Composition-dependent Diffusion Coefficient Research Articles

Utilising physics-informed neural networks for optimisation of diffusion coefficients in pseudo-binary diffusion couples

ABSTRACT We propose a numerical inverse method based on physics-informed neural networks (PINN) for calculating composition-dependent diffusion coefficients in pseudo-binary (PB) diffusion couples in multicomponent alloys. Traditional methods rely solely on experimental diffusion profiles as design targets, which can lead to unreliable estimates. In contrast, PINN uses a combination of available data and physics-based constraints to obtain optimised design parameters and exact solutions for constrained governing differential equations. The constraints in PINN include governing partial differential equations, initial and boundary conditions, and any other equality/inequality relations obeyed by physical parameters. Our study shows the necessity of experimentally estimated intrinsic diffusion coefficients for the prediction of reliable composition-dependent mobility data. In the absence of such data, different combinations of unknown intrinsic diffusion coefficients can also produce reasonable approximations of diffusion profiles and interdiffusion coefficients while wrongly predicting more fundamental quantities (i.e. intrinsic diffusivities). Our method utilises PINN to simultaneously obtain optimised design parameters (diffusion coefficients) and exact solutions for governing diffusion equations. The implementation of PINN uses experimentally estimated diffusion coefficients obtained by the PB diffusion couple method, in addition to diffusion profiles, as design targets. The method is extended to non-ideal PB diffusion profiles, such as conventional diffusion couples in which all the components develop diffusion profiles, by incorporating additional constraints such as zero composition gradient of certain component(s) and experimentally estimated interdiffusion coefficients at extrema in the composition profiles. PINN is found to be a promising approach for obtaining reliable estimates of diffusion coefficients in multicomponent alloys.

Correlation between structures and atomic transport properties of compound forming liquid Cu-In alloys

The Lebowitz solution of the Percus-Yevick hard-sphere mixture is invoked with the square-well potential function to determine Ashcroft-Langreth type three partial correlation functions in binary Cu-In liquid alloys. These computed partial correlation functions have been employed to generate Bhatia-Thornton fluctuations in the considered melts. Using these structural parameters we investigate the temperature and composition-dependent diffusion coefficients of considered melts. The computed values of concentration-concentration fluctuations, in the long-wavelength limit, and the Warren-Cowley short-range order parameter, show the formation of chemical compounds between unlike atoms in the In-rich region of Cu-In melts. We find a very good agreement between theoretically formulated and computed data of with the corresponding experimental values. Computed results of and suggest that hetero coordination is favorable over homo coordination in the investigated melts. The validity of the Stokes-Einstein relation was observed over a wide range of temperatures and compositions in the investigated melts. Further, a new correlation between two body pair excess entropy and the Stoke-Einstein relation has been formulated for square-well binary liquid alloys. Surface tension and compressibility as a function of In composition have been computed through microscopic structural functions of the alloys. Computed surface tension and ratio of mutual to intrinsic diffusion (D m /D id ) are found comparable to available simulated data. The computed values of shear viscosity are in good agreement with the experimental data. Calculated results suggest that the combination of hard sphere potential with square-well tail under mean spherical model approximation is one of the good method for determining the structures and transport coefficients of compound forming binary liquid alloys.

Diffusion, defects and understanding the growth of a multicomponent interdiffusion zone between Pt-modified B2 NiAl bond coat and single crystal superalloy

Composition-dependent diffusion coefficients are determined in B2-Ni(CoPt)Al system following the pseudo-binary and pseudo-ternary diffusion couple methods, which would not be possible otherwise in a quaternary inhomogeneous material fulfilling the conditions to solve the equations developed based on the Onsager formalism. The end-member compositions to produce ideal/near-ideal diffusion profiles are chosen based on thermodynamic details. The pseudo-binary interdiffusion coefficients of Ni and Al decrease in the presence of Co but increase in the presence of Pt. The pseudo-ternary interdiffusion coefficients indicate that the main interdiffusion coefficients increase significantly in the presence of Pt. Marginal changes of the cross interdiffusion coefficients substantiate a minor change of the diffusional interactions between the components. The thermodynamic driving forces show opposite trends with respect to composition as compared to the changes of the interdiffusion coefficients advocating a dominating role of the Pt(Co)-induced modifications of point defect concentrations. DFT-based calculations revealed that Pt alloying increases the Ni vacancy concentration and decreases the activation energy for the triple defect diffusion mechanism. These findings explain the increase in the thickness of the interdiffusion zone between the B2-Ni(Pt)Al bond coat and the single crystal superalloy René N5 because of Pt addition. Furthermore, the EPMA and TEM analyses reveal the growth of refractory elements-enriched precipitates.

Influence of Composition Dependent Diffusion Coefficient, Viscosity and Relaxation Time on Evaporative Rayleigh-Bénard-Marangoni Instabilities Induced by Solvent Evaporation in a Polymer Solution

In this study, a linear stability analysis is performed for both monotonic and oscillatory modes within a horizontal polymer solution layer, which solely the solvent evaporates into air. The approach is based on general thermodynamic principles and also on the physics of the gas phase and its interactions with the liquid phase. Due to evaporation, the solvent mass fraction changes and cooling occurs at the liquid-gas interface. This can trigger solutal and thermal Rayleigh-Benard-Marangoni instabilities in the system. For the monotonic mode, the effects of composition dependent diffusion coefficient and dynamic viscosity on the onset of Rayleigh-Benard-Marangoni convection are studied. Moreover, the effect of different total heights of the liquid-gas system on the behavior of convection onset is considered. The results show that a variable diffusion coefficient and a variable viscosity can notably change the onset of instability for a polyisobutylene (PIB)/toluene solution. Our model for the monotonic mode is also satisfactorily compared with an experimental study. For the oscillatory mode, where the relaxation time is also composition dependent, we observe that very thin layers will be susceptible to an oscillatory instability when drying occurs in the system. Finally, an approximate model is derived exploiting the fact that the solutal Marangoni is by far the most dominant instability mechanism here. A negligible difference with respect to the full model confirms the predominance of the solutal Marangoni mechanism.

Pseudo-binary and pseudo-ternary diffusion couple methods for estimation of the diffusion coefficients in multicomponent systems and high entropy alloys

ABSTRACTThe benefits of using the pseudo-binary and pseudo-ternary diffusion couple methods in multicomponent inhomogeneous systems are demonstrated by estimating different types of composition-dependent diffusion coefficients. These are important for understanding the basic atomic mechanism of diffusion and complex compositional evolutions. These were otherwise considered impossible during the last many decades. Without any options previously, sometimes the average values over a composition range of random choice were estimated, which are not the material constants but depend on the composition range and also the end member compositions. The steps and analyses for utilising the pseudo-binary and pseudo-ternary methods are first described in the Ni-Co-Fe-Mo system by producing the ideal diffusion profiles fulfilling the concepts behind these methods. Following, the discussion is extended to the systems related to medium (Ni-Co-Cr) and high (Ni-Co-Fe-Mn-Al) entropy alloys. In fact, this is the first report showing a correct experimental method that should be followed for the estimation of the interdiffusion and intrinsic diffusion coefficients in inhomogeneous high entropy alloys. In the end, the limitations of following these methods because of the generation of non-ideal diffusion profiles are discussed based on experimental results. The steps are also suggested to avoid such complications. These methods are easy to adopt for research engineers. Most importantly, these give an opportunity to validate the data estimated following newly proposed numerical methods by different groups with experimentally estimated diffusion coefficients, which were not possible earlier.

Generalized Fickian approach for phase separating fluid mixtures in Smoothed Particle Hydrodynamics

In the preparation process of porous polymer membranes, multi-component mass transfer during phase separation leads to the formation of pores. In this paper, we study the Generalized Fickian approach for a phase separating ternary fluid mixture using the Smoothed Particle Hydrodynamics (SPH) method. The thermodynamics are based on the Flory–Huggins equation of state for polymer mixtures with different molar weight and density of the components.We present a validation of the discrete balance equation and study the effect of different molar weight and density on the accuracy of the method separately by considering phase separation in a periodic box in two dimensions. We analyze the mass fraction distribution and the time-evolving reduction of the amount of interface using a mapping scheme based on the rheological Doi–Ohta model. We find good agreement with the diffusion-dominant power law for coarsening. Finally, we apply the proposed SPH model on a realistic Polyvinylidenfluorid/Dimethylformamid/water system. The corresponding, composition-dependent diffusion coefficients are estimated using molecular dynamics simulations.

Impact of variation in multicomponent diffusion coefficients and salinity in CO2-EOR: A numerical study using molecular dynamics simulation

CO2 injection in depleted or partially depleted oil reservoirs entails a three phase flow system governed by physical processes such as molecular diffusion and solubility. Using numerical modelling, the aims of this paper are two-fold. (i) We investigate the impact of variations in the magnitude of diffusion of CO2 into oil on dissolution of CO2 in brine, and quantify the sensitivity of the simulation outputs (recovery factor and amount of CO2 stored in water and oil phases) by use of different sets of diffusion coefficients throughout the simulation based on the variations in the compositions of the fluids. (ii) We investigate whether CO2 dissolution in brine in a water-flooded system will be a competing or limiting factor for enhanced oil recovery by molecular diffusion of CO2 into oil. To this end, we use molecular dynamics (MD) simulation to determine composition-dependent diffusion coefficients for a multicomponent fluid system in a synthetic fractured reservoir that undergoes CO2 injection. In total we consider 5 components interacting in the reservoir model, namely, CO2, CH4, C4H10, C6H14 and C10H22. The fracture-matrix interaction is simplified with the dual-porosity assumption. Our results show that (i) molecular diffusion not only enhances oil recovery but also enhances CO2 dissolution in water. The enhancement, nevertheless, depends on the values of the multicomponent diffusion coefficients and may exhibit an optimal condition for dissolution due to the impact of CO2 diffusion and entrapment into matrix oil. (ii) The amount of CO2 stored in oil is strongly affected by variation in molecular diffusion coefficients (we observe up to %13 difference). (iii) The results show that there is 4% discrepancy between estimates of the recovery factor for simulation cases that are run with different values of diffusion coefficients. Therefore it is important to account for compositions-dependent diffusion coefficients in simulation of CO2-enhanced oil recovery processes.

Effect of including a gas layer on the gel formation process during the drying of a polymer solution.

In this paper, we study the influence of the upper gas layer on the drying and gelation of a polymer solution. The gel is formed due to the evaporation of the binary solution into (inert) air. A one-dimensional model is proposed, where the evaporation flux is more realistically described than in previous studies. The approach is based on general thermodynamic principles. A composition-dependent diffusion coefficient is used in the liquid phase and the local equilibrium hypothesis is introduced at the interface to describe the evaporation process. The results show that the high thickness of the gas layer reduces evaporation, thus leading to longer drying times. Our model is also compared with more phenomenological descriptions of evaporation, for which the mass flux through the interface is described by the introduction of a Peclet number. A global agreement is found for appropriate values of the Peclet numbers and our model can thus be considered as a tool allowing to link the value of the empirical Peclet number to the physics of the gas phase. Finally, in contrast with other models, our approach emphasizes the possibility of very fast gelation at the interface, which could prevent all Marangoni convection during the drying process.

Modelling the evolution of composition-and stress-depth profiles in austenitic stainless steels during low-temperature nitriding

Nitriding of stainless steel causes a surface zone of expanded austenite, which improves the wear resistance of the stainless steel while preserving the stainless behaviour. During nitriding huge residual stresses are introduced in the treated zone, arising from the volume expansion that accompanies the dissolution of high nitrogen contents in expanded austenite.An intriguing phenomenon during low-temperature nitriding is that the residual stresses evoked by dissolution of nitrogen in the solid state, affect the thermodynamics and the diffusion kinetics of nitrogen dissolution. In the present paper solid mechanics was combined with thermodynamics and diffusion kinetics to simulate the evolution of composition-depth and stress-depth profiles resulting from nitriding. The model takes into account a composition-dependent diffusion coefficient of nitrogen in expanded austenite, short range ordering (trapping) of nitrogen atoms by chromium atoms, and the effect of composition-induced stress on surface concentration and diffusive flux. The effect of plasticity and concentration-dependence of the yield stress was also included.

The Cyclic Carburization Process by Bi-velocity Method

AbstractThe paper presents numerical analysis of the cyclic external fluxes on diffusion process. It is shown, that the cycles characteristics strongly affects the concentration profiles and stress distribution in material. Calculations were performed using the bi-velocity method with the stress field factor. Moreover, the more general model describing the cyclic carburization process influenced by self stress was formulated. Numerical solutions were obtained assuming composition dependent diffusion coefficients. The results of simulation were compared with the experimental data. The model proposed in this paper may be applied to other diffusion controlled processes with the external flux of the diffusive component.

On the atomic interdiffusion in Mg–{Ce, Nd, Zn} and Zn–{Ce, Nd} binary systems

Abstract

Solvation Dynamics in a Prototypical Ionic Liquid + Dipolar Aprotic Liquid Mixture: 1-Butyl-3-methylimidazolium Tetrafluoroborate + Acetonitrile

Solvation energies, rotation times, and 100 fs to 20 ns solvation response functions of the solute coumarin 153 (C153) in mixtures of 1-butyl-3-methylimidazolium tetrafluoroborate ([Im41][BF4]) + acetonitrile (CH3CN) at room temperature (20.5 °C) are reported. Available density, shear viscosity, and electrical conductivity data at 25 °C are also collected and parametrized, and new data on refractive indices and component diffusion coefficients presented. Solvation free energies and reorganization energies associated with the S0 ↔ S1 transition of C153 are slightly (≤15%) larger in neat [Im41][BF4] than in CH3CN. No clear evidence for preferential solvation of C153 in these mixtures is found. Composition-dependent diffusion coefficients (D) of Im41(+) and CH3CN as well as C153 rotation times (τ) are approximately related to solution viscosity (η) as D, τ ∝ η(p) with values of p = -0.88, -0.77, and +0.90, respectively. Spectral/solvation response functions (Sν(t)) are bimodal at all compositions, consisting of a subpicosecond fast component followed by a broadly distributed slower component extending over ps-ns times. Integral solvation times (⟨τ(solv)⟩ = ∫(0)(∞)Sν(t) dt) follow a power law on viscosity for mixturecompositions 0.2 ≤ x(IL) ≤ 1 with p = 0.79. With recent broad-band dielectric measurements [J. Phys. Chem. B 2012, 116, 7509] asinput, a simple dielectric continuum model provides predictions for solvation response functions that correctly capture thedistinctive bimodal character of the observed response. At x(IL) ∼ 1 predicted values of ⟨τ(solv)⟩ are smaller than those observed by a factor of 2-3, but the two become approximately equal at x(IL) = 0.2. Predictions of a recent semimolecular theory [J. Phys. Chem. B 2011, 115, 4011] are less accurate, being uniformly slower than the observed solvation dynamics.

Point defects and diffusion in ordered alloys: An ab initio study of the effect of vibrations

We present a detailed investigation of the influence of atomic vibrations on the point defect and diffusion properties of ordered metallic alloys, by means of ab initio calculations with density-functional theory. Considering the case of Ni2Al3 which provides a rich panel of defect-related properties, our study reveals that the behaviour of this compound is largely monitored by self-interstitials, whereas such defects are usually ignored in metallic compounds. The vibration free energies are obtained for the full set of point defects of Ni2Al3, showing that these quantities are strongly defect-dependent, and significantly modify the free energy of the compound in an intricate composition-dependent manner. The second key-issue is the first ab initio full analysis of attempt frequencies, via the coupling of vibration analysis and saddle-point search for significant atomic jumps. This analysis indicates that attempt frequencies range over several orders of magnitude and exponentially increase with migration energies. We show the importance of these factors in reaching realistic composition-dependent diffusion coefficients.

Dimensional Analysis of Chronoamperometry: Proof and Verification of Cottrellian Behavior of Systems with Composition-Dependent Diffusion Coefficients

Chronoamperometry with a Dirichlet boundary condition and semi-infinite linear diffusion exhibits Cottrellian behavior, that is, the product it1/2 is constant as a function of time as long as the system is initially homogeneous, a conclusion that can be reached using only dimensional analysis; no detailed mathematical analysis is required. The generality of this result is known to include purely diffusional systems and systems in which transport also involves migration. In the present work, it is shown that Cottrellian behavior obtains, even when the system diffusion coefficients are a function of system composition, regardless of the exact form of that function. These conclusions are confirmed by simulations of examples for purely diffusional systems as well as for systems with migration. Some experimental examples from the literature are cited.

Nonlinear Stress Effects in Diffusion

According to classical Nernst-Einstein equation the diffusive flux is proportional to the driving force. However, this linear law is not valid if the driving force is very large. Attempts in the literature for the derivation of an “improved relation” till now were mostly restricted to the cases when the diffusion coefficient was independent of the composition. On the other hand, even if there are no externaldriving forces (other than related to the chemical driving force) present, deviations from the Fick I law are expected (transition from parabolic to linear growth-behaviour) on nanoscale for composition dependent diffusion coefficients. General description for the case when the driving forces and the diffusion asymmetry are large, is treated. The special case of large pressure gradients is discussed in detail and their effects on the deviation form the parabolic growth law on nanoscale will be analyzed. Effect of a pressure gradient on the crossover thickness between parabolic and linear regimes and on the interface transfer coefficient, K, is also treated.

Ternary Mutual Diffusion Coefficients from Error-Function Dispersion Profiles: Aqueous Solutions of Triton X-100 Micelles + Poly(ethylene glycol)

Taylor dispersion has gained widespread popularity for measuring diffusion in liquids. The usual procedure is to inject small volumes of solution containing solute at concentration c + Δc into carrier streams of composition c. Binary mutual diffusion coefficients D are evaluated from the Gaussian distribution of the dispersed solute measured at the outlet of a long capillary tube. As a result of strong dilution of the injected solute with the carrier solution, obtaining favorable signal-to-noise ratios for the measured profiles can require unacceptably large Δc values for solutions with strongly composition-dependent diffusion coefficients or broad dispersion profiles produced by slowly diffusing solutes. For these systems, D can be reliably evaluated from error-function profiles generated by changing the solution flowing into dispersion tube from composition c − (Δc/2) to c + (Δc/2). There are no dilution factors, so Δc can be orders of magnitude smaller than the values employed in conventional pulse...

Strategy for Applying Microanalytical Techniques

A combination of microanalysis techniques is used to reveal with great precision the composition and microstructure of typical features in engineering materials. In this paper small particles, very thin films, interfaces and their contiguous region are considered. Small silicon nitride particles in an iron matrix produced by nitriding are characterized quantitatively with High Resolution Electron Microscopy (HREM) and Electron Probe Micro Analysis (EPMA) using Wavelength Dispersive X-ray Spectrometry (WDS). Thin aluminium-oxide films grown on aluminium substrates by thermal oxidation at low pressure are investigated with HREM and X-ray Photoelectron Spectroscopy (XPS). Their thickness, composition and degree of crystallinity are established as a function of oxidation temperature and time. At and near the oxide-layer/MCrAlY-coating (M=Ni and/or Co) interface the aluminium and chromium depletion behaviour of the coating material due to high-temperature oxidation is studied with high-resolution Scanning Electron Microscopy (SEM), EPMA and Auger electron spectroscopy (AES). The composition-depth profiles obtained for the coating material are used to determine the composition dependent diffusion coefficients. These properties are crucial for the prediction through computational modelling of the coating material oxidation behaviour and enable the development of new coating materials.

Stationary dynamics approach to analytical approximations for polymer coexistence curves.

Phase separation in polymer blends is an important process. However, the compositions of the coexisting phases can only be predicted by numerical methods. We provide simple analytical expressions which serve as good approximations for the compositions after phase separation of binary homopolymer blends. These approximations are obtained by a stationary dynamics approach: we calculate the compositions of two polymer mixtures such that the stationary diffusion between these distinguishable mixtures vanishes. For the diffusion equations we employ composition-dependent diffusion coefficients, as derived according to the slow- and fast-mode theory from the Flory-Huggins free energy. The analytical results are in good agreement with exact (numerically calculated) binodal compositions. Our coexistence curves are more accurate than some conventional approximations. Another advantage of the stationary dynamics approach is that it is not only applicable to binary polymer blends or polymer solutions, but also to symmetrical multicomponent blends. The same diffusion coefficients may be used to obtain the exact spinodal compositions in multicomponent systems.

Influence of composition on the diffusivity of carbon in steels—I. Non-alloyed austenite

A quantitative expression for the variation with concentration of the diffusion coefficient of carbon in austenite was derived by modifying the absolute reaction theory for diffusion proposed by Fisher et al. and accounting for carbon resonance. By considering the kinetic energy transfer between carbon atoms during resonance, it was demonstrated that the activity coefficient of the activated complex γ M, in Fisher's formula, should be inversely related to the carbon concentration rather than being constant or proportional to the solute carbon activity coefficient, γ, as assumed by previous investigators. The reported carbon diffusion coefficients in austenite, measured over a wide composition and temperature range, were systematically analyzed on the basis of the present theory. Excellent agreement was obtained between the theory and the experimental data by describing the composition-dependent diffusion coefficient of carbon in austenite as: D=D 0(1−βγ) γ γM βγ 1−βY +1+Y d 1n(γ/γM dY

Thermochemistry of the formation of nitrogen-rich surface layers on transition metal nitrides: A study of the VN1− x− N2 couple

The formation of nitrogen-rich surface layers on substoichiometric transition metal nitrides upon cooling from their equilibrium state at high temperatures down to room temperature is described in terms of thermodynamics and diffusion equations. The formalism is applied to the VN 1− x -N 2 couple. Using a surface composition-dependent diffusion coefficient for nitrogen in VN 1− x and a simplified superposition algorithm, the diffusion profiles can be approximated. The frequency factor introduced for the diffusion coefficient (Arrhenius-type equation) is log ( D 0) = −14.6 × (1− x) + 7.8 for Q = 125.5 kJ mol −1. The compositional shift effected by nitrogen uptake at the surface of compact plates of VN 1− x was investigated by X-ray diffraction and measurements of the superconducting transition temperature. Powders show a shift of the bulk composition towards stoichiometric values. The nitrogen uptake can significantly affect the results of measurements of solid-state and thermodynamic properties vs composition. With the applied formalism the diffusion profiles in the solid of a gas-solid couple (nitrogen-nitride, oxygen-oxide, hydrogen-hydride) can be predicted for given experimental conditions, provided that diffusion and p, T, (1 − x) data are known.