Theory Of Diffusion-controlled Growth Research Articles

Precipitation kinetics of gamma double prime phase during direct aging treatment of Inconel 718 superalloy additively manufactured by selective laser melting

While direct aging of Inconel 718 nickel-based superalloy from the as-built state is known as a promising practice to tune the mechanical properties, the precipitation kinetics of γ″ phase, as the main precipitating phase, has received less attention. Accordingly, in the present work, post-processing in the form of the direct aging treatment of Inconel 718 superalloy manufactured by the laser powder bed fusion additive manufacturing, also known as selective laser melting (SLM), was systematically investigated by microstructural analysis and hardness measurements. It was revealed that the peak aged condition for direct aging can be achieved at much shorter holding times compared to aging after homogenization/solutionizing heat treatment. Moreover, the Johnson–Mehl–Avrami–Kolmogorov (JMAK) analysis based on both hardness measurements and microstructural analysis were applied to unravel the γ″ precipitation kinetics, which respectively led to the precipitation activation energies of 160 and 157 kJ/mol, which are consistent with the activation energy for the diffusion of Nb in IN718 superalloy manufactured by SLM. Moreover, the Lifshitz–Slyozov–Wagner (LSW) diffusion-controlled growth theory was modified to become applicable for investigating the precipitation kinetics during the evolution of the volume fraction of the precipitating phase, which also led to the activation energy of 157 kJ/mol. These results can also shed light on the procedures used for investigating the kinetics of phase transformations during aging of additively manufactured Inconel 718 superalloy.

Simulation of bcc-Cu precipitation in ternary Fe-Cu-M alloys

The precipitation of bcc-Cu in the ferrite matrix of Fe-Cu-M alloys, where M is Al, Mn and Ni, is simulated using the Kampmann-Wagner-Numerical (KWN) model with the classical nucleation theory (CNT) and the theory of diffusion-controlled growth by Zener and Kirkaldy. The instantaneous local equilibrium interfacial concentrations of Cu and M were calculated at every time interval, which revealed that the influence of M diffusion is insignificantly small no matter it is faster or slower than the diffusion of Cu. Whilst the capillarity-corrected Cu concentration at the interface in the matrix essentially dictated the growth and coarsening, the concentration of M in the particle affected by capillarity also had an appreciable influence on the number of Cu particles. Al decreased the Cu solubility in the bcc-Fe matrix, increased the number of Cu particles, and decreased the matrix supersaturation fastest. Mn decreased the number and increased the mean radius of bcc-Cu particles due to the increase in Cu solubility to a large extent. The effects of Ni on particle number were less pronounced than one would expect from experimental observations in multicomponent steel. The small changes in particle/matrix interfacial energy and strain energy of nucleus can alter significantly the evolution of particle number and size during aging.

Effects of heating and cooling rates on δ↔γ phase transformations in duplex stainless steel by in situ observation

ABSTRACTUltra high temperature confocal laser scanning microscope is applied to observe the characteristics of the γ↔δ phase transformations at different heating and cooling rates on the surface of duplex stainless steels. It is found that the migration of the δ/γ IB is always the main form of the phase transformation and leads to the continuous decline and final disappearance of the retained γ-phase during the γ→δ phase transformation at different heating rates. Heating rate does not radically later the mode of transformation under the current composition system. Interestingly, the δ-phase inside the γ-phase acts as the nucleation cores and grows more rapidly at the slow heating rate than at the rapid heating rate. During the δ→γ phase transformation, the γ-cells prefer to precipitate along the δ/δ grain boundaries with a flaky pattern, and their fronts are jagged in shape at the slow cooling rate, but in needle-like feature at the rapid cooling rate. The γ-cells also nucleate inside the δ-ferrite grains with a flaky pattern at the slow cooling rate and with a needle-like pattern at the rapid cooling rate. Furthermore, at different cooling rates, the growth speed in the longitudinal direction is always much faster than that in the lateral direction. More importantly, the effects of cooling rates on the precipitating morphologies of γ-phase are clarified based on the experimental results and the diffusion controlled growth theory.

Crystallization kinetics of an Au-based metallic glass upon ultrafast heating and cooling

The crystallization behavior of Au50Cu25.5Ag7.5Si17 metallic glass during continuous heating/cooling and isothermal treatment is investigated by ultrafast calorimetry. The relationship between the transformation and the time/temperature scales of the alloy is established experimentally in a wide temperature range from glass transition temperature to the melting point. Temperature-dependent viscosity in the expanded supercooled liquid region is derived from the thermodynamic and kinetic analysis of the crystallization behavior of the metallic glass. The nucleation rate and growth velocity for crystallization of the supercooled liquid are analyzed by applying classical nucleation theory and diffusion-controlled growth theory.

Effect of cold rolling and first precipitates on the coarsening behavior of γ″-phases in Inconel 718 alloy

The coarsening behaviors of γ″-phase particles in Inconel 718 alloy aged at 750, 800, and 850°C were investigated by scanning electron microscopy (SEM). Detailed observations and quantitative measurements were conducted to characterize the coarsening behavior of the γ?-phase under various aging conditions. The experimental results indicate that the existence of the δ-phase retards the formation and coarsening of the γ″-phase, without influencing its final particle size or amount. Moreover, when cold rolled with a reduction of 50%, the dimensions of the γ″ particles in Inconel 718 alloy decrease with increasing aging time. Furthermore, the coarsening behavior of the γ″-phase in the Inconel 718 alloy after a normal aging treatment (sample A) and that of the primary δ-phase (sample B) follow the Lifshitz–Slyozov–Wagner (LSW) diffusion-controlled growth theory; the thus-obtained activation energies for the γ″-phase are 292 kJ·mol-1 and 302 kJ·mol-1, respectively.

Some difficulties in the theory of diffusion-controlled growth in substitutionally alloyed steels

The theory for the diffusion-controlled growth of ferrite in steels that also contain substitutional solutes is fraught with difficulties when it comes to transformation at large supersaturations, where the bulk compositions of the ferrite and austenite do not differ much, but where local-equilibrium is nevertheless maintained at the transformation front. This requires the existence of a narrow variation in substitutional solute content in the austenite at the interface (so-called ‘concentration spike’) - so narrow that it has no physical meaning. Drawing on the theory for spinodal reactions, it is demonstrated here that there is a substantial penalty associated with the creation of such sharp changes in composition. Therefore, the spikes would never occur in practice. The actual distribution of solute would be over distances orders of magnitude larger than currently calculated, leading to slower growth rates than are predicted currently. The consequences of this conclusion place doubt both on the transition from local to paraequilibrium, and whether the latter state exists at all for reconstructive transformations.

Theoretical analysis of protein effects on calcium phosphate precipitation in simulated body fluid

The effects of protein on the nucleation and growth rate of calcium phosphate precipitation, including hydroxyapatite (HA) and octacalcium phosphate (OCP) precipitation, in simulated body fluid were analyzed based on the classical nucleation and diffusion-controlled growth theory. To obtain quantitative results for analysis, a model system of bovine serum albumin and lysozyme added to revised simulated body fluid was studied. The parameters, such as the concentration of protein binding to calcium and phosphorous ions and the viscosity of the revised simulated body fluid with protein additions, were experimentally measured to assist quantitative analysis. The analysis focused on the effects of protein on calcium phosphate nucleation including changes in the ion concentration, interfacial energies and pH values of the simulated body fluid system. The protein effects on calcium phosphate growth were also analyzed based on a diffusion-controlled growth model. The analysis results indicate that proteins present in simulated body fluid act to increase the free energy of nucleation and reduce the nucleation rates by reducing the effective level of supersaturation in simulated body fluid. The proteins also affect calcium phosphate nucleation by changing the interfacial energies in simulated body fluid. However, the analysis indicates that the effects of protein on the interfacial energies could be to either promote or retard the nucleation. The protein effects on the nucleation induced by altering the pH value of the simulated body fluid can be neglected. The analysis also indicates that protein addition in revised simulated body fluid does not change the basic thermodynamic nature of HA and OCP formation; that is, HA precipitation is more thermodynamically favourable than OCP while OCP precipitation is more kinetically favourable than HA. Proteins presenting in simulated body fluid always exhibit an inhibitory effect on the growth rate of calcium phosphate precipitates because proteins increase the diffusion resistance of calcium and phosphate ions by increasing the revised simulated body fluid viscosity.

Non-isothermal precipitate growth and the principle of additivity

Abstract A theory of diffusion-controlled growth and dissolution of precipitates in binary alloys, taking account of temperature variations, is proposed. Analytical solutions are derived for planar and spherical interfaces by a method which assumes chemical equilibrium to prevail at the moving interface. The diffusion equations are solved by using the stationary-interface approximation, which restricts the concentration field to have no memory of the past motion of the interface. Predictions of growth and dissolution rates for plate-shaped and spherical particles are formulated and compared using data from the Al-Si equilibrium phase diagram. Solutions by analytical methods are compared with results obtained more easily by using the additivity rule. Conditions for additivity are discussed on the basis of numerical examples. Since the exact moving-boundary solution is unknown, the validity of the stationary-interface approximation could not be discussed. However, isothermal results imply that this approximation is reasonable when precipitates with relatively high concentrations are growing or shrinking in dilute solutions.

Non-isothermal precipitate growth and the principle of additivity

Abstract A theory of diffusion-controlled growth and dissolution of precipitates in binary alloys, taking account of temperature variations, is proposed. Analytical solutions are derived for planar and spherical interfaces by a method which assumes chemical equilibrium to prevail at the moving interface. The diffusion equations are solved by using the stationary-interface approximation, which restricts the concentration field to have no memory of the past motion of the interface. Predictions of growth and dissolution rates for plate-shaped and spherical particles are formulated and compared using data from the Al-Si equilibrium phase diagram. Solutions by analytical methods are compared with results obtained more easily by using the additivity rule. Conditions for additivity are discussed on the basis of numerical examples. Since the exact moving-boundary solution is unknown, the validity of the stationary-interface approximation could not be discussed. However, isothermal results imply that this approxim...

Coarsening behavior of γ″ precipitates in modified inconel 718 superalloy

The growth of precipitate particles in precipitation-hardened alloys is very significant, because it can control the long-time mechanical properties of these alloys. The kinetics of this coarsening process have been accounted for in the theory of diffusion-controlled growth, based on the growth theory of Lifshitz and Slyozov and Wagner. The growth of {gamma}{prime} precipitate particles, which have an ordered fcc structure and precipitate coherently in many Ni-containing fcc alloys, follows the d {proportional_to} t{sup 1/3} law. Another commonly observed strengthening phase in many nickel alloys is {gamma}{double_prime}, which has a bct structure and precipitates coherently with a disk shape. A well-known example of this class of alloy, is the widely used commercial alloy, Inconel 718. Here, the measured coarsening kinetics of separate precipitation of {gamma}{double_prime} is found to be linearly related to the cube root of the aging time at 700 and 730 C. However, the coarsening behavior of {gamma}{double_prime} in a compact morphology of {gamma}{prime} + {gamma}{double_prime} in the modified alloy is different from that of Inconel 718. Initially the coarsening rate is very slow, but eventually it coarsens rapidly. The actual {gamma}{double_prime} particle size distribution of the modified alloy is skewed towards large particle size and differs significantlymore » from the theoretically predicted distribution. TEM observations also confirm the preferential dissolution of {gamma}{double_prime} during coarsening, which could be attributed to the presence of an easy-diffusion path or lies to in a short-circuit diffusion.« less

Light Scattering from Diffusion-Controlled Phase Separations in Glass

Light-scattering measurements were made on a number of phase-separated glasses in which the number and size of the particles of the new phase were varied. Plots of scattering intensities versus scattering angles showed two unique features: (1) higher scattering intensities were found in the back angles and (2) in some cases sharp maxima in intensities were found. In the first case the experimental data were in excellent quantitative agreement with Goldstein's theory for scattering from systems in which the growth of the new phase is diffusion controlled. In the second case the data were explained using Rayleigh-Gans theory of light scattering for spheres and Frank's theory for diffusion-controlled growth. The results confirmed the existence as well as the extent of the composition gradient around the growing particles as predicted by diffusion-controlled growth theory.