Coarsening Mode Research Articles

Steady states and coarsening in one-dimensional driven Allen-Cahn system.

We study the steady states and the coarsening dynamics in a one-dimensional driven nonconserved system modeled by the so-called driven Allen-Cahn equation, which is the standard Allen-Cahn equationwith an additional driving force. In particular, we derive equationsof motion for the phase boundaries in a phase ordering system obeying this equationusing a nearest-neighbor interaction approach. Using the equationsof motion we explore kink binary and ternary interactions and analyze how the average domain size scale with respect to time. Further, we employ numerical techniques to perform a bifurcation analysis of the one-period stationary solutions of the equation. We then investigate the linear stability of the two-period solutions and thereby identify and study various coarsening modes.

Phase-field simulation of γʹ coarsening behavior in cobalt-based superalloy

Because the microstructure characteristics of γʹ strengthened cobalt-based superalloys are critical for anticipated next generation superalloys, the morphological evolution and coarsening behavior of γʹ precipitates during isothermal aging in Co-Al-W ternary alloys are investigated using the phase-field method. Accurate γʹ evolution over the entire aging process is simulated through coupling with the antiphase domain and the elastic energy. With a change in γʹ volume fraction from 38% to 66%, the coarsening rate of γʹ precipitates shows an approximately linear increase, primarily due to the increased diffusion potential difference between the γ matrix and γʹ precipitate. The coalescence between adjacent precipitates is promoted in the high γʹ volume fraction, resulting in polyhedral precipitates and particle size distribution (PSD), which agrees with MLSW theory. The morphology transition mechanism of γʹ is showed through the evolutions of interfacial energy and elastic energy, and the effects of microstructure features on the coarsening rate and coarsening mode are discussed. Results of this study provide guidance for the development of Co-based superalloys with optimized microstructures.

Gradient-dynamics model for liquid drops on elastic substrates.

The wetting of soft elastic substrates exhibits many features that have no counterpart on rigid surfaces. Modelling the detailed elastocapillary interactions is challenging, and has so far been limited to single contact lines or single drops. Here we propose a reduced long-wave model that captures the main qualitative features of statics and dynamics of soft wetting, but which can be applied to ensembles of droplets. The model has the form of a gradient dynamics on an underlying free energy that reflects capillarity, wettability and compressional elasticity. With the model we first recover the double transition in the equilibrium contact angles that occurs when increasing substrate softness from ideally rigid towards very soft (i.e., liquid). Second, the spreading of single drops of partially and completely wetting liquids is considered showing that known dependencies of the dynamic contact angle on contact line velocity are well reproduced. Finally, we go beyond the single droplet picture and consider the coarsening for a two-drop system as well as for a large ensemble of drops. It is shown that the dominant coarsening mode changes with substrate softness in a nontrivial way.

Effect of driving on coarsening dynamics in phase-separating systems

We consider the Cahn–Hilliard (CH) equation with a Burgers-type convective term that is used as a model of coarsening dynamics in laterally driven phase-separating systems. In the absence of driving, it is known that solutions to the standard CH equation are characterized by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e. clusters or drops/holes or islands are obtained) followed by the coarsening process, where the average size of the structures grows in time and their number decreases. Moreover, two main coarsening modes have been identified in the literature, namely, coarsening due to volume transfer and due to translation. In the opposite limit of strong driving, the well-known Kuramoto–Sivashinsky equation is recovered, which may produce complicated chaotic spatio-temporal oscillations. The primary aim of the present work is to perform a detailed and systematic investigation of the transitions in the solutions of the convective CH equation for a wide range of parameter values, and, in particular, to understand in detail how the coarsening dynamics is affected by an increase of the strength of the lateral driving force. Considering symmetric two-drop states, we find that one of the coarsening modes is stabilized at relatively weak driving, and the type of the remaining mode may change as driving increases. Furthermore, there exist intervals in the driving strength where coarsening is completely stabilized. In the intervals where the symmetric two-drop states are unstable they can evolve, for example, into one-drop states, two-drop states of broken symmetry or even time-periodic two-drop states that consist of two traveling drops that periodically exchange mass. We present detailed stability diagrams for symmetric two-drop states in various parameter planes and corroborate our findings by selected time simulations.

Validation of an alloy design strategy for stable Fe–Cr–Al–Nb-X ferritic alloys using electron microscopy and atom probe tomography

The Fe–Cr–Al–Nb ferritic alloys strengthened by A2B Laves phases, with superior oxidation and corrosion resistance, are being considered for high temperature operation within fossil-fired steam power plants to increase process efficiency and reduce CO2 emissions. In this study, new sets of alloys based on Fe–30Cr–3Al–1Nb (in weight percent) were designed with (i) a high microstructural stability of Laves phase precipitates in a BCC-Fe matrix and (ii) reduced precipitate free zones along the grain boundaries, targeting an operating temperature of 700 °C. Two alloys with titanium or tungsten additions were down-selected through thermodynamic calculations with a design strategy to maximize the amount of Laves phase at 700 °C and lower the BCC solvus temperature. To validate this design strategy, the candidate alloys were cast, heat treated and aged at 700 °C, and the resulting microstructure was characterized using scanning electron microscopy and atom probe tomography. The alloys with titanium addition (1 wt%) showed monotonic precipitate coarsening. On the other hand, the alloys with tungsten addition (6 wt%) showed a reduced coarsening rate of Laves phase precipitates in the matrix up to 1008 h aging, whereas an acceleration of the precipitate coarsening after that. Atom probe tomography of this alloy revealed the formation of a core-shell precipitate structure with a Nb/Si rich core and W-rich shell, while the alloy with Ti-addition did not show the core-shell structure. The detailed characterization results revealed that the core-shell structure was strongly correlated with the observed two-step coarsening mode in the tungsten containing alloy.

Ag₃Sn Compounds Coarsening Behaviors in Micro-Joints.

As solder joints are being scaled down, intermetallic compounds (IMCs) are playing an increasingly critical role in the reliability of solder joints, and thereby an in-depth understanding of IMCs microstructure evolutions in micro-joints is of great significance. This study focused on coarsening behaviors of Ag3Sn compounds in Sn-3.0Ag-0.5Cu (SAC305) micro-joints of flip chip assemblies using thermal shock (TS) tests. The results showed that the Ag3Sn compounds grew and rapidly coarsened into larger ones as TS cycles increased. Compared with such coarsening behaviors during thermal aging, TS exhibited a significantly accelerating influence. This predominant contribution is quantitatively determined to be induced by strain-enhanced aging. Moreover, based on observations for Ag3Sn microstructure evolutions during TS cycling, one particular finding showed that there are two types of coarsening modes (i.e., Ostwald ripening and Necking coalescence) co-existing in the Ag3Sn coarsening process. The corresponding evolutions mechanism was elucidated in a combination of simulative analysis and experimental validation. Furthermore, a kinetic model of the Ag3Sn coarsening was established incorporating static aging and strain-enhanced aging constant, the growth exponent (n) was calculated to be 1.70, and the predominant coarsening mode was confirmed to be the necking coalescence.

Elucidating the effect of intermetallic compounds on the behavior of Mg–Gd–Al–Zn magnesium alloys at elevated temperatures

Abstract

Coarsening modes of clusters of aggregating particles.

There are two modes by which clusters of aggregating particles can coalesce: The clusters can merge either (i) by the Ostwald ripening process, in which particles diffuse from one cluster to the other while the cluster centers remain stationary, or (ii) by means of a cluster translation mode, in which the clusters move toward each other and join. To understand in detail the interplay between these different modes, we study a model system of hard particles with an additional attraction between them. The particles diffuse along narrow channels with smooth or periodically corrugated walls, so that the system may be treated as one-dimensional. When the attraction between the particles is strong enough, they aggregate to form clusters. The channel potential influences whether clusters can move easily or not through the system and can prevent cluster motion. We use dynamical density functional theory to study the dynamics of the aggregation process, focusing in particular on the coalescence of two equal-sized clusters. As long as the particle hard-core diameter is nonzero, we find that the coalescence process can be halted by a sufficiently strong corrugation potential. The period of the potential determines the size of the final stable clusters. For the case of smooth channel walls, we demonstrate that there is a crossover in the dominance of the two different coarsening modes, which depends on the strength of the attraction between particles, the cluster sizes, and the separation distance between clusters.

Using two-dimensional time resolved light scattering to study the cure reaction induced phase separation process of epoxy-amine-polyethersulfone blend with secondary phase separation

The generalized two-dimensional correlation analysis based on time-resolved light scattering patterns (2D TRLS) has been employed to study the phase separation process of an epoxy-amine-polyethersulfone blend in which the secondary phase separation takes place. The results of the 2D TRLS provided more detailed information that was not readily observed in the 1D TRLS patterns. (i) During the first process of phase separation, the sequential order of coarsening in size of the domains among the larger and smaller ones has been reversed between the diffusion regime and the hydrodynamic regime. (ii) The change of the larger domains in size, due to the hydrodynamic flow in the late stage of the first phase separation process, keeps on taking place earlier than that of the new domains appeared in the secondary phase separation process. (iii) During the secondary phase separation process the size growth of the smaller domains takes place earlier than that of the larger ones, probably due to the assumption that the coarsening mode could decrease the interface tension more quickly.

Microscopic phase-field simulation for coarsening behavior of L12 and DO22 phases of Ni75CrxAl25−x alloy

Based on the microscopic phase-field dynamic model and the microelasticity theory, the coarsening behavior of L12 and DO22 phases in Ni75CrxAl25−x alloy was simulated. The results show that the initial irregular shaped, randomly distributed L12 and DO22 phases are gradually transformed into cuboidal shape with round corner, regularly aligned along directions [100] and [001], and highly preferential selected microstructure is formed during the later stage of precipitation. The elastic field produced by the lattice mismatch between the coherent precipitates and the matrix has a strong influence on the coarsening kinetics, and there is no linear relationship between the cube of the average size of precipitates and the aging time, which does not agree with the results predicted by the classical Lifshitz-Slyozov-Wagner. The coarsening processes of L12 and DO22 phases are retarded in elastically constrained system. In the concurrent system of L12 and DO22 phases, there are two types of coarsening modes: the migration of antiphase domain boundaries and the interphase Ostwald ripening.

Long-wave theory of bounded two-layer films with a free liquid–liquid interface: Short- and long-time evolution

We consider two layers of immiscible liquids confined between an upper and a lower rigid plate. The dynamics of the free liquid–liquid interface is described for arbitrary amplitudes by an evolution equation derived from the basic hydrodynamic equations using long-wave approximation. After giving the evolution equation in a general way, we focus on interface instabilities driven by gravity, thermocapillary and electrostatic fields. First, we study the linear stability discussing especially the conditions for destabilizing the system by heating from above or below. Second, we use a variational formulation of the evolution equation based on an energy functional to predict metastable states and the long-time pattern morphology (holes, drops or maze structures). Finally, fully nonlinear three-dimensional numerical integrations are performed to study the short- and long-time evolution of the evolving patterns. Different coarsening modes are discussed and long-time scaling exponents are extracted.

Phase separation in the presence of spatially periodic forcing

We have investigated theoretically phase separation in the presence of spatially periodic forcing. From the analytic and numeric study of a suitably generalized Cahn-Hilliard equation in one and two dimensions, we find that the forcing amplitude necessary to generate a periodic kink-type state from small random initial conditions depends weakly on wave number. This amplitude is much larger than the one necessary to stabilize the periodic state, i.e., to prevent late-stage coarsening, once it is established. Surprisingly, the destabilizing mode is of long-wave type, which is in contrast to the well-known most rapidly growing coarsening mode in the unforced system. In the Allen-Cahn equation with nonconserved order parameter the relevant modes are always long wave. It appears feasible to observe these effects by imposing a temperature modulation by optical grating which then couples to concentration modulation via the (Ludwig-)Soret effect.

Coarsening mode and microstructure evolution of Al–In hypermonotectic alloy during rapidly cooling process

A numerical model reflecting the real physical processes well is developed to predict the coarsening mode of the second phase droplets and the microstructural evolution under the common action of nucleation, diffusional growth, collision coagulation during rapid cooling Al–In hypermonotectic alloys.

Three-dimensional model of precipitation of ordered intermetallics

The development of the two-phase (f.c.c.+L1 2) coherent microstructure in the prototype Ni–Al superalloy is studied by using the three-dimensional computer simulation technique. The dynamics and morphology of the microstructure evolution are described by our three-dimensional version of the stochastic time-dependent kinetic equation which explicitly includes the coherency strain, elastic anisotropy and L1 2 ordering of the preciptate phase. The input parameters, the crystal lattice misfit, elastic moduli, interfacial energy and equilibrium compositions of the coexisting phases are taken from the published independent measurements. The simulation results demonstrate that the strain accommodation in the microstructure evolution results in the cuboidal-like precipitates faceted by the {100} planes. The size of the precipitates obtained in the simulation is of the order of 50 nm. The important conclusion is that the precipitates are always single-domain particles with no antiphase boundaries. This effect is associated with the ordered structure of precipitates. It causes the slowing down of the coarsening kinetics since it excludes the agglomeration of the out-of-phase precipitates in one particle. As has been shown previously, the latter is a very important coarsening mode in an absence of ordering.

Coarsening of Incoherent α-Fe Particles in a Cu–Fe Alloy

The coarsening of α-Fe particles in a Cu-Fe alloy was examined. Using the Lifshitz-Slyozov-Wagner (LSW) theory of Ostwald ripening, the interfacial energy between α-Fe and Cu estimated to be 3.3 j/m 2 . A coarsening also studied when α- and γ-Fe particles coexisted. This coarsening mode was different from that of the usual Ostwald ripening. When the coarsening processes of α- and γ-Fe particles werefollowed separately, however, the growth and shrinkage of the Fe particle can be understood from the LSW theory

Some effects of partial recrystallisation on 14C dating Late Pleistocene corals and molluscs

The issue of sea level during the last interstadial revolves largely around the problem of achieving reliable 14C dates for shell carbonate from Late Pleistocene shallow marine and littoral deposits. A set of 27 samples were collected from Late Pleistocene reefs in New Guinea, and measurements made of 13C, 14C, plus the degree of recrystallisation (determined by X-ray diffraction). The original fibrous aragonite structure of the samples (corals and clams) is seen in thin section to recrystallise in two quite different modes. The carbon isotope results strongly suggest that one mode, the sparry calcite recrystallisation, represents an open geochemical system which allows contamination by more recent 14C, while the subtle coarsening mode of recrystallisation represents a closed system, often yielding reliable results. The reliability of the latter can be validated if a similarly recrystallised sample, known to be outside the range of 14C dating, shows a background 14C count.

Shape instabilities of eutectic composites at elevated temperatures

A calculation of the growth of shape instabilities is presented for a rod composite microstructure. It is assumed that migration of the interface is driven by capillarity and controlled by diffusive transport in the matrix. If the interface is faceted or if the rod fraction exceeds 20 per cent, a two dimensional coarsening mode occurs more rapidly than spheroidization of the rods. The time for a significant number of rods to dissolve by two dimensional growth is found to depend on the initial distribution of rod sizes. If faults are present coarsening may also occur by fault migration. In the case of faults coarsening occurs most rapidly at the beginning and then slows down because of the increased rod size and because of fault annihilation. The effects of interface kinetics, surface energy anisotropy and stress on the coarsening of composite structures are discussed.