Concentration Of Solute Atoms Research Articles

Unravelling precipitation behavior and mechanical properties of Al–Zn–Mg–Cu alloy

We investigate the effect of aging temperature on precipitation behavior and mechanical properties of an Al–7.6Zn–2.7Mg–2.0Cu–0.1Zr–0.07Ti (wt.%) alloy by evaluating the matrix's microhardness, electrical resistivity, and tensile properties: additionally, employing X-ray diffraction (XRD), differential scanning calorimetry (DSC), transmission electron microscopy (TEM), and atom-probe tomography (APT) to characterize this alloy. The nanoprecipitates forming under peak-aging conditions vary with aging temperature, forming coherent GPI zones at 80 °C, GPII zones with minor η' at 120–150 °C, and η'/η with minor GP zones at 180–220 °C. GPI and GPII zones forming at 80–150 °C contain similar concentrations of solute atoms (11Zn–9Mg–(<1.0)Cu (at.%)), whereas the η'/η nanoprecipitates forming at 180 °C contain larger concentrations of solute atoms (28Zn–24Mg–3.4Cu (at.%)). The strength of the peak-aged alloy decreases with increasing aging temperature owing to the increasing size and decreasing number density of the nanoprecipitates. Under peak-aging conditions, precipitation strengthening originates mainly from dislocation shearing at 80–150 °C and from Orowan bypassing at temperatures above 180 °C. The shearable to non-shearable transition of the nanoprecipitates at 180 °C reduces the strain hardening rate, thereby decreasing the alloy's ductility.

The influence of magnesium compounds on the properties of silicon nitride ceramics

AbstractSilicon nitride (Si3N4) ceramics using four compound sintering aids Er2O3–MgO, Er2O3–MgF3, Er2O3–Mg2Si, and Er2O3–Mg3N2. The effects of binary sintering aid types and temperatures on the microstructure, mechanical properties, and thermal conductivity of Si3N4 ceramics were investigated. The depolymerization effect of fluorine atoms introduced by MgF2 sintering additives on the silicate melt structure reduced the viscosity of the liquid phase. However, the existence of volatile phase leads to low density of Si3N4 ceramics. While the additional Si and N atoms increase the concentration of solute atoms, which was conducive to sintering densification of Si3N4 ceramics. Although Mg2Si and Mg3N2 sintering aids produced higher viscosity liquid phase compared with MgF2, denser Si3N4 ceramics were obtained at 1900°C. The Si3N4 ceramics doped with 5 mol% Er2O3 and 2 mol% Mg3N2 at 1900°C showed the excellent properties: including the highest Vickers hardness (16.8 GPa), flexural strength (912 MPa), and fracture toughness (8.1 MPa m1/2), as well as the desirable thermal conductivity (83 W/(m K)).

Heat treatment effects on microstructure and properties of Cu–Ti–Fe alloys

High-strength Cu–Ti–Fe alloys were prepared and investigated in this study. The alloy was subjected to cold deformation and aging treatment, and revealed the transformation mechanism of the microstructure and properties. Cold deformation of 60 %, aging at 450 °C for 240 min, can optimize the hardness and conductivity of the Cu–Ti–Fe alloy. The hardness and electrical conductivity of Cu-1.5Ti-0.3Fe alloy increased from 196.8 HV to 7.8 % IACS to 269.3 HV and 19.8 % IACS after aged. Cu4Ti and Fe2Ti are the mainly precipitated phases in Cu–Ti–Fe alloy, which effectively obstructs dislocations slipping and delays recrystallization, and the most important strengthening mechanism. The concentration of solute atoms increased could further improve the grain boundary strengthening, solution strengthening, and deformation strengthening.

Effect of solute concentration on microstructure evolution during static recrystallization in Mg-0.2%Ce alloy using cellular automata

A new cellular automata model has been developed to evaluate the effect of solute drag on the microstructure and texture evolution during static recrystallization of Mg and its alloys in the temperature range of 200 – 400 °C. In the present model, the site saturation nucleation was assumed and all the nuclei are present at the grain boundaries. The growth of each nucleus depends on the solute concentration and dislocation density difference between neighboring grains. The microstructural analysis of the newly developed cellular automata model showed the evolution of grain size at different temperatures compared to the equilibrium microstructure containing the equilibrium concentration of solute atoms at the grain boundaries. Further, the texture evolution showed that there is an increase in the texture strength with an increase in the annealing temperature when the solute concentration was lower than the equilibrium concentration at the boundaries, whereas the texture strength decreased with increasing the annealing temperature when the solute concentration was higher than the equilibrium solute concentration at the boundaries. The present model shows that solute drag pressure increased continuously with increasing the annealing temperature when the solute concentration was lower than the equilibrium concentration at the boundaries. However, the solute drag pressure decreased with an increase in the annealing temperature when the solute concentration was higher than the equilibrium solute concentration at the boundaries.

Synergistic effect of multi-element co-segregation on mechanical properties of Mg twin grain boundary

Properties of grain boundaries (GBs) are pronouncedly influenced by alien elemental segregation. Synergistic effect of co-segregation behavior on mechanical properties of Mg twin GB was systematically investigated with first-principles calculations. Ten elements (Ni, Mn, Zn, Al, Ag, Ti, Li, Zr, Y, Ca) for single segregation and four solute combinations (Al–Ca, Al–Y, Zn–Ca, Zn–Y) for co-segregation were chosen respectively, the segregation energies and solubility energies with different elemental concentrations were calculated to evaluate the thermodynamic stability of corresponding segregated GBs. These solute atoms have unique effects on the stability of the system, and the interaction between solute atoms also has a synergistic effect on the stability of the system. It is found that the segregation tendency is enhanced and the GB structure tends to be stable with the increase of the concentration of solute atoms at the appropriate sites, regardless of single-element segregation or co-segregation. The redistribution of GB charges due to co-segregation significantly affects the binding and mechanical properties of GBs. With the increase of the co-segregation concentration, the lamellar charge is transferred between the matrix and the twins, which further enhances the electron hybridization and leads to a notable enhancement of interface binding. This shows that the appropriate combination of solute elements can effectively improve the mechanical properties of the interface. The manuscript reveals the theoretical understanding of the effect of elemental segregation on the mechanical properties of GB, and provides design ideas for the tuning of mechanical properties of magnesium alloy by GB engineering.

Deformation mechanisms and mechanical properties of the high-strength and ductile Al-Zn-Mg-Cu alloys processed by repetitive continuous extrusion forming process with different heat treatments

Continuous severe plastic deformation (SPD) processes, which overcome the limitations of economic efficiency and sample size for classical SPD approaches, are promising industry technologies for manufacturing aluminum alloy products with refined microstructures and enhanced properties. In the present work, the repetitive continuous extrusion forming (R-Conform) process combined with auxiliary thermal treatments was implemented in Al-Zn-Mg-Cu alloys, which had poor workability. By combining microstructural observation, mechanical testing and finite element simulation, the influence of auxiliary heat treatments on the mechanisms of grain refinement and strengthening for Al-Zn-Mg-Cu alloys processed by R-Conform were systematically investigated. The results indicated that the grains of Al-Zn-Mg-Cu alloys were significantly refined by a continuous dynamic recrystallization (CDRX) during the R-Conform process with special thermomechanical conditions. The increased supersaturated solid solution concentration and constituent precipitate fraction caused by prehomogenization were found to accelerate CDRX for grain refinement through the pinning effect on dislocation motion. Due to the simultaneous effect of shear deformation and online solution, the R-Conformed alloys contained massive lattice distortions and high concentrations of solute atoms. As a result, the processed Al-Zn-Mg-Cu alloys combined with prehomogenization and direct aging treatments possibly possess a good combination of strength and ductility due to the integrated functions of fine grains, massive lattice distortion, and evenly distributed nanosized precipitates.

In-situ TEM characterization of basal dislocations between nano-spaced long-period stacking ordered phases in MgYZn alloy

Long-period stacking ordered (LPSO) phases play strong barriers to the motion of non-basal dislocations and deformation twins, effectively improving the strength of Mg alloys. Corresponding to the crystallographic character of LPSO phases, it is generally believed that LPSO phases would not affect the motion of basal dislocations. Here, we perform in-situ TEM nanoindentation testing in a single crystal specimen of Mg97Y2Zn alloy that contains LPSO phases with various spacing within 100 nm, and report on a significantly low activity of basal dislocations in Mg matrix between LPSO phases with spacing less than 20 nm. Atomic HADDF-STEM results reveal a high concentration of solute atoms near LPSO phases, which could provide strong solute strengthening effect. These findings may provide possible guidance to design ultra-high strength of Mg alloys containing nano-spaced LPSO phases.

Molecular Dynamics Simulation of Solution Strengthening of Si and Cu Atoms in Aluminum Alloy

For heat‐treatable aluminum alloys, solid solute elements play key role in material strengthening. Al–Mg–Si alloy is a typical heat‐treatable alloy; Cu and Si atoms are its main solid solution atoms. To reveal the strengthening mechanism, the interaction between the edge dislocations and the Cu and Si solute atoms of different concentration in aluminum matrix is investigated by molecular dynamics (MD) simulation. Results indicate that Cu atoms provide a more effective strengthening due to the stronger pinning effect. The increment of critical resolved shear stress (ΔCRSS) is a function of concentration of solid solute atoms. When more than two types of solid solution atoms coexist in matrix, the final increment of the ΔCRSS is determined by the interactive effects of the atoms instead of the direct sum of all items. The pinning of Cu solid solute atoms can lead to two Shockley partial dislocations merging to an edge dislocation.

Influence of the twin boundary and Cr segregation on the spalling of Ni-base alloys by large-scale molecular dynamic simulations

Large-scale nonequilibrium molecular dynamics simulation is used to study the effect of grain size and Cr segregation at the twin boundary (TB) on the spalling fracture mechanism of nickel based alloys. In particular, loading waves are designed so that the maximum tensile stress first appears in the grain interior for all the crystals with different grain sizes. In contrast to traditional understandings, no monotonous relationship between the spall strength and the grain size appears in our results. The spall strength is found to depend on the wave attenuation distance measured from the first maximum tensile stress position to the spalled TB as well as the accompanied microstructure evolutions. The number of spalling plane increases with the increase of TB or the decrease of the grain size. As the grain size continues to decrease, a greater impact strength is required to cause spallation fracture at multiple TBs. In this case, the spall strength becomes insensitive to the first maximum tensile stress position. With the increment of solute atom concentration, the number of the spalling plane increases when the solution is segregated. But it decreases when the solution is uniformly distributed. Such a result is explained by segregation-enhanced energy dissipation and interactions between the waves and the microstructures nearby TBs. In particular, the shock wave would induce a local lattice reorientation nearby the TB depending on its segregation degree and the lattice reorientation would modify the slip manner of stacking faults and, thus, affect void nucleation and growth. The lattice reorientation would also contribute to the generation of sub-grain boundaries inside the grains in terms of the movement of stacking faults. Void nucleation at the sub-GB is the main cause of fracture at the grain interior.

Microhardness and tensile strength of electrochemically synthesized nickel-cobalt binary alloy sheets exfoliated from a dumbbell-shaped titanium cathode

Nanocrystalline nickel–cobalt (Ni–Co) binary alloy sheets were fabricated through electroforming in an acidic aqueous bath using exfoliation from a metallic titanium cathode. Cobalt content in Ni–Co alloy sheets ranged from 28.8 at% to 72.0 at% depending on experimental parameters, such as cathodic overpotential and bath composition. The surface roughness (R a) of the electroformed alloy sheets significantly decreased down to 1.5 μm as saccharin sodium dihydrate was added as an additive to the acidic aqueous solution bath. X-ray diffraction profiles and transmission electron microscopy images indicated that the electroformed Ni–Co alloy sheets have a nanocrystalline structure (grain size ≈ 30 nm). The lattice constant of the electroformed Ni–Co alloy sheets increased with an increase in cobalt content (i.e. solute atom concentration). The mechanical properties were significantly improved because of the synergistic effects of crystal grain refinement and solid solution strengthening. The microhardness and tensile strength of the electroformed Ni–Co alloy sheets reached 609 kgf mm−2 and 1757 MPa (X Co = 49.9 at%), respectively. The tensile strength of the electroformed Ni–Co alloy sheets in this study significantly exceeded that of solidified Ni–Co alloys (approximately 370 MPa). Therefore, this study offers a technique to enhance the mechanical properties of electroformed Ni–Co alloy sheets.

A multiplicative finite strain deformation for diffusion-induced stress: An incremental approach

Understanding the evolution of lithium concentration and stresses in electrode materials during electrochemical cycling is of paramount importance in studying the fundamental mechanisms contributing to the structural degradation of lithiated electrode materials. In this work, we use the change rate of the volume of a host material in analyzing diffusion-induced volume expansion and obtain the volume expansion as an exponential function of the concentration of solute atoms and the coefficient of linear expansion in contrast to the widely used linear relation between the diffusion-induced volume expansion and the coefficient of linear expansion. Using the principle of multiplicative decomposition in the framework of elastic deformation and thermodynamic free energies, we derive the constitutive relation for diffusion-induced finite deformation of elastic materials. The symmetric Piola-Kirchhoff stress is an exponential function of the coefficient of linear expansion and the concentration of solute atoms. Incorporating the change of strain field with the change in the number of solute atoms in Helmholtz free energy, we derive the chemical potential of the solute atoms in the host material the same as the one given by Li et al. (Zeitschrift für PhysikalischeChemie, 49 (1966) 271–290). Numerical results for the lithiation of a spherical silicon electrode reveal that the spatial distributions of lithium concentration and stresses for small dimensionless time (dilute solution) exhibit similar trends to the corresponding ones calculated from the theory of linear elasticity and nonlinear characteristics for large dimensionless time (concentrated solution).

Effect of low-temperature annealing on electrical resistivity and mechanical properties of laser-powder bed fused AlSi10Mg alloy

The AlSi10Mg sample was additive-manufactured using laser powder bed fusion (L-PBF), and subsequent low-temperature annealing (LTA) at 413K or 473 K was applied. Vickers hardness tests show the peak at around 160 HV by 473 K annealing and around 145 HV by 413 K annealing. The flow stress of tensile and compression deformations are also affected by LTA, and peak aging condition gives the highest 0.2% proof stress. 0.2% proof stress obtained by tensile tests is around 270 MPa, whereas that obtained by compression tests is between 326 MPa and 386 MPa. Electrical resistivity at 293 K changes from approximately 80 nΩm down to around 60 nΩm, whereas that at 77 K changes from around 40 nΩm down to around 20 nΩm. The estimated concentration of solute Si in the as-built specimen is between 2.5 mass% and 2.9 mass%, corresponding to the previous results using energy-dispersive X-ray spectroscopy. The lattice constant obtained from XRD increased from 0.4043 nm up to 0.4048 nm by LTA. Both electrical resistivity and XRD measurement indicate a decrease in the concentration of solute atoms. Microstructure observations revealed numerous nano-sized precipitates of Si phase with a plate-like or granular morphologies in α-Al grains due to LTA.

Nondestructive Ultrasonic Evaluation of Ferritic CrW Steel upon Long-term Aging

The microstructural evolution of heat-resistant ferritic CrW steel during long-term aging was characterized by ultrasonic wave velocity measurement. Long-term aging was conducted at 700°C for 4000 h. The ultrasonic wave velocity increased monotonically upon long-term aging. The ultrasonic wave velocity increased rapidly during the initial 1000 h of aging and moderately thereafter. The MSUB23/SUBCSUB6/SUB carbide and Fe2W Laves phases underwent coarsening through the Ostwald ripening mechanism, causing a decrease in Cr and W solute atom concentration of the iron matrix during 4000-h aging. The dislocation density also decreased because of the heat treatment. The width of martensite lath increased owing to dislocation recovery at the lath boundaries. The ultrasonic wave velocity depends on the Young’s modulus of the material, which may influence microstructural features such as dislocation substructure, precipitates, solute atom concentration, and martensite lath boundaries. This study utilizes ultrasonic wave velocity to quantify the damage level and microstructural changes during the long-term aging of ferritic CrW steel.br/

Interfacial precipitation of the in-situ TiB2-reinforced Al–Zn–Mg–Cu–Zr composite

The interfacial precipitation of reinforced particles has a critical influence on particle-reinforced aluminum-matrix composites. In this study, TiB2 interfacial precipitation of 6 wt% TiB2/Al–10Zn–2.3 Mg–2Cu–0.12Zr composite was studied. The results demonstrate that the TiB2 surface has a considerable impact on precipitation during different states of the composite preparation process (including heat treatments). The TiB2 surface acted as a nucleation site for D023–Al3Zr and was wrapped with the T phase during solidification; a good orientation relationship existed between TiB2 and D023–Al3Zr: (0 00 1)TiB2//(1‾ 03‾)D023–Al3Zr, (1 01‾ 0)TiB2//(1 05‾)D023–Al3Zr, and [1 21‾ 0]TiB2//[0 1 0]D023–Al3Zr. After homogenization or solution, the TiB2 wall hindering solute atoms diffused into the matrix, particularly for Mg and Cu, causing the T phase to dissolve incompletely, and the residual T phase on the TiB2 surface transformed to the S phase layer at a longer holding time. Additionally, the hindering effect of the TiB2 walls results in the enhancement of the local high solute atom concentration and provides a high precipitation driving force. Therefore, the interfacial phase, Mg(Cu2-XZnX), preferentially precipitated at the D023–Al3Zr and S phase interface during peak aging. In this process, the periodic misfit dislocations of the interface facilitate the diffusion of solute atoms and reduce the nucleation energy barrier, promoting interfacial precipitation. Eventually, the interfacial precipitates layer relieved the local stress concentration but consumed the solute atoms, which was considered to be beneficial to plasticity at the cost of strength weakening to achieve excellent comprehensive mechanical properties.

Derivation of phase field model for the second nearest interatomic potential of D022 structure and sedimentation process in N0.75AlxV0.25−x alloy

This paper obtains a new phase field method that can simultaneously calculate the interaction potential between the first and second neighbor atoms of the D022 structure Ni3V phase in the Ni0.75AlxV0.25−x alloy, and analyzes the interaction potential between the two neighbor atoms of the D022 structure with the change trend of temperature and concentration. The feasibility of using the phase field method to invert the interaction potential between two neighboring atoms of the D022 structure in the Ni0.75AlxV0.25−x alloy laws and the precipitation of D022 and the types of interfaces formed by D022 and L12 during the aging process to be used in the microscopic phase field equation is studied. The results show that with the increase/decrease of the temperature or the concentration of solute atoms, the first neighbor interaction potential between the first neighbor atoms of the D022 structure increases/decreases and the neighbor interaction potential between the second neighbor atoms of the D022 structure decreases/increases; Ni0.75AlxV0.25−x alloys form two ordered phase interfaces between L12 and D022 phases during high temperature precipitation; at the same temperature and the concentration, the calculated interatomic potential is similar to the interatomic potential obtained by other methods. The calculated interatomic potential is substituted into the phase field model, and the simulated alloy precipitation results are consistent with the experimental results.

Improvement in the mechanical properties of as-rapidly solidified Al-Zn-Mg-Cu alloys with Zr addition before and after aging

High cooling rate can increase the concentration of solute atoms in the solidification structure to produce a supersaturated solid solution. As-rapidly solidified Al-Zn-Mg-Cu alloys with different Zr mass fractions were prepared. The microstructure and tensile properties of as-cast and aged alloys showed that the increase of Zr element did not significantly refine the grain size. But it obviously improved the mechanical properties of alloys before and after aging. The addition of Zr in the as-cast alloys improved the mechanical properties of the alloy by solution strengthening. In the aged alloys, the mechanical properties were enhanced mainly by the precipitation of L12-Al3Zr, which has a coherent relationship with the Al matrix and refined the size of θʹ-Al2Cu. With superfluous Zr content, the coarsening of L12-Al3Zr precipitations harmed the mechanical properties.

Microstructure evolution and precipitation behavior of Al-Mg-Si alloy during initial aging

The microstructure evolution and precipitation behavior of Al-Mg-Si alloy during initial aging were studied using hardness testing, conductivity testing, differential scanning calorimetry (DSC), and high resolution transmission electron microscopy (HRTEM). The results show that the precipitation sequence of the Al-Mg-Si alloy during initial aging can be represented as: supersaturated solid solution → spherical Mg/Si clusters → needle-like Guinier Preston (GP) zone → β″. Clusters are completely coherent with the Al matrix. The GP zone with relatively complete independent lattice parameters that differ slightly from the Al matrix parameters, is oriented along the direction of <111>Al and lying on {111}Al plane. The strength of the Al-Mg-Si alloy is greatly enhanced by the three-dimensional strain field that exists between the β″ phase and the two {200}Al planes. After aging at 170 °C for 6 h, the hardness reaches the peak of 127 HV and remains for a long time. At this stage, the electrical conductivity keeps relatively stable due to the formation of coherent precipitates (Mg/Si clusters/GP zones) and the reduction in solute atom concentration in the Al matrix. The severe coarsening and decreased number density of the β″ phase during the over-aging stage result in a significant decrease in the hardness.

Impact of the pulse induced current on the microstructure and mechanical properties of the 7075-T6 aluminum alloy

The effect of the pulse induced current on the mechanical properties and microstructure of the 7075-T6 aluminum alloy was thoroughly studied by transmission electron microscopy, X-ray diffraction, and electron backscattered diffraction measurements. The experimental results demonstrated that the strength of the alloy decreases slightly, whereas the elongation increases significantly. On top of that, after the application of the induced electro-pulsing treatment, the dislocation density of the alloy decreases, while the grain size and the second phase particles grow slightly. In addition, the induced current also causes the Cube, Copper and S textures to decrease. On the other hand, the Goss texture is enhanced. The current density and the local temperature distribution of the sample were simulated by employing the ANSYS software. From the acquired outcomes, we can draw the conclusion that the recorded changes in the mechanical properties and microstructure of our sample can be attributed to the combination of the joule phenomenon and the pure electric effects of the current. The Joule heat and current increase the vacancy concentration and diffusion coefficient of solute atoms and promote dislocation recovery, precipitation phase growth, and grain rotation. • Induced electro-pulsing treatment (IEPT) is used to change the properties of 7075-T6 aluminum alloy. • After IEPT, the plasticity of the alloy is significantly improved while the strength is slightly decreased. • It is found that induced current can reduce dislocation density, promote grain rotation and change texture type.

Annealing effects on the microstructure, strength, and electrical resistivity in a dual-phase Cu-15 wt%Ag alloy

The precipitation behavior and the microstructural evolution of a dual-phase Cu-15 wt%Ag alloy during annealing were investigated. We discussed the effects of dislocation density, grain size, solute atom concentration, and precipitation morphology on strength and electrical conductivity. The results showed that continuous precipitation and discontinuous precipitation simultaneously occurred during the annealing of the Cu-15 wt%Ag alloy. It was accompanied by pronounced recovery and recrystallization, significantly decreasing the strength and electrical resistivity of the cold-rolled Cu-15 wt%Ag alloy. After 48 h of annealing at 350 °C, the fully recrystallized Cu-15 wt%Ag alloy exhibited a higher strength (299 MPa) and a lower electrical resistivity (1.92 μΩ cm) compared to the samples annealed at higher temperatures. The higher strength was attributed to smaller grain size and higher volume fraction of the continuous precipitation zone. Nanoscale Ag precipitates formed by continuous precipitation provided more extensive precipitation strengthening than sub-micrometer Ag precipitates formed by discontinuous precipitation. In addition, reducing the dislocation density and the concentration of solid atoms by controlling the annealing conditions was an effective strategy to weaken the dislocation scattering and impurity scattering effect in the Cu-15 wt%Ag alloy. The electrical resistivity of the fully crystallized Cu-15 wt%Ag alloy was linearly correlated to the concentration of Ag solute atoms. The electrical resistivity of the annealed Cu-15 wt%Ag alloy increased by 0.117 μΩ cm when 1 at.% Ag was dissolved in the Cu solid solution.

A finite element simulation on fully coupled diffusion, stress and chemical reaction

The oxidation of metallic materials at high temperatures is accompanied by atomic/ionic diffusion and stress. There exist stress effects on atomic/ionic diffusion and oxidation rate as well as diffusion-induced stress. In this work, we extend the models of diffusion-induced stress and reaction-induced stress to develop a coupled diffusion, reaction and mechanics model in the framework of linear elasticity by introducing a kinetic relationship between reaction rate and hydrostatic stress. This model demonstrates the effects of hydrostatic stress on chemical equilibrium constant and reaction rate, which are associated with reaction-induced stress. Using the diffusion-reaction-mechanics model, numerical analyses of three different cases are performed: a plate-like structure with fixed boundaries and constant concentration of solute atoms on surfaces, the growth of an oxide layer of a Cr–Fe alloy and the carburization of semi-infinite stainless steel. The numerical results reveal the presence of large compressive stresses and stress gradient in the oxide scale. Comparing the numerical results to experimental data available in literature, we illustrate the need of incorporating the interaction between stress and diffusion and the stress dependence of reaction rate in the growth analysis of the oxide layer and the carburized layer. It is expected that this model is applicable to other systems, such as lithium-ion and sodium-ion batteries.