Gibbs Energy Balance Research Articles

Carbon solute drag effect on the growth of carbon supersaturated bainitic ferrite: Modeling and experimental validations

The carbon partitioning and lengthening rate of bainitic ferrite (αb) are excellent experimental parameters to estimate our level of understanding of the mechanism of bainitic transformation from a continuum perspective and our ability to capture it in analytical expressions. For Fe-C alloys and relatively simple steels the classical Zener-Hillert theory captures the bainitic transformation rather well but mispredicts the level of carbon in solution in the bainite and overestimates the lengthening rates for transformations at lower temperatures. To address this issue, this paper presents a new thermo-kinetic model based on the Zener-Hillert theory and the Gibbs energy balance concept to simulate the lengthening behavior of αb in the Fe-C and low alloyed steels. The model incorporates the effect of the temperature dependent carbon diffusion within the migrating interface via a temperature dependent ferrite/austenite interfacial energy and a temperature dependent diffusion coefficient but does not impose local equilibrium across the interface. The good agreement between the model predictions and nine sets of published experiments indicates that both the carbon supersaturation in αb and the slower lengthening rate are caused by carbon diffusion within the migrating interface. It is found that the degree of carbon supersaturation in αb increases significantly with decreasing temperature. Consequently, the enhanced carbon solute drag effect, resulting from carbon diffusion within the interface, strongly retards the lengthening rates of αb at lower temperatures. Transformation strain is shown to have a modest effect on the lengthening rates but to lower the degree of carbon supersaturation.

Microstructures in a carburized steel after isothermal pearlitic treatment

The influence of carbon concentration variations on pearlite formation (20 h at 600 °C) in a case-carburized steel is investigated. The resultant microstructure shows three distinct regions: carburized case, a transition region, and the original core. The microstructural transition from the case to the core regions is observed to be relatively sharp. The investigated region of the carburized case (0.9 wt.% C) contains two types of pearlite: ferrite + cementite and ferrite + M23C6, where the pearlitic aggregate with M23C6 shows faster formation kinetics. The kinetics of pearlite formation in the transition region (0.3 wt.% C) is very slow and is observed with only M23C6 carbide. Only around 40% austenite decomposes into pearlite in the transition region, which, in comparison to the carburized case region of 0.9 wt.% C is a fraction that is lower by a factor of two. Pearlite is absent in the investigated core region (0.16 wt.% C). The microstructure in this region is predominantly martensite and pro-eutectoid ferrite, with a fraction of ferrite well below the equilibrium fraction. Ferrite formation in this region is limited by the redistribution of mainly Ni, Mn, and Cr, and their resulting solute drag effect on the austenite/ferrite interface. A thermodynamic and kinetic argumentation of these observations is provided with the help of thermodynamic data, precipitation simulations, and a general mixed-mode Gibbs energy balance model.

Prediction of Retained Austenite Fraction in Quenching-and-Partitioning (Q&P) Steels Using the Gibbs Energy Balance Approach

Prediction of Retained Austenite Fraction in Quenching-and-Partitioning (Q&P) Steels Using the Gibbs Energy Balance Approach

Analysis of transformation stasis by the Gibbs energy balance (GEB) approach modified to consider carbon redistribution into remaining austenite

The Gibbs energy balance (GEB) concept for modeling the transformation stasis during isothermal ferrite formation is revisited. Modification has been made to take into account the carbon redistribution into remaining austenite and the increased carbon solubility in ferrite transformed at low temperature. Both the original and modified models adequately describe the transformation stasis in ferrite formation in alloy systems considered in earlier studies; however, the modifications proposed in the present study are critical for the reasonable description of the effect of Mn content on the transformation stasis in a medium carbon steel where the redistribution of carbon has more significant influence.

A Gibbs energy balance model for the isothermal ferrite-to-austenite transformation

A Gibbs energy balance (GEB) model for the ferrite (α)-to-austenite (γ) transformation is developed based on the mixed-mode concept and applied to study the isothermal α→γ transformation at the intercritical temperature of 760 °C in Fe-C-Mn and Fe-C-Mn-Si alloys. Both transformation kinetics and stable γ-volume fraction are found to decrease due to the Gibbs energy dissipation caused by the Mn diffusion inside the interface. The decreased interface mobility leads to a slower transformation rate, but has a negligible effect on the stable γ-volume fraction. The transformation kinetics predicted by the GEB model compares reasonably well with the dilatometric measurement.

Thermodynamic model for metalorganic vapor-phase epitaxy of N-polar group-III nitrides in step-flow growth mode: Hydrogen, competitive adsorption, and configuration entropy

A thermodynamic model for metalorganic vapor-phase epitaxy (MOVPE) of the N-polar $(000\overline{1})$ binary group-III nitrides (AlN, GaN, and InN) in the step-flow growth mode is proposed based on the Burton, Cabrera, and Frank (BCF) theory. The coverages of the group-III adatoms are thermodynamically evaluated under competitive adsorption with hydrogen, which is used as a carrier gas or dissociated from the ${\mathrm{NH}}_{3}$ source gas during MOVPE. The chemical potentials of the group-III and H adatoms on N-polar group-III nitride surfaces are modeled using the respective bond energies with the surface N atoms of the nitride and the vibrational frequencies of the adatoms. The coverages of the coadsorbed group-III and H adatoms are calculated using the Langmuir adsorption isotherm with these chemical potentials. The configuration entropy of the group-III adatoms bridges the gap between the thermodynamic model and the BCF theory. The coverage of the group-III adatoms plays a role like partial pressure of the group-III gas in the thermodynamics. The equilibrium coverage of the group-III adatoms and the equilibrium pressure of the ${\mathrm{NH}}_{3}$ gas are evaluated from the conditions of Gibbs energy balance between the sources (group-III adatom and ${\mathrm{NH}}_{3}$ gas molecule) and products (group-III nitride and $3/2\phantom{\rule{0.28em}{0ex}}{\mathrm{H}}_{2}$ gas molecules) and of speed balance between group-III and N incorporation into step kinks. Fair agreements with the experimentally optimized growth conditions for MOVPE of N-polar GaN and InN are obtained by this method. Among the examined binary group-III nitrides, AlN growth is hardly affected by ${\mathrm{H}}_{2}$ gas pressure, GaN growth is controlled well by ${\mathrm{H}}_{2}$ gas pressure, and InN growth is strongly inhibited by ${\mathrm{H}}_{2}$ gas. A criterion for selecting the ${\mathrm{NH}}_{3}$/group-III flow ratio for maximum products/cost and minimum waste of the materials is demonstrated using the growth model and the estimated growth parameters. The offcut angle dependence of the growth rate on the vicinal substrates is also investigated.

A method for exergetic analysis of a real kraft biomass boiler

Detailed model was proposed in this work for exergy analysis of a real biomass boiler. The model was based on an equilibrium model which uses Gibbs Energy minimization and energy balance to calculate the flue gas composition, the temperature of the furnace and the temperature of the flue gases along the boiler. The exergetic efficiency was calculated by the indirect method, and the exergy destruction in each component of the boiler and each material stream was evaluated. The Gibbs minimization model predicted successfully the complete combustion of the biomass and can be adapted to another thermochemical processes. The exergy analysis results showed that the furnace and the water walls have the higher exergy destruction, accounting for 47% and 30% of the total exergy destruction, respectively. In other parts of the boiler, the exergy destruction was higher in the water and steam flows than in the flue gases, and the results indicate that exergy destruction was higher in streams at lower temperatures. The global exergetic efficiency was 42.47%.

Kinetics and microstructural modeling of isothermal austenite-to-ferrite transformation in Fe-C-Mn-Si steels

During the multi-stage processing of advanced high-strength steels, the austenite-to-ferrite transformation, generally as a precursor of the formation of other non-equilibrium or metastable structures, has a severe effect on the subsequent phase transformations. Herein, a more flexible kinetic and microstructural predictive modeling for the key austenite-to-ferrite transformation of Fe-C-Mn-Si steels was developed, in combination with the classical nucleation theory, the general mixed-mode growth model based on Gibbs energy balance, the microstructural path method and the kinetic framework for grain boundary nucleation. Adopting a bounded, extended matrix space corresponding to a single ferrite grain, both soft-impingement and hard-impingement can be naturally included in the current modeling. Accordingly, this model outputs the ferrite volume fraction, the austenite/ferrite interface area per unit volume, and the average grain size of ferrite, which will serve as the input parameters for modeling the subsequent bainite or martensite transformations. Applying the model, this work successfully predicts the experiment measurement of the isothermal austenite-to-ferrite transformation in Fe-0.17C-0.91Mn-1.03Si (wt%) steel at different temperatures and explains why the final-state average grain size of ferrite has a maximum at the moderate annealing temperature. Effectiveness and advantages of the present model are discussed arising from kinetics and thermodynamics accompanied with nucleation, growth and impingement.

A Gibbs Energy Balance Model for Growth Via Diffusional Growth-Ledges

Growth ledges are commonly observed on interphase boundaries during diffusional phase transformations and are of great importance for understanding inter-sheet spacing of interphase precipitates. A simple model based on Gibbs Energy Balance (GEB) for describing growth kinetics via diffusional growth-ledges of height λ is presented for the case of ferrite growth into austenite. The model is validated against the case of austenite to ferrite transformation involving interphase precipitation in a V, Mn, Si alloyed HSLA steel where, λ is assumed to be equal to the inter-sheet spacing of interphase carbide precipitates. The presented model provides a computationally efficient and versatile method for predicting the ledge height, λ, and the growth kinetics of ferrite from initial nucleation through to final soft impingement considering the evolution of solute drag at growth ledge risers. It is suggested that the intrinsic mobility of growth ledge risers is: M_{m}^{aR} = 0.58(exp\frac{-140x10^{3}{RT})mmol.J^{-1}s^{-1}, with R the gas constant and T the absolute temperature in K.

An upper limit on Gibbs energy dissipation governs cellular metabolism.

The principles governing cellular metabolic operation are poorly understood. Because diverse organisms show similar metabolic flux patterns, we hypothesized that a fundamental thermodynamic constraint might shape cellular metabolism. Here, we develop a constraint-based model for Saccharomyces cerevisiae with a comprehensive description of biochemical thermodynamics including a Gibbs energy balance. Non-linear regression analyses of quantitative metabolome and physiology data reveal the existence of an upper rate limit for cellular Gibbs energy dissipation. By applying this limit in flux balance analyses with growth maximization as the objective function, our model correctly predicts the physiology and intracellular metabolic fluxes for different glucose uptake rates as well as the maximal growth rate. We find that cells arrange their intracellular metabolic fluxes in such a way that, with increasing glucose uptake rates, they can accomplish optimal growth rates but stay below the critical rate limit on Gibbs energy dissipation. Once all possibilities for intracellular flux redistribution are exhausted, cells reach their maximal growth rate. This principle also holds for Escherichia coli and different carbon sources. Our work proposes that metabolic reaction stoichiometry, a limit on the cellular Gibbs energy dissipation rate, and the objective of growth maximization shape metabolism across organisms and conditions.

Analysis of the interaction between moving α/γ interfaces and interphase precipitated carbides during cyclic phase transformations in a Nb-containing Fe-C-Mn alloy

The interaction between moving α/γ interfaces and interphase precipitated (IPd) carbides during the austenite (γ) to ferrite (α) and the ferrite (α) to austenite (γ) transformation has been systematically investigated through cyclic phase transformation experiments for a 0.1C-1.5Mn alloy containing 0.1 wt% Niobium (Nb) and its Nb-free counterpart. Shifts in the critical reaction temperatures during continuous heating and cooling are observed, which are attributed to the pinning force (PF) originating from the IPd carbides present. By applying the Gibbs energy balance (GEB) model to analyze experimental results, the PF was derived to be about 15 J/mol for the α→γ transformation and about 5 J/mol for the γ→α transformation, respectively, both of which are quite small compared to chemical driving force of phase transformations. Moreover, various modified Zener pinning equations have also been used to predict the PF, and it was found that these values are comparable with those obtained from experiments, which suggests that the classical Zener theory still has promising potential for carbide-interface interaction analysis.

Predicting the cooperative effect of Mn–Si and Mn–Mo on the incomplete bainite formation in quaternary Fe–C alloys

ABSTRACTPredicting the effect of alloying elements on the degree of incomplete austenite to bainite transformation in low carbon steels is of great industrial importance. This study introduces an extended Gibbs energy balance model which makes use of an additive approach to calculate the coupled effect of Mn, Si and Mo on the fraction of bainitic ferrite after the incomplete transformation in multicomponent steels. The model predicts significant effects of Mn and Mo and the negligible effect of Si levels on the fraction of bainitic ferrite. This is attributed to the high value of dissipation of Gibbs energy caused by interfacial diffusion of Mn and Mo and low values caused by Si diffusion. The model predictions for quaternary Fe–C–Mn–Si system are comparable with the experimentally measured values of bainite fraction. For the Fe–C–Mn–Mo system, the agreement is less accurate and the accuracy decreases with increasing Mo content, which is attributed a substantial carbide formation but interaction effects between Mn and Mo or a temperature dependent binding energy cannot be ruled out.

Determination of the intrinsic α/γ interface mobility during massive transformations in interstitial free Fe-X alloys

Kinetics of the austenite (γ) to ferrite (α) transformation and the reverse ferrite (α) to austenite (γ) transformation in a series of Fe-X (X = Ni, Mn and Co) binary alloys has been experimentally and theoretically investigated. A transition from partitioning to partitionless transformation has been predicted to occur during both the γ→α and α→γ transformations by a so called Gibbs Energy Balance (GEB) model, in which the chemical driving force is assumed to be equal to the energy dissipation due to interface friction and diffusion of X inside the migrating interfaces. The transition temperature is found to depend on the kind of X and its concentration, which is in good agreement with experimental results. The intrinsic mobility of the α/γ interface has been derived from the kinetic curves of both the γ→α and α→γ transformations in the investigated alloys, and its value seems to be marginally affected by the transformation direction and alloying elements.

Predicting the Transition between Upper and Lower Bainite via a Gibbs Energy Balance Approach

The transition temperature between upper bainite and lower bainite is calculated with an extended Gibbs energy balance model, which is able to quantitatively describe the evolution of carbon supersaturation within bainitic ferrite sheaves during the entire thickening process. The nucleation rate of intra-lath cementite precipitation on a dislocation is calculated based on of the degree of carbon supersaturation. Upper bainite and lower bainite are thus distinguished by the effective nucleation density and therefore a numerical criterion can be set to define the transition. The model is applied to Fe-xC-1Mn/2Mn/1Mo ternary alloys. Results show that the transition temperature increases with bulk carbon content at lower carbon concentration but decreases in the higher carbon region. This prediction agrees very well with the experimental observations in Mn and Mo alloyed systems. Moreover, the highest transition temperature and the carbon content at which it occurs in the Fe-xC-2Mn system are in good agreement with reported experimental data. The inverse “V” shaped character of the carbon concentration–transition temperature curve indicates two opposite physical mechanisms operating at the same time. An analysis is carried out to provide an explanation.

Effect of Mo Addition on the Transformation Stasis Phenomenon During the Isothermal Formation of Bainitic Ferrite

The effect of Mo addition on the transformation stasis (TS) phenomenon during bainitic transformation has been investigated in Fe-0.2C-1.5Mn-1.5Si-(0.2, 1.5, 3)Mo (wt pct) alloys. Both diffusionless and diffusional models failed in predicting the TS that is strongly affected by Mo concentration in experiment. The Gibbs energy balance approach could capture TS in alloys with 0.2 and 1.5 wt pct Mo, but significantly underestimate bainite fraction at stasis for alloy with 3 wt pct Mo, which is deduced to arise from the so-called anti-coupled solute drag effect.

The Effect of Interfacial Element Partitioning on Ferrite and Bainite Formation

The formation of bainitic ferrite and that of grain boundary ferrite in low alloy steels have been two of the most important and interesting research topics in the field of solid state ferrous phase transformation for several decades, and various aspects of these two transformations have been discussed extensively in the literature. Recently, a so-called Gibbs energy balance (GEB) model was proposed by the authors to evaluate alloying element effects on the growth of bainitic ferrite and grain boundary ferrite. The model predicts a growth mode transition from paraequilibrium, negligible partitioning to partitioning during the isothermal formation of bainitic ferrite and grain boundary ferrite. Transformation stasis and bay phenomenon are well explained by the GEB model and both of them are found to be due to alloying element diffusion at the interface. This overview gives a summary of the authors’ recent progress in the understanding of the growth of bainitic ferrite and grain boundary ferrite, with particular focus on the growth mode transition, the transformation stasis phenomenon and the bay phenomenon.

A 2D analysis of the competition between the equiaxed ferritic and the bainitic morphology based on a Gibbs Energy Balance approach

Upon austenite decomposition, either of two ferritic morphologies may result depending on the steel composition and the transformation temperature, i.e. equiaxed (i.e. quasi spherical) or plate-like ferrite. A 2-D model based on the Gibbs Energy Balance (GEB) model and the Zener–Hillert equation is presented to explain the conditions leading to either morphology. To this aim the kinetics of isotropic growth and plate-lengthening, plate-thickening are modeled with appropriate assumptions regarding the interface conditions respectively. The competition between the two morphologies is determined by the fastest minimization of the total Gibbs energy of the system during transformation. The model is first applied to a ternary alloy containing an austenite forming element (Fe-0.23C-1.86Mn) and the predicted morphology competition is in good agreement with experimental observations. In the refined model, the effect of nucleation on the growth of a plate is rationalized. For the plate-like bainitic transformation, a new concept of ‘single plate-thickening stasis’ is proposed, in addition to the commonly discussed ‘global stasis of the bainitic transformation’, i.e. the incomplete bainitic transformation. A general model of the global transformation kinetics bridging the single plate stasis and the global stasis is depicted. Finally, without changing any of the model assumptions or transition conditions the model is applied to a ternary alloy containing a ferrite forming element (Fe-0.375C-1.48Si) The predicted morphology competition is fully in line with the experimental observations for samples subjected to isothermal decomposition at different temperatures.

On the transition between grain boundary ferrite and bainitic ferrite in Fe–C–Mo and Fe–C–Mn alloys: The bay formation explained

The transition between grain boundary ferrite (GBF) and bainitic ferrite (BF) formation in Fe–C–Mo and Fe–C–Mn alloys has been investigated from a diffusional point of view by a so called Gibbs energy balance approach based on solute drag theory, in which the chemical driving force is assumed to be equal to energy dissipation due to alloying elements diffusion inside the interface. The GEB approach predicted that with decreasing temperature the transformation is dominated by GBF and BF formation, respectively, which provided a rationale for the transition between GBF and BF formation. The presence of transformation bay phenomenon in Fe–C–Mo alloys was well captured by the GEB approach, and the bay phenomenon was found to be due to strong solute drag effect of Mo. No bay phenomenon was predicted in Fe–C–Mn alloys. The effect of Mo and Mn on bainite start temperature has also been well predicted by the GEB approach.

A general mixed-mode model for the austenite-to-ferrite transformation kinetics in Fe–C–M alloys

Using the Gibbs energy balance (GEB) concept, a concise GEB model assuming a planar interface is proposed to study the austenite-to-ferrite transformation kinetics in Fe–C–M alloys (where M is a substitutional alloying element). Compared with the binary mixed-mode model, the newly developed GEB model is a more general mixed-mode model, and can physically describe the mixed-mode kinetics during the austenite-to-ferrite transformation in binary, ternary and even higher alloy systems. The GEB model can also predict kinetic transitions during the austenite-to-ferrite transformation, and the classical paraequilibrium and Local equilibrium are found to be its two specific cases. Furthermore, the GEB model provides a physical understanding of the effect of solute drag on the effective interface mobility. Experiments in Fe–C–Mn and Fe–C–Ni alloys indicate that the transformation stasis phenomenon can also occur during the austenite-to-ferrite transformation. The degree of incomplete transformation increases with increasing Mn or Ni concentration, while Mn has a stronger effect on the degree of incomplete transformation than Ni does due to its higher partitioning coefficient. The GEB model describes well the austenite-to-ferrite transformation stasis phenomenon.

Predicting the Effect of Mo, Ni, and Si on the Bainitic Stasis

The “transformation stasis” phenomenon during the isothermal bainitic ferrite formation in a series of Fe-C-X (X = Mo, Ni, Si ) alloys has been analyzed. Both the Gibbs energy balance (GEB) approach and the \(T_0\) model have been applied to model the “transformation stasis” phenomenon, and their predictions are compared with experimental results. The \(T_0\) model failed in predicting the transformation stasis (TS) phenomenon for the alloys investigated here, while the GEB predictions are in very good agreement with experimental data. It is found that Mo has a very strong effect on the TS phenomenon, while the effect of Si is found to be negligible. Ni has an intermediate effect.