Lifshitz Slyozov Wagner Theory Research Articles

Natural time analysis together with non-extensive statistical mechanics shorten the time window of the impending 2011 Tohoku [formula omitted]9 earthquake in Japan

Applying natural time analysis (NTA) to the Japanese seismic data, we have found that the system enters the critical stage upon the occurrence of M=4.2-5.0 earthquakes from 08:36 to 13:14 local time (LT) on 10 March 2011, i.e., almost one day before the M9 Tohoku earthquake on 11 March 2011. In addition, here, we find that just after this period the entropy change ΔS under time reversal of NTA along with the Tsallis entropy of non-extensive statistical mechanics (NESM) show distinct simultaneous changes. This simultaneous appearance enables the shortening of the time window of the impending mainshock to several hours. Furthermore, upon the occurrence of the M7.3 foreshock at 11:45 LT on 9 March 2011, the following fact emerged: The Tsallis entropy of NESM exhibited a scaling behavior with a characteristic exponent 1/3 that conforms to Lifshitz–Slyozov–Wagner theory for phase transitions as had been also observed in NTA for the fluctuations of the entropy change ΔS under time reversal upon a M7.8 earthquake occurrence in Japan on 22 December 2010. The latter obeyed the form A(t−t0)c where c is approximately equal to 1/3 and the pre-factors A are proportional to the scale i (number of events) while t0 is almost 0.2 days after this M7.8 earthquake.

Unveiling the effect of Hf-(Mo,W) addition on the coarsening, agglomeration, and dissolution behaviour of γ'-Ni3Al precipitates in model Ni-based superalloys

In compositional design of Ni-based superalloys, atomic size of alloying elements appears as the primary factor affecting alloy strengthening. Depending on atomic size, the phase partitioning preferences of alloying elements not only control the coarsening and dissolution characteristics of γ' precipitates, but also govern their morphological development upon aging. The current study, therefore, investigates the influence of larger atomic radius X = Hf element addition and its co-additions with smaller atomic radii refractory X = Mo(W) elements on the microstructural and mechanical properties of model Ni80Al15Hf5 and Ni80Al15Hf2.5Mo(W)2.5 superalloys. Accordingly, it is concluded that the replacement of 2.5 at% X = Hf addition with the same amount of X = Mo(W) additions reduces and increases γ' participation tendency of X = Hf and X = Mo(W) elements, respectively. Agreeing with classical Lifshitz-Slyozov-Wagner (LSW) kinetics, γ'-Ni3Al-Hf-W precipitates coarsen at a slower rate compared to their γ'-Ni3Al-Hf-Mo counterparts, both begin to agglomerate at longer aging times, i.e., 256 h. On the other hand, readily coarsened, aligned, and agglomerated γ'-Ni3Al-Hf precipitates start to dissolve in γ matrix after 16 h, which hence disobey LSW theory. Most probably, higher γ'/γ interfacial energy contribution of faster-diffusivity X = Hf element leads to this three-step temporal evolution of γ'-Ni3Al-Hf precipitates, i.e., (i) coarsening, (ii) alignment/agglomeration, and (iii) dissolution, to occur much earlier and accelerates the strength degradation. Conversely, X = Hf-W co-addition, which causes better mechanical strength than both X = Hf and X = Hf-Mo additions, reduces the degree of strength worsening possibly due to the beneficial contributions of X = W element at γ'/γ interfaces.

Microstructural evolution and stress rupture behaviour of a Ni-Based wrought superalloy during thermal exposure

The microstructural evolution of a wrought Ni-based superalloy GH742 during long-term thermal exposure at 1023 K for durations up to 10,000 h was investigated. The influence of the microstructure degradation on the stress rupture properties at 923 K/823 MPa and 1023 K/510 MPa was systematically analysed. Scanning electron microscopy and electron probe microanalysis were utilised to study the material microstructures before and after the thermal exposure. The results show that small flower-like secondary γ′ precipitates changed their morphology from flower-like to cubic with rounded corners, while large flower-like secondary γ′ precipitates started to split. Tertiary spherical γ′ precipitates in the fine-equiaxed-grain region dissolved into the γ matrix. However, coarsening of tertiary γ′ precipitates occurred in large-fibrous-grain regions. The coarsening kinetics appeared to well conform to the Lifshitz–Slyozov–Wagner theory. Stress rupture tests at 923 K/823 MPa and 1023 K/510 MPa showed a sharp decrease in time to failure within the first 1,000 h of prior thermal exposure, which was then almost constant for the next 9,000 h of thermal exposure. This was attributed to the evolution of γ′ precipitates.

3D Phase Field Modeling of Multi-Dendrites Evolution in Solidification and Validation by Synchrotron X-ray Tomography.

In this paper, the dynamics of multi-dendrite concurrent growth and coarsening of an Al-15 wt.% Cu alloy was studied using a highly computationally efficient 3D phase field model and real-time synchrotron X-ray micro-tomography. High fidelity multi-dendrite simulations were achieved and the results were compared directly with the time-evolved tomography datasets to quantify the relative importance of multi-dendritic growth and coarsening. Coarsening mechanisms under different solidification conditions were further elucidated. The dominant coarsening mechanisms change from small arm melting and interdendritic groove advancement to coalescence when the solid volume fraction approaches ~0.70. Both tomography experiments and phase field simulations indicated that multi-dendrite coarsening obeys the classical Lifshitz–Slyozov–Wagner theory , but with a higher constant of n = 4.3.

Precipitation kinetics and morphology evolution of the Co3(Al, W) phase in a medium supersaturation Co–Al–W alloy

The effects of aging temperature on the precipitation of the γ′-Co3(Al, W) phase were studied by utilizing transmission electron microscopy, scanning electron microscopy and atom probe tomography, with the aim of exploring the continuum kinetics of the γ′ phase in the medium supersaturation Co–9Al–8W (at.%) alloy from early precipitation to later coarsening. The results showed that the γ′ phases with small size and large number density caused a rapid increase in alloy microhardness at 1073 K when the aging time was less than 136 h, implying that precipitation strengthening clearly occurred. Due to the high thermodynamic driving force and small critical radius of nucleation, the γ′ phases expeditiously precipitated within 0.08 h of aging. Additionally, the partitioning coefficients of the elements showed that the partitions of Al and W in the γ′ phase were more obvious at 1073 K. The time exponents of the average particle radius were close to the value of 1/3 predicted by the Lifshitz–Slyozov–Wagner theory. The particle size distribution of the γ′ phase was more consistent with the Brailsford–Wynblatt model in the early growth-dominated stage, whereas it matched well with the trans-interface diffusion-controlled model in the later concomitant growth and coarsening stages. This research revealed the intimate relationship among the morphology, mechanical properties and evolution kinetics of the γ′ phase in a medium supersaturation Co–Al–W alloy.

Reverse Coarsening and the Control of Particle Size Distribution through Surfactant

The minimization of surface area, as a result of the minimization of (positive) surface energy, is a well-known driving force behind the spontaneous broadening of (nano) particle size distribution. We show that surfactant molecules binding to particle surfaces effectively decrease the surface energy and may change its sign. In this case, contrary to the expected broadening behavior, a minimum of free energy is achieved at the maximum surface area for all particles, i.e., when the particles are identical. Numerical simulations based on the classical Lifshitz–Slyozov–Wagner theory with surfactant-induced surface energy renormalization confirm the collapse of the particle size distribution. As the particle size evolution is much slower than particle nucleation and growth, the manipulation of surface energy with in-situ replacement of surfactant molecules provides a method for controlling particle size distribution with great potential for creating mono-disperse nanoparticles, a key goal of nanotechnology.

Anomalous effect of lattice misfit on the coarsening behavior of multicomponent L12 phase

The multicomponent L12 phase in high entropy alloys (HEAs) has shown excellent thermal stability. However, the understanding of its intrinsic origins is limited to the Lifshitz-Slyozov-Wagner (LSW) theory. In the present study, we investigate the influence of lattice misfit, which is not considered in LSW theory, on the coarsening kinetics of the multicomponent L12 phase. Different from that in most traditional alloys, the coarsening constant of multicomponent L12 particles decreases with the increasing lattice misfit in HEAs. This finding provides a new view for understanding the sluggish coarsening kinetics of L12 phase in HEAs.

Effect of Interactions Among Elements on Diffusion Process Associated with γ′ Coarsening in a Ni-Based Single-Crystal Superalloy

Coarsening of cuboidal γ′ precipitates and relevant diffusion process in Ni-based single-crystal superalloy CMSX-4 were investigated at 1000, 1020 and 1040 °C for specific times. The γ′ coarsening kinetics followed a cubic rate law with time and was presumably controlled by bulk diffusion of elements in γ matrix. The associated diffusion activation energy was experimentally determined to be about 300 kJ/mol when it is considered the temperature-dependent thermo-physical parameters in modified Lifshitz–Slyozov–Wagner theory. The influence of temperature on γ/γ′ microstructure is briefly discussed based on pseudo-binary [Ni]–[Al] phase diagram. Interactions among elements can effectively raise the local vacancy formation and vacancy–atom exchange barriers close to γ- and γ′-partitioning elements, respectively. Thus, it can significantly reduce the inter-coupling migrations of atoms during the macroscopic cross-diffusion process associated with γ′ coarsening of Ni-based superalloys.

Anomalous Effect of Lattice Misfit on the Coarsening Behavior of Multicomponent L12 Phase

The multicomponent L12 phase in high entropy alloys (HEAs) has shown excellent thermal stability, but understandings on its intrinsic origins are limited to the Lifshitz-Slyozov-Wagner (LSW) theories. In the present study, we investigated the influence of lattice misfit, which is not considered in LSW theory, on the coarsening kinetics of the multicomponent L12 phase. Different from those in most traditional alloys, the coarsening constants of multicomponent L12 particles decrease with the increasing lattice misfit in HEAs. This finding provides a new view for understanding the sluggish coarsening kinetics of L12 phase in HEAs.

Local approach for coarsening of precipitates

Coarsening of precipitates is very often treated by the classical Lifshitz–Slyozov–Wagner (LSW) mean-field coarsening theory. The theory provides simple evolution equations for volumes of precipitates. A new model based on application of the Thermodynamic Extremal Principle extends the LSW theory by terms accounting for mutual interaction of individual precipitates during coarsening. The model is demonstrated on an example showing that only one interaction term is dominant, which leads to a set of linear equations with a non-diagonal matrix for the rates of precipitate volumes. Of course, the set of equations must be solved numerically.

Microstructural evolution and mechanical properties of thixoformed 7075 aluminum alloy prepared by conventional and new modified SIMA processes

Abstract The microstructural evolution during semi-solid processing and thixoformability of a 7075 alloy prepared by conventional and new modified strain induced melt activation (SIMA) processes were comparatively investigated in this paper. The semi-solid slurries were thixoformed at 600 °C, at which temperature the solid fraction was estimated to be 0.8. The coarsening process of the semi-solid samples was described using Lifshitz–Slyozov–Wagner theory and the effect of pre-deformation on the coarsening kinetics of the solid particles was discussed. The coarsening rate constant of the new modified SIMA sample showed a remarkable decrease compared to that of the conventional SIMA sample. Microstructural and mechanical investigations indicated that the sample with a near-equiaxed microstructure deforms through the plastic deformation of solid grains mechanism. However, the sliding of solid grains and flow of liquid incorporating solid grains mechanisms were dominant in the sample with a globular microstructure. Also, it was observed that the yield and ultimate strengths and hardness of the sample prepared by the new modified SIMA process after thixoforming and T6 heat treatment increased by about 15 %, 10 % and 25 % respectively, compared to those of the conventional SIMA sample.

Effect of Coherent Lattice Mismatch on the Morphology and Kinetics of Ordered Precipitates

The morphological evolution and precipitation kinetics of ordered precipitates under the influence of lattice mismatch were studied by using the phase field model. The variation of the average aspect ratio of the precipitates and their distributions was investigated quantitatively, and the relationship of particles shape and particles fraction with the variation of the aspect ratio was determined. As the lattice mismatch increases, the coarsening rate constants and the particles number density decrease, the particles size distribution (PSD) becomes narrower, and the peak value of the PSD becomes lower than the value obtained by the Lifshitz–Slyozov–Wagner theory, while it is close to the value obtained by the Brailsford–Wynblatt theory. The change rate of the particles number density shows four-stage characteristics and decreases as the lattice mismatch and time increase, indicating that the precipitation and coarsening are retarded for a large lattice mismatch and that a large lattice mismatch results in fewer particles being dispersed in the matrix phase. The morphology and the particles number density show that a suitable lattice mismatch can enhance the mechanical properties of the alloys.

Microstructural Degradation and the Effects on Creep Properties of a Hot Corrosion-Resistant Single-Crystal Ni-Based Superalloy During Long-Term Thermal Exposure

In the present study, the kinetics of microstructural degradation during long-term thermal exposure (LTTE) and the effects on creep deformation mechanisms of a hot corrosion-resistant single-crystal Ni-based superalloy with a low γ′ volume fraction and γ/γ′ lattice misfit were investigated in detail. The kinetic of γ′ coarsening in the experimental alloy conforms well to the Lifshitz–Slyozov–Wagner theory during LTTE at 900 °C up to 10,000 hours. The evolution of γ/γ′ lattice misfit during the LTTE was also investigated by a first attempt. The focused research emphasized on the influences of γ/γ′ lattice misfit evolution after the LTTE on the microstructural degradation, dislocation motion, and different creep mechanisms during high-temperature low-stress creep and high-temperature high-stress creep. The results show that the decreasing of the absolute values of γ/γ′ lattice misfit and change of γ′ size and morphology after the LTTE contribute to the weakening of barrier to the dislocation cutting process into γ′ precipitates during creep and the sharp reduction of stress-rupture lifetime at 950 °C/280 MPa after 1000 hours exposure. As the applied stress decreased to 230 MPa at 950 °C, the creep mechanisms change from the dislocation cutting through γ′ precipitates at high applied stress to the dislocation glide and climb around γ′ precipitates. The dislocation glide and climb by-pass deformation mechanism were not significantly influenced by the change of γ′ precipitates morphology and magnitude of γ/γ′ mismatch within 1000 hours thermal exposure, and the minimum creep rate and creep lifetime after 1000 hours thermal exposure were similar to that of the original heat-treated samples.

Synergistic Effect of Nitrogen and Refractory Material on TiN Formation and Equiaxed Grain Structure of Ferritic Stainless Steel

The effect of nitrogen content on the formation of an equiaxed solidification structure of Fe-16Cr steel was investigated. Moreover, two different kinds of refractory materials, i.e., alumina and magnesia, were employed to control the type of oxide inclusion. The characteristics of TiN(-oxide) inclusions were quantitatively analyzed in both molten steel and solidified samples. When the melting was carried out in the alumina refractory, the grain size continuously decreased with increasing nitrogen content. However, a minimum grain size was observed at a specific nitrogen content (approx. 150 ppm) when the steel was melted in the magnesia refractory. Most of the single TiN particles had a cuboidal shape and fine irregularly shaped particles were located along the grain boundary due to the microsegregation of Ti at the grain boundary during solidification. The type of TiN-oxide hybrid inclusion was strongly affected by the refractory material where Al2O3-TiN and MgAl2O4-TiN hybrid-type inclusions were obtained in the alumina and magnesia refractory experiments, respectively. The formation of oxide inclusions was well predicted by thermochemical computations and it was commonly found that oxide particles were initially formed, followed by the nucleation and growth of TiN. When the nitrogen content increased, the number density of TiN linearly increased in the alumina refractory experiments. However, the number of TiN exhibits a maximum at about [N] = 150 ppm, at which a minimum grain size was obtained in the magnesia refractory experiments. Therefore, the larger the number density of TiN, the smaller the primary grain size after solidification. The number density of TiN in the steel melted in the magnesia refractory was greater than that in the steel melted in the alumina refractory at given Ti and N contents, which was due to the lower planar lattice disregistry of MgAl2O4-TiN interface rather than that of Al2O3-TiN interface. When ∆TTiN (= difference between the TiN precipitation temperature and the liquidus of the steel) was 20 K to 40 K, the number density of effective TiN was maximized and thus, the grain size was minimized after solidification. Finally, although most of the TiN particles were smaller than 1 μm in the molten steel samples irrespective of the nitrogen content, TiN particles larger than 10 μm were observed in the solidified samples when the nitrogen content was greater than 150 ppm. The growth of TiN particles during melting and solidification was well predicted by the combinatorial simulation of the ‘Ostwald ripening model’ based on the Lifshitz–Slyozov–Wagner theory in conjunction with the ‘Diffusion controlled model’ using Ohnaka’s microsegregation equation.

High-Temperature Structural Stabilities of Ni-Based Single-Crystal Superalloys Ni–Co–Cr–Mo–W–Al–Ti–Ta with Varying Co Contents

It has been recently pointed out that the compositions of industrial alloys are originated from cluster-plus-glue-atom structure units in solid solutions. Specifically for Ni-based superalloys, after properly grouping the alloying elements into Al, Ni-like (\( \overline{\text{Ni}} \)), γ′-forming Cr-like (\( \overline{\text{Cr}}^{{\gamma^{\prime } }} \)) and γ-forming Cr-like (\( \overline{\text{Cr}}^{\gamma } \)), the optimal formula for single-crystal superalloys is established [Al–\( \overline{\text{Ni}} \) 12](Al1 \( \overline{\text{Cr}}^{{\gamma^{\prime } }}_{0.5} \overline{\text{Cr}}^{\gamma }_{1.5} \)). The Co substitutions for Ni at the shell sites are conducted on the basis of the first-generation single-crystal superalloy AM3, formulated as [Al–Ni12−x Co x ](Al1Ti0.25Ta0.25Cr1W0.25Mo0.25), with x = 1.5, 1.75, 2 and 2.5 (the corresponding weight percents of Co are 9.43, 11.0, 12.57 and 15.71, respectively). The 900 °C long-term aging follows the Lifshitz–Slyozov–Wagner theory (LSW theory), and the Co content does not have noticeable influence on the coarsening rate of γ′. The microstructure and creep behavior of the four (001) single-crystal alloys are investigated. The creep rupture lifetime is reduced as Co increases. The alloy with the lowest Co (9.43 Co) shows the longest lifetime of about 350 h at 1050 °C/120 MPa, and all the samples show N-type rafting after creep tests.

Probing Novel Microstructural Evolution Mechanisms in Aluminum Alloys Using 4D Nanoscale Characterization.

Dispersions of nanoscale precipitates in metallic alloys have been known to play a key role in strengthening, by increasing their strain hardenability and providing resistance to deformation. Although these phenomena have been extensively investigated in the last century, the traditional approaches employed in the past have not rendered an authoritative microstructural understanding in such materials. The effect of the precipitates' inherent complex morphology and their 3D spatial distribution on evolution and deformation behavior have often been precluded. This study reports, for the first time, implementation of synchrotron-based hard X-ray nanotomography in Al-Cu alloys to measure kinetics of different nanoscale phases in 3D, and reveals insights behind some of the observed novel phase transformation reactions. The experimental results of the present study reconcile with coarsening models from the Lifshitz-Slyozov-Wagner theory to an unprecedented extent, thereby establishing a new paradigm for thermodynamic analysis of precipitate assemblies. Finally, this study sheds light on the possibilities for establishing new theories for dislocation-particle interactions, based on the limitations of using the Orowan equation in estimating precipitation strengthening.

Influence of Zinc on Coarsening of δ-Ni2Si Particles, Aging Behavior and Hardness in a Cu-Ni-Si Alloy

In the present paper, the aging precipitation and coarsening of disk-like δ-Ni2Si particles in Cu and Cu-10Zn alloys aged at 450 °C have been investigated by hardness, electric resistivity measurement and transmission electron microscopy observation. The coarsening dynamics of the average diameter of the δ-Ni2Si particles coincides with the t 1/3 time law for both alloys. The coarsening of the diminution of supersaturation related to aging time t coincides with the t −1/3 time rule. Adding Zn to the Cu-Ni-Si alloy increases the growth and coarsening rate of the particles mainly because of the increased diffusivity D of the δ-Ni2Si particles in the matrix. The value of D of the δ-Ni2Si particles in the Cu-xZn (x = 0, 10 wt.%) matrix and the Cu/δ-Ni2Si interfacial energy γ are independently calculated by using the Lifshitz–Slyozov–Wagner theory which was extended to include disk-like particles by Boyd and Nicholson. The values of D and γ increase from 0.77 × 10−19 to 2.21 × 10−19 m2/s and 0.19 to 0.63 J/m2, respectively, when Zn is added to the Cu-Ni-Si alloy. These calculations and the analysis show that the properties of Cu-Ni-Si-Zn alloy can significantly be enhanced by reducing the aging temperature.

Initial precipitation and coarsening of the γ phase in inverse Ni–Al alloys

Coalescence coarsening and Ostwald ripening as well as the kinetics of γ precipitation in the γ′ matrix of inverse Ni–Al alloys are investigated by phase field simulations. The ordering of the γ′ matrix during the initial phase transformation of inverse Ni–Al alloys is faster than clustering, and disordering of the γ precipitates is faster than dissolution. As the Al content increases from 20.2 to 20.8 at.% Al, the initial precipitation and growth of γ precipitates decelerates, and the volume fractions of γ precipitates decreases with higher Al content. The rate constants of steady-state coarsening decrease with decreasing volume fraction of γ precipitates, the particle size distributions (PSDs) are broader than the PSDs predicted by Lifshitz–Slyozov–Wagner theory at steady-state coarsening, and the PSD becomes broad with increasing volume fraction of γ precipitates. The simulated morphology and kinetics variation of the γ precipitates are similar to previous experimental results.

Ripening kinetics of bubbles: A molecular dynamics study.

The ripening kinetics of bubbles is studied by performing molecular dynamics simulations. From the time evolution of a system, the growth rates of individual bubbles are determined. At low temperatures, the system exhibits a t1/2 law and the growth rate is well described by classical Lifshitz-Slyozov-Wagner (LSW) theory for the reaction-limited case. This is direct evidence that the bubble coarsening at low temperatures is reaction-limited. At high temperatures, although the system exhibits a t1/3 law, which suggests that it is diffusion-limited, the accuracy of the growth rate is insufficient to determine whether the form is consistent with the prediction of LSW theory for the diffusion-limited case. The gas volume fraction dependence of the coarsening behavior is also studied. Although the behavior of the system at low temperatures has little sensitivity to the gas volume fraction up to 10%, at high temperatures it deviates from the prediction of LSW theory for the diffusion-limited case as the gas volume fraction increases. These results show that the mean-field-like treatment is valid for a reaction-limited system even with a finite volume fraction, while it becomes inappropriate for a diffusion-limited system since classical LSW theory for the diffusion-limited case is valid at the dilute limit.

Precipitation kinetics of ordered γ′ phase and microstructure evolution in a Ni[sbnd]Al alloy

The precipitation kinetics of ordered γ′ (L12-Ni3Al) phase from nucleation and growth to coarsening in a NiAl alloy is studied by phase field simulation. The initial nucleation of the γ′ phase is quantitatively described by the variations of composition and order parameter profiles of the initial Al-enriched cluster, and the coarsening kinetics is clarified with the time exponent of the average particle radius of γ′ phase. The precipitation of the γ′ phase from the supersaturated γ solid solution goes through the non-stoichiometric ordered phase, stoichiometric ordered phase, growth and coarsening. The time exponent of the average radius of the γ′ phase with coherent equilibrium is smaller than that indicated by the Lifshitz–Slyozov–Wagner theory at the coarsening stage, and it decreases from 1.35 of the nucleation and growth stage to 0.22 of the coarsening stage. The peak of the particle size distribution decreases and the width of the particle size distribution increases as the aging progresses. As the average radius of the γ′ phase increases, the average aspect ratio of the γ′ phase undergoes a slowly increase to a fast increase from nucleation to growth stage, then goes into a relatively stable state at the coarsening stage.