reached by controlling the oxygen/hydrogen treatment and the annealing temperature.

Molecular dynamics simulation and thermodynamics calculation on surface segregation in Ni-Cu nano-films under stress

The surface segregation of Cu atoms in a Ni-Cu system was investigated using molecular dynamics simulations. Thermodynamic calculations were performed to verify the results of the molecular dynamics simulations. For the thermodynamic calculations, a model for evaluating the influence of stress on surface segregation was developed using the modified Darken model in combination with the broken-bond model. Using molecular dynamics simulations, it was found that the enrichment of Cu atoms occurred for a free-standing Ni-10 at.% Cu film consisting of 20 layers. Simultaneously, the stress distribution across the Ni-Cu thin film is obtained. The thermodynamic calculation results show that the influence of stress on the surface segregation cannot be ignored because of the considerable surface stress. Surface tension stress promotes the surface segregation of copper in Cu-Ni alloys due to the larger lattice parameter of copper than nickel, which leads to the reduction of surface strain energy. When the thickness is greater than 31 nm (or the number of layers exceeds 89), the size effect disappears, i.e., the surface concentration doesn’t increase with the increase of thickness. The calculation results obtained by the Bragg-William equation used for the surface segregation in equilibrium are in good agreement with the thermodynamic calculation and molecular dynamics simulation results.

Mutual diffusion coefficients of ethanol + isooctane and n-butanol + isooctane mixtures and tests of a new local-composition prediction model for diffusion in binary liquids

The mutual diffusion coefficients of ethanol + isooctane and n-butanol + isooctane mixtures were measured by digital holographic interferometry. The measurements were performed at atmospheric pressure with temperatures ranging from 288.15 K to 318.15 K and mass fractions ranging from 0.05 to 0.95, and the effects of changes in temperature and composition were discussed. Based on the local composition theory, a new model is proposed for predicting mutual diffusion coefficients. Modified from the Vignes model, this model can be used to predict the mutual diffusion coefficient in ideal or non-ideal mixtures within a given deviation range. A comparison of the prediction results for the new model and five other models shows that the new model is far superior to the Darken model and Vignes model, and also has some advantages over the other models.

Thermodynamics and kinetics of surface/interface segregation in the stressed ultrathin alloy film on inert substrate

A general method is proposed for the simulation of thermodynamics and kinetics of surface/interface segregation in ultrathin alloy films deposited on inert substrates. This method takes into account the effects of the finite size of the system, the interlayer atomic interaction and the heteroepitaxial strain, and is based on the combination of the regular solution broken bond-type model and the modified Darken model. As an example, the method is applied to simulate the equilibrium surface/interface segregation in ultrathin Cu(111)–0.25at.% Ag alloy film deposited on the sapphire substrate, and the kinetics of its change. It is shown that heteroepitaxial strain strongly affects the chemical compositions of the film surface and its interface with the substrate.

AES study of Cu and S surface segregation in a ternary Ni-Cu(S) alloy in combination with a linear heating method

Surface segregation of Cu and S in a Ni-18.7 at.%Cu alloy with 0.0007 at.%S was studied by means of Auger electron spectroscopy. A linear temperature programmed heating method was employed to in-situ heat the sample in the temperature range from 423 K to 1121 K at a constant rate of 0.02 K/s. It was found that the Cu started to segregate at 560 K and reached the maximum surface coverage values of 41.4 at.% at 896 K. Once the Cu reached the maximum surface coverage, it started to desegregate until Cu was replaced by S. At that moment, S segregation reached the maximum surface coverage values of 17.1 at.%. The modified Darken model simulations for both the kinetic and equilibrium in a ternary system were compared with the measured surface segregation profile results yielding the segregation parameters. The segregation parameters for both Cu and S in Ni-Cu(S) ternary alloy, namely the segregation energy (ΔG0), the interaction parameter (Ω) and the diffusion parameters (pre-exponential factor (Do) and the activation energy (Q)) were determined as: ΔGCu0 = −36.0 kJ/mol, ΔGS0 = −136.0 kJ/mol, ΩCu-Ni = 7.6 kJ/mol, ΩS-Ni = 28.1 kJ/mol and ΩCu-S = −10.3 kJ/mol, DCu in Ni = 8.6 × 10−14exp(-145.2 kJ/RT) m2/s and DS in Ni = 9.2 × 10−2exp(-224.0 kJ/RT) m2/s.

Effective potential theory for diffusion in binary ionic mixtures.

Self-diffusion and interdiffusion coefficients of binary ionic mixtures are evaluated using the effective potential theory (EPT), and the predictions are compared with the results of molecular dynamics simulations. We find that EPT agrees with molecular dynamics from weak coupling well into the strong-coupling regime, which is a similar range of coupling strengths as previously observed in comparisons with the one-component plasma. Within this range, typical relative errors of approximately 20% and worst-case relative errors of approximately 40% are observed. We also examine the Darken model, which approximates the interdiffusion coefficients based on the self-diffusion coefficients.

Equilibrium and Kinetics of Methane and Ethane Adsorption in Carbon Molecular Sieve

Abstract The comprehensive study involving uptake and kinetics behaviour of methane and ethane was carried out on a commercial carbon molecular sieve. The gravimetric technique was applied to measure the adsorption isotherms at different temperatures. The kinetic behaviours of methane and ethane were investigated by extracting the uptake data from isotherms at different pressure steps. Considering CMS as a homogenous microporous adsorbent, only two main resistances: surface barrier (at the pore mouth) and diffusion resistance (inside the pores) were considered in analysing the kinetic data. The kinetic analysis showed dominance of surface barrier resistance at all temperatures and pressures investigated in this study. Modified Darken equation was used to calculate the sorbates’ activation energies for adsorption in CMS pores. Compared to Darken equation, the modified Darken model predicted the concentration dependency of the surface rate constants more accurately, especially at higher sorbates’ concentrations. The activation energy of methane and ethane was greater than the heat of adsorption demonstrating a large impedance and barrier at the pore surface to the methane and ethane molecules. The kinetic selectivity of methane over ethane was calculated from the combined surface barrier/diffusion model parametric analysis. The results generally showed a greater selectivity to methane over ethane, i. e., the higher mass transfer rates.

Influence of the “surface effect” on the segregation parameters of S in Fe(100): A multi-scale modelling and Auger Electron Spectroscopy study

The article takes a new look at the process of atomic segregation by considering the influence of surface relaxation on the segregation parameters; the activation energy (Q), segregation energy (ΔG), interaction parameter (Ω) and the pre-exponential factor (D0). Computational modelling, namely Density Functional Theory (DFT) and the Modified Darken Model (MDM) in conjunction with Auger Electron Spectroscopy (AES) was utilized to study the variation of the segregation parameters for S in the surface region of Fe(100). Results indicate a variation in each of the segregation parameters as a function of the atomic layer under consideration. Values of the segregation parameters varied more dramatically as the surface layer is approached, with atomic layer 2 having the largest deviations in comparison to the bulk values. This atomic layer had the highest Q value and formed the rate limiting step for the segregation of S towards the Fe(100) surface. It was found that the segregation process is influenced by two sets of segregation parameters, those of the surface region formed by atomic layer 2, and those in the bulk material. This article is the first to conduct a full scale investigation on the influence of surface relaxation on segregation and labelled it the “surface effect”.

Local equilibrium in the dissolution and segregation kinetics of Ag on Cu(1 1 1) surface

The local equilibrium in the kinetic processes of dissolution and segregation of Ag on Cu(111) surface is addressed. The measured dissolution and segregation kinetic data of Ag on Cu(111) surface at temperature of 450°C are well fitted by the modified Darken model. The segregation parameters, i.e. segregation energy, interaction and diffusion parameters, in the Cu(111)(Ag) binary system are obtained upon fitting. Using the obtained segregation parameters, the discontinuous transition of Ag surface concentration against the bulk concentration of the surface neighboring layers deduced from the local equilibrium model is quantitatively interpreted.

Equilibrium and kinetic surface segregations in Cu–Sn thin films

Equilibrium and kinetic surface segregations in Cu–Sn thin films are simulated based on the modified Darken model for a finite sized system. The simulations are carried out for all compositional ranges and film thicknesses. The segregation energy and the interaction parameter for the simulation are determined respectively by the Miedema model and from the thermodynamic data used for the Cu–Sn phase diagram calculation. The strong negative interaction parameter restrains Sn surface segregation in the Cu–Sn system. The size effect on surface segregation depends on the segregation energy, interaction parameter, initial bulk concentration, temperature and film thickness.

Surface segregation measurements of In and S impurities from a dilute Cu(In,S) ternary alloy

The surface segregation of In and S from a dilute Cu(In,S) ternary alloy were measured using Auger electron spectroscopy coupled with a linear programmed heater. The alloy was linearly heated and cooled at constant rates. Segregation data of a linear heat run showed surface segregation of In that reached a maximum surface coverage of 25% followed by S, which reached a coverage of 30%. It was found that after In had reached a maximum surface coverage, it started to desegregate as soon as the S enriched the surface until In was completely replaced by S. The segregation parameters, namely, the pre‐exponential factor (D0), activation energy (Q), segregation energy (ΔG˚) and interaction energy (Ω) were extracted from the measured segregation data for both In and S segregation in Cu by simulating the measured segregation data with a theoretical segregation model (modified Darken model). The segregation parameters obtained for In segregation in Cu are D0 = 1.8 ± 0.5 × 10−5 m2 s−1, Q = 184.3 ± 1.0 kJ.mol−1, ΔG˚ = −61.4 ± 1.4 kJ.mol‐1, ΩCu−In = 3.0 ± 0.4 kJ.mol−1; for S segregation in Cu the parameters are D0 = 8.9 ± 0.5 × 10−3 m2 s−1, Q = 212.8 ± 3.0 kJ.mol−1, ΔG˚ = −120.0 ± 3.5 kJ.mol−1, ΩCu−S = 23.0 ± 2.0 kJ mol−1 and the In and S interaction parameter is ΩIn−S = −4.0 ± 0.5 kJ.mol−1. The initial parameters used for the Darken calculations were extracted from fits performed with the Fick's and Guttmann model. Copyright © 2013 John Wiley & Sons, Ltd.

Local Equilibrium in the Dissolution Kinetics of Ag on Cu(111) Surface

The local equilibrium in the kinetic process of dissolution is addressed. The measured dissolution kinetic data of Ag on Cu(111) surface are well fitted by the modified Darken model. Using the obtained fitting parameters, the discontinuous transition of Ag surface concentration against the bulk concentration of the neighboring layers deduced from the local equilibrium model is quantitatively interpreted.

Influence of Bulk Concentration on the Discontinuous Transition in Surface Segregation

The discontinuous transition of surface concentration due to the change of bulk concentration in isosteric/isothermal surface segregation has been discussed quantitatively according to the modified Darken model. The criteria for observing the discontinuous transition in surface segregation has been derived on the basis of the Bragg-William expression.

Size effects on surface segregation in Ni–Cu alloy thin films

Size effects on both equilibrium and kinetic surface segregations in Ni–Cu alloy thin films are simulated based on the modified Darken model for a finite sized film system. The simulations are carried out for all compositional ranges and film thicknesses. The segregation energy used for the simulation is taken from that of the corresponding Ni–Cu bulk material, which could be obtained by fitting the experimental data and/or calculated according to the Miedema model. The size effect on segregation depends on segregation energy, initial bulk concentration, temperature and film thickness.

A combinatorial approach for extracting thermo-kinetic parameters from diffusion profiles

A strategy for extracting thermodynamic and diffusion parameters from interdiffusion profiles is presented. The approach is based on the Darken model or its development by Manning. Diffusion profiles are calculated numerically using the Radau 5 integrator. Optimization of the parameters is performed by simultaneously fitting simulated profiles to many experimental ones, representing a whole system, using the Levenberg–Marquardt method. The approach is validated for Cu–Fe–Ni system at 1273 K.

Predictive Darken Equation for Maxwell-Stefan Diffusivities in Multicomponent Mixtures

This Article presents the derivation and validation of a rigorous model for the prediction of multicomponent Maxwell-Stefan (MS) diffusion coefficients. The MS theory provides a sound framework for modeling mass transport in gases and liquids. Unfortunately, MS diffusivities are concentration dependent, and this needs to be taken into account in practical applications. There is therefore a considerable interest in models describing the concentration dependence of MS diffusivities. While current practice employs empirical models for this purpose, recent work on molecular simulations favor the physically based Darken equation. The Darken equation, however, is limited to binary mixtures and is not predictive. In this study, a multicomponent Darken model for MS diffusivities is derived from linear response theory and the Onsager relations. In addition, a predictive model for the required self-diffusivities in the mixture is proposed, leading to the predictive Darken-LBV model. We compare our novel model to the existing generalized Vignes equation and the generalized Darken equation using molecular dynamics (MD) simulation. Two systems are considered: (1) ternary and quaternary systems in which particles are interacting using the Weeks-Chandler-Andersen (WCA) potential; (2) the ternary system n-hexane-cyclohexane-toluene. Our results show that, in all studied systems, the novel predictive Darken-LBV equation describes the concentration dependence better than the existing models. The physically based Darken-LBV model provides a sound and robust framework for prediction of MS diffusion coefficients in multicomponent mixtures.

Thermodynamically Consistent Formulation of the Sedimentation in Solids and Liquids

We propose a theoretical framework to derive a consistent set of equations describing sedimentation, i.e. diffusion induced by the gravity (centrifugal) field, in a multi component system. This procedure starts from the combined 1) mass conservation law, 2) equation of motion, 3) volume continuity equation and 4) Nernst-Planck flux formulae. The method known as bi-velocity or Darken model is based on the postulate of a unique transport mechanism for volume and mass. The concentration profile in the equilibrium state is found by solving the nonlinear set of equations. Example applications of this theory applied to Bi-Sb system is evidence of sedimentation of substitutional atoms in condensed matter via diffusion mechanisms.

Compensating impurity in high purity silicon single crystal investigated by photo-thermal ionization spectroscopy

The optimum photo-thermal ionization range near liquid helium temperature for the shallow impurities in n type high purity silicon single crystal was determined experimentally. At temperature within this range, the high resolution photo-thermal ionization spectra (PTIS) of this sample were measured. Positive PTIS responses from the majority residual impurity phosphor (P) and compensating impurity boron (B) were observed by simultaneously irradiating the sample with far infrared and band-edge light. Under action of applied external magnetic fields, the positive response from compensating impurity B changes to negative response. The posibility of these phenomena being the result of temperature effect was excluded, which revealed the deficiency of Darken model and supported the model of rapid recombination of minority carriers for PTIS response of compensating impurity in elemental semiconductors.