Parabolic Growth Research Articles

High-Temperature Oxidation of Steel under Linear Flow Rates of Air and Water Vapor—An Experimental Determined Set of Data

High-temperature oxidation phenomena play a crucial role in steel production and processing. The development of new alloying concepts, the modification of production routes due to the decarbonization of steelmaking, and increasing quality demands will additionally stimulate research activities on high-temperature oxidation. Within the scope of this publication, the oxidation behavior of an unalloyed steel was investigated by means of thermogravimetric (TG) and metallographic analysis. The TG measurements were performed with a simultaneous thermal analyzer in combination with a water vapor generator and a mass spectrometer for quality control. Experiments were conducted in a temperature range from 900 to 1200 °C under various oxidation atmospheres, containing synthetic air and water vapor, with different linear flow rates of the oxidizing medium. According to the thermogravimetric data, an evaluation method is introduced to obtain direct kinetic data for linear and parabolic scale growth. The influence of the experimental parameters on high-temperature oxidation is discussed in detail.

The effect of annealing on the microstructures and oxidation behaviors of AlCoCrFeNi complex concentrated alloys

This paper reports the effects of annealing at 1050 °C in an inert atmosphere on the microstructures and high temperature oxidation behaviors of Alx(CoCrFeNi)1−x CCAs, where x = 8, 12, 15, 20, and 30 (at%) complex concentrated alloys. In all cases, annealing stabilized multi-phase microstructures consisting of a combination of BCC, B2, and/or FCC phases with the final phase constitution being dependent on the Al content in the alloy. All the alloys exhibited brief periods of transient oxidation, followed by various degrees of parabolic oxide growth. The as-cast alloys, which were observed to predominantly be Cr2O3 formers, oxidation resistance increased with increasing Al content. After annealing, all of the alloys transitioned into Al2O3 formers but with increased mass gain up to 20 at% Al. Comparison of the experimental results with thermodynamic models of alloy phase equilibria and oxidation products suggests that an increase in the volume fractions of Al-rich phases near the alloy surfaces as a result of annealing, increases oxidation resistance by providing Al reservoirs for the formation of Al2O3. The oxidation mechanisms are discussed relative to existing M-Cr-Al oxide formation models.

Simplified Paralinear Oxidation Analyses

Paralinear oxidative behavior, i.e., concurrent parabolic scale growth (kp) and linear scale volatility (kl or kv), was analyzed by an alternative to the Tedmon equation. A convenient COSP for Windows cyclic oxidation program analyzed published data for Cr, NiCr, Pb, SiC, Si3N4, and BN. All of these exhibit scale volatility due to CrO3, CrO2(OH)2, PbCl2, Si(OH)4, or HBO2. The ‘cyclic’ model used an iterative constant outer layer loss formalism, whereby a normalized spall constant, Q0 /Δt, defined the scale volatility rate, kl (or kv). Optimized trial inputs (fitting maximum mass gain (ΔWmax at tmax) and time to reach ΔW = 0) generated accurate kp and kl values. COSP models replicated ideal paralinear form. Simple approximations for kp and kl are also found as ~ 4.1 (ΔWmax2/ tmax) and ~ 1.2 (ΔWmax/tmax), respectively, for most scales. Alternatively, ΔWmax and tmax can be predicted by kp and kl. High or low volatility extremes (e.g., burner tests, short times, or Al2O3 scales) may mask classic paralinear weight change features.

Investigation on the temperature-dependent diffusion growth of intermetallic compounds in the Mg-Al-Zn system: Experiment and modeling

With the rapid development of Mg alloys, deeper understanding to the thermodynamic and diffusional kinetic behavior of intermetallic compounds (IMCs) is important for studying the effect of alloying elements to the microstructure evolution. Specially, a systematic quantitative investigation on the diffusional growth of IMCs is of great necessity. However, the works studying the elemental diffusion behaviors of multiple-element IMCs are rare in magnesium alloy systems. The current work takes the ternary Mg-Al-Zn system as research target, and combines the diffusion couple technique, phase stability diagrams, in-situ observation technique and numerical inverse method to investigate the temperature-dependent kinetic coefficients. The parabolic growth constant (PGC) and interdiffusion coefficients for Mg solid-solution phase and γ-Mg17Al12, β-Mg2Al3, ε-Mg23Al30, MgZn2, Mg2Zn3, τ-Mg32(Zn, Al)49 and ϕ-Mg5Zn2Al2 IMCs in the Mg-Al-Zn alloy system are determined. By comparing the current experimental with calculation results, the rate-controlling factor of the temperature-dependent diffusion growth of ϕ, τ and ε ternary IMCs in the Mg-Al-Zn system is further discussed in detail.

The impact of co-deposited silicon impurity on oxidation of silicon carbide coatings in the air and in steam at high temperatures

We report a study on the oxidation behaviour of stoichiometric and hyper-stoichiometric silicon carbide (SiC) with silicon impurity. In air, silica growth was parabolic at 1200 °C and 1600 °C on both SiCs but was faster on hyper-stoichiometric SiC. However, the oxidation kinetics was reversed in the steam due to the generation of a larger quantity of gaseous products and pore networks in silica on stoichiometric SiC, which resulted in easy ingress of steam and a reaction-controlled linear oxidation behaviour. In contrast, the pores in silica on hyper-stoichiometric SiC were mostly isolated, resulting in diffusion-controlled parabolic growth of the silica layer.

Comparison of unirradiated and irradiated AGR-2 TRISO fuel particle oxidation response

The silicon carbide (SiC) coating in a tristructural isotropic (TRISO) particle acts as a barrier to fission product release during reactor operation and accident scenarios. Oxidation and subsequent failure of the SiC layer during a rare air ingress event is a proposed mechanism for fission product release in a high-temperature gas-cooled reactor (HTGR). Although previous oxidation studies have analyzed unirradiated TRISO particle response, this study compared the oxidation behavior of irradiated and unirradiated TRISO particles from the second Advanced Gas Reactor Fuel Development and Qualification Program irradiation experiment (AGR-2). Particles with exposed SiC were subjected to six varying oxidizing tests in the Furnace for Irradiated TRISO Testing (FITT), examined for failure fraction with the Irradiated Microsphere Gamma Analyzer (IMGA) and characterized with focused ion beam and scanning/transmission electron microscopy techniques to analyze the oxide layer. Uncorrelated unirradiated particle failures throughout the series of exposures suggests that external factors inherent to the experiment increased particle failure sensitivity. However, irradiated particle observations indicated an increased failure response at 400 h 1400 °C in both 2% and 21% O2 atmospheres above failure associated with external factors. Oxide thickness measurements after 400 h at 1400 °C revealed a greater oxidation rate than predicted by parabolic growth, which was attributed to the increased complexity of the oxide structure at longer exposure times. Altering the atmosphere from 21% to 2% O2 reduced the average oxide thickness by approximately 12%–14% in both irradiated and unirradiated particles at 400 h 1400 °C. Overall, the minor variations observed between irradiated and unirradiated particles in this study led to the conclusion that unirradiated TRISO particles can be used to approximate irradiated TRISO oxidation kinetics.

Effect of ultrafine microstructure on interdiffusion-driven phase transformations in Ni-Sn sandwich diffusion couples

Solid-state diffusion in materials is greatly influenced by microstructural features such as grain boundaries, dislocations, and second phase particles. However, a systematic investigation of structure-kinetics correlation during interdiffusion is largely missing. Herein, a novel sandwich diffusion couple approach was utilized to demonstrate the effects of microstructure on interdiffusion-driven phase formation in the Ni-Sn system. Pure Ni samples were prepared by cold rolling (CR) and spark plasma sintering (SPS) with different microstructures. Two sandwich diffusion couples were prepared– a) NiAM/Sn/NiCR and b) NiCR/Sn/NiSPS, for phase growth analysis at each interface after annealing at 200 °C for 96 h. The intermetallic phase Ni3Sn4 formed at the NiAM/Sn, NiCR/Sn, and NiSPS/Sn interfaces had the thickness of 6, 14, and 41 µm, respectively, consistent with larger parabolic growth constant for the NiSPS/Sn interfaces. The enhanced kinetics at the SPS interface could be attributed to the presence of ultrafine-grained (UFG) (∼320 nm) microstructure dominated by high-angle boundaries.

Experimental investigation on mechanical characteristics of red sandstone under graded cyclic loading and its inspirations for stability of overlying strata

The horizontal stress in the fractured blocks of a fractured zone is in a stress relief status under repeated mining disturbances. To investigate the effect of the horizontal stress in fractured blocks on the stability of the overlying strata, uniaxially graded cyclic loading and unloading experiments on red sandstone with different initial stress level were carried out. First, the evolution of the loading and unloading elastic modulus, irreversible strain and load-unload response ratio were analyzed. Then, the strain energy evolution and damage variable characteristics were examined. Finally, the stability of a voussoir beam structure under repeated mining disturbances was discussed. The results showed that with increasing cycle number, the elastic modulus showed a nonlinear decreasing trend, the load-unload response ratio always fluctuated at approximately 1, and the irreversible strain showed a rapid and steady increasing trend. The proportion of elastic energy tended to increase linearly with increasing cycle number, while the proportion of dissipated strain energy tended to decrease. Moreover, the damage variable, based on the dissipation strain energy definition, showed a parabolic growth trend with increasing cycle number. The irreversible strain and damage variables showed an “increasing–decreasing-increasing” trend as the initial stress level increased. Finally, a stability criterion of the voussoir beam structure based on the effect of repeated mining disturbances on horizontal thrust was proposed. This research has important implications for overlying strata stability.

Superior oxidation resistance of the chemically complex but structurally simple Ti-Al-Ta-Ce-Si-La-B-nitride

Alloying (Ti,Al)N with Ta, CeSi2 or LaB6 is beneficial for both hardness and thermal stability. Here we show that their different mechanisms allow for a cumulative improvement when alloyed together, which is especially pronounced for the oxidation resistance. During isothermal oxidation treatments in ambient air at 900 °C (for up to 25 h), Ti0.44Al0.44Ta0.12N allows for a parabolic scale growth rate constant kp of 6.039 × 10-5 µm2/s. This is reduced to 2.074 × 10-5 and even 0.399 × 10-5 µm2/s when alloying with 2 mol% CeSi2 respectively CeSi2 + LaB6 (1 mol% each). The oxide scale growth kinetics for the latter can be even better described by a logarithmic or cubic law. In the as-deposited state the CeSi2 and CeSi2 + LaB6 alloyed (Ti,Al,Ta)N are single-phase fcc structured providing an indentation hardness of 32.6 ± 1.5 and 37.8 ± 1.5 GPa, combined with an indentation modulus of 496 ± 22 and 496 ± 14 GPa, respectively. After vacuum-annealing at 1000 °C, their hardness is still 33.0 ± 1.6 and 34.8 ± 1.1 GPa, and noticeable formation of hexagonal AlN and TaNx phases can only be detected by X-ray diffraction when annealed at temperatures above 1200 °C.Using CeSi2 and LaB6 instead of their elemental form is furthermore beneficial for the target production itself, as Ce and La are highly reactive.

Low-temperature, short-time, wafer-level bonding for Cu/Sn/Cu solid-state-diffusion interconnect in 3-D integration

In this paper, Cu/Sn/Cu solid-state diffusion (SSD) under low temperature is proposed and investigated for three-dimensional (3-D) integration. Cu and Sn films were deposited by high-efficiency and low-cost physical vapor deposition to fabricate 40-μm-pitch daisy-chain structures. Subsequently, the Cu bump surface was treated with Ar (5% H2) plasma. The Cu/Sn/Cu structure was bonded face to face at 200 °C for 15 min The interfacial composition of the as-bonded dies comprised five layers, Cu/Cu3Sn/Cu6Sn5/Cu3Sn/Cu, with no Sn remaining and no overflow. After annealing at 200 °C for 15 min under N2 atmosphere, as the Cu6Sn5 completely transformed into Cu3Sn, the microstructure changed to stable three layers: Cu/Cu3Sn/Cu. Additionally, the average bonding shear strength reached 27.0 MPa, which is higher than that for conventional Cu/Sn SSD bonding. The measured bonding resistance value was maintained at the theoretical value. Moreover, the parabolic growth constant of Cu3Sn reached 1.86 × 10−15 m2/s. Our study demonstrates the feasibility of using Cu/Sn/Cu SSD for low-temperature, short-time, wafer-level bonding.

Decay estimate and blow-up for a fourth order parabolic equation modeling epitaxial thin film growth

<abstract><p>In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived.</p></abstract>

Analysis of a finite-volume scheme for a single-species biofilm model

An implicit Euler finite-volume scheme for a parabolic reaction-diffusion system modeling biofilm growth is analyzed and implemented. The system consists of a degenerate-singular diffusion equation for the biomass fraction, which is coupled to a diffusion equation for the nutrient concentration, and it is solved in a bounded domain with Dirichlet boundary conditions. By transforming the biomass fraction to an entropy-type variable, it is shown that the numerical scheme preserves the lower and upper bounds of the biomass fraction. The existence and uniqueness of a discrete solution and the convergence of the scheme are proved. Numerical experiments in one and two space dimensions illustrate, respectively, the rate of convergence in space of our scheme and the temporal evolution of the biomass fraction and the nutrient concentration.

Comparison and Analysis of Diffusion Models: Growth Kinetics of Diiron Boride Layers on ASTM A283 Steel

Hard-coated surfacing of a few micrometers is widely applied to increase the efficiency of tools, e.g., for cutting, forming, and casting applications. Therefore, the base thermodiffusion surface treatment is a practical solution to these issues by hardening surface layers with interstitial elements such as carbon, nitrogen, and boron. In particular, within this study, the growth kinetics of an iron boride layer on ASTM 283 steel were investigated with two diffusion models of the powder-pack boriding technique in the temperature range of 1123-1273 K with different treatment periods. The first model, called the steady-state diffusion model, used the modified version of the mass balance equations at the Fe2B/substrate growth interface, the parabolic growth law, and the solution of Fick's second law without time dependence. At the same time, the second diffusion model was based on Goodman's method, also called the integral heat balance method. Afterward, the diffusion coefficient of boron in the Fe2B phase was calculated by fitting the experimental data to the models. Nevertheless, the estimated value for the activation energy of ASTM A238 steel in both diffusion models was coincident (168.2 kJ∙mol-1). A mathematical analysis was implemented by means of a power series (Taylor series) to explain this similarity. The SEM examinations showed a solid tendency to saw-tooth morphology at the growth interface with the formation of the Fe2B layer, whose presence was verified by XRD analysis. The tribological characterizations, including the tests of Rockwell-C indentation, pin-on-disc, and Vickers hardness test method, were used to analyze the antiwear features of the Fe2B layers. Finally, this value of energy was compared to the literature for its experimental validation.

Multilayer interface tracking model of pure tungsten oxidation

We present a numerical model to predict oxide scale growth on tungsten surfaces under exposure to oxygen at high temperatures. The model captures the formation of four thermodynamically-compatible oxide sublayers, WO2, WO2.72, WO2.9, and WO3, on top of the metal substrate. Oxide layer growth is simulated by tracking the oxide/oxide and oxide/metal interfaces using a sharp-interface Stefan model coupled to diffusion kinetics. The model is parameterized using selected experimental measurements and electronic structure calculations of the diffusivities of all the oxide subphases involved. We simulate oxide growth at temperatures of 600∘C and above, extracting the power law growth exponents in each case, which we find to deviate from classical parabolic growth in several cases. We conduct a comparison of the model predictions with an extensive experimental data set, with reasonable agreement at most temperatures. While many gaps in our understanding still exist, this work is a first attempt at embedding the thermodynamic and kinetic complexity of tungsten oxide growth into a comprehensive mesoscale kinetic model that attempts to capture the essential features of tungsten oxidation to fill existing knowledge gaps and guide and enhance future tungsten oxidation models.

A new insight on the diffusion growth mechanism of intermetallic compounds in Al-Er system

The diffusion growth of intermetallic compounds in Al-Er alloys are closely related to the properties of the alloys. The current work aims at explaining the dominance of Al3Er in the Al-Er alloys precipitation phases and the interface thin layer phenomenon by diffusion couple technique, estimating the parabolic growth constant and diffusion activation energy of intermetallic compound in Al-Er diffusion couples to provide theoretical guidance for the design of new Al-Er alloys. In this work, Al-Er diffusion couples were successfully prepared by casting-cladding method in the atmosphere. The growth of Al-Er intermetallic compounds at diffusion couple interface during annealing were observed and recorded by High-Temperature Laser-Scanning Confocal Microscopy at 673, 698, 723 and 748 K respectively. The results show that the growth characteristics of Al-Er intermetallic compounds were accord with layer-terraced growth during annealing. The thickness of intermetallic compound was linear with the square root of time at experimental temperature. The intermetallic compound layer was composed of Al3Er and a very thin AlEr phase. The parabolic growth constants of Al3Er phase at 673, 698, 723 and 748 K were 1.017 × 10−14, 1.609 × 10−14, 3.111 × 10−14 and 4.76 × 10−14 respectively. The activation energy of Al3Er phase was (88.4 ± 5.3) kJ/mol and the pre-exponential factor was 7.126 × 10−8 m2/s.

Non-uniform high-temperature oxidation behavior of twin wire and arc additive manufactured Ni3Al-based alloy

Ni3Al-based alloy fabricated by twin wire and arc additive manufacturing exhibited periodic remelting bands (RB) with unique microstructural characteristics. In this work, the deposited Ni3Al-based alloy was oxidized at 1100 °C for different times to investigate the high-temperature oxidation difference caused by RB. The differences in the microstructure, grain orientation, oxidation behavior, and oxidation mechanism between RB and non-RB were systematically studied. The results showed that these RB, epitaxially grown on the non-RB during each deposition layer, displayed similar Goss textures but a higher proportion of γ + γ′ dual-phase microstructure compared to non-RB. The oxidation of the deposited Ni3Al-based alloy obeyed parabolic growth kinetics at 1100 °C for 10 h and the oxidation scales were consisted of external NiO layer, intermediate NiAl2O4 layer with dispersed Al2O3 and TiO2 particles and internal Al2O3 layer. The outward diffusion of Al and Ti elements during the oxidation process led to the decomposition of γ′ precipitations and the formation of γ′-reduced layer beneath the Al2O3 layer. The cross-section microstructure of oxidation scales indicated that the RB displayed thicker scales and poorer oxidation resistance than non-RB. The oxidation rate of the deposited Ni3Al-based alloy was controlled by the outward transport rate of the total cations (Ni2+, Al3+, and Ti4+) through the Al2O3 layer. Initial oxidation and diffusion interface analysis demonstrated that the γ + γ′ dual-phase had a higher outward transport rate of the cations, which was responsible for the thicker oxide scales on the RB.

Boron-dependent self-healing behavior and mechanical properties of polymer-derived amorphous SiBCN monoliths

The pyrolysis characteristics of precursors, mechanical and anti-oxidation properties of the resulting amorphous SiBCN monoliths with different boron contents were investigated. An increase of boron content results in the decrease of ceramic yields, bulk densities, open porosities and mechanical properties of SiBCN monoliths. The boron-lean SiBCN monolith has the highest hardness and elastic modulus of 13.85 GPa and 142.65 GPa, respectively, due to the presence of more high-energy Si–N and Si–O bonds, and the promoted structural rigidity of the network with more C–B and B–N bonds. The boron-moderate and boron-rich SiBCN monoliths show a better self-healing performance at 1350 °C oxidation in spite of porous oxide scales. The oxide scales exhibit a parabolic growth law with rate constants of 0.22, 6.40 and 10.65 μm 2 /h for boron-lean, boron-moderate and boron-rich SiBCN monoliths, respectively.

Effect of Chromium Content on the Oxidation Behavior of a Ta Stabilized γ'-Strengthened Polycrystalline Co-30Ni-10Al-4W-4Ti-2Ta Alloy.

The high-temperature oxidation behaviors of polycrystalline Co-30Ni-10Al-4W-4Ti-2Ta superalloys with Cr contents ranging from 1 to 5 at.% are characterized at 900 °C to provide benchmark data for the alloy design of the CoNi-based superalloys. The mass gain curves for all three alloys exhibit parabolic growth, and the addition of 5Cr at.% is sufficient to decrease the oxidation rate by two orders of magnitude compared to the Cr-free alloy. Furthermore, cross-sectional findings reveal that these three alloys form qualitatively similar oxide scales composed of an outer oxide layer of Co3O4 and CoAl2O4 phase on top of an Al2O3 scale, following the inner oxide layers of Cr2O3, TiO2, and TiTaO4, and internally oxidized Al2O3 precipitate. The alloy forms a chromium-rich oxide scale as the Cr addition increased, and the concentration of Cr in the scale/alloy interface increases, promoting the growth of Cr2O3, while CoAl2O4 and Co3O4 nucleation is inhibited. The results further indicate that Cr has a superior effect on improving the oxidation resistance of CoNi-based alloys and that a higher content of Cr can assist the formation of a continuous Al2O3, Cr2O3, and TiTaO4 layers, which in turn hampers outer Co and Ni, and inward oxygen flux.

Thermodynamic and kinetic analysis of ternary intermetallics formation in Al–Cu–La alloy

The thermodynamic and kinetic analysis of rare earth (RE) intermetallics formation in Al–Cu–La alloy was carried out in this study. The formation enthalpy, excess entropy, excess Gibbs free-energy and activity of each component of Al–Cu–La ternary alloy were calculated combining Miedema model with Toop model. The calculated results show that the formation enthalpy, excess entropy and excess Gibbs free-energy of Al–Cu–La ternary alloy are all negative in the whole alloy composition range. The activities of Al, Cu and La are all tiny in the central zone of composition triangle of Al–Cu–La ternary alloy, indicating that Al, Cu and La components are easy to form ternary intermetallics. Moreover, the parabolic growth kinetics equation of RE intermetallics was derived integrating the Fick's second law with the mass balance law. According to the parabolic growth rate, the thickness of the RE intermetallics increases in proportion to the square root of the diffusion time.

A novel nitridation- and pesting-resistant Cr-Si-Mo alloy

Cr-13.5Si-32.2Mo (at%) solidifies as metastable σ-phase with a dendritic microstructure during arc melting. Heat treatment leads to a decomposition into a fine-lamellar microstructure of (Cr,Mo)ss and (Cr,Mo)3Si. A protective Cr2O3 layer with low growth rates was formed upon cyclic oxidation at 800, 1100 and 1200 °C for up to 100 h. In addition to the Cr2O3 layer, the formation of an internal silicon oxide layer is found for 1200 °C. Oxide scales show parabolic growth rates at 1100 and 1200 °C. Up to 100 h, the oxidation occurred remarkably free of nitridation, spallation or pesting.