Onset Of Morphological Instability Research Articles

A Two-Dimensional Phase-Field Investigation on Unidirectionally Solidified Tip-Splitting Microstructures

The onset of morphological instabilities along a solidifying interface has a tendency to influence the microstructural characteristics of cast alloys. In the present study, the initiation as well as the mechanism of microstructural pattern formation is investigated by a quantitative phase-field approach. For energetically isotropic interfaces, we show that the presence of grain boundary grooves promotes the initiation of morphological instabilities, and with progressive solidification, they subsequently amplify into tip-splitting microstructures. We also demonstrate that the grain boundary groove shape influences the amplification of the ridge-shaped instability near the pit region. The structural transition of tip splitting to dendritic microstructures is showcased through the effect of interfacial anisotropy. In addition, the prediction of the tip-splitting position is discussed through an analytical criterion, wherein the sign of the surface Laplacian of interfacial curvature dictates the formation of crest and trough positions in a tip-splitting pattern. In complete agreement with the sharp-interface theory, our phase-field simulations validate the analytically obtained tip-splitting position and suggest that the two tips evolve symmetrically on either side of the hindered concave region. Furthermore, the role of lattice anisotropy on the tip-splitting phenomenon is also discussed in detail.

4D synchrotron X-ray tomographic quantification of the transition from cellular to dendrite growth during directional solidification

Solidification morphology directly impacts the mechanical properties of materials; hence many models of the morphological evolution of dendritic structures have been formulated. However, there is a paucity of validation data for directional solidification models, especially the direct observations of metallic alloys, both for cellular and dendritic structures. In this study, we performed 4D synchrotron X-ray tomographic imaging (three spatial directions plus time), to study the transition from cellular to a columnar dendritic morphology and the subsequent growth of columnar dendrite in a temperature gradient stage. The cellular morphology was found to be highly complex, with frequent lateral bridging. Protrusions growing out of the cellular front with the onset of morphological instabilities were captured, together with the subsequent development of these protrusions into established dendrites. Other mechanisms affecting the solidification microstructure, including dendrite fragmentation/pinch-off were also captured and the quantitative results were compared to proposed mechanisms. The results demonstrate that 4D imaging can provide new data to both inform and validate solidification models.

Onset of morphological instability in two binary liquid layers

We consider the linear stability of a horizontal liquid bilayer subject to vertical heating. The two layers consist of a binary liquid that has undergone a phase transition, resulting in a horizontal interphase boundary between two phases with different compositions. We perform linear stability calculations to determine the critical values for the applied temperature difference across the system that is necessary to produce instability using both numerical computations and small-wavenumber approximations. We focus on an instability primarily due to the phase change, which can occur in the absence of buoyancy and surface-tension-driven convection. We find both direct and oscillatory modes of instability, either of which can persist to small wavenumbers that allow approximate analytical descriptions. The interaction of flow with a deforming phase boundary plays a critical role in the instability, and the results are compared to morphological stability results that can be obtained in the absence of flow.

Coupled morphological and thin film instabilities generated by inclusions in crystal growth

The interaction of a foreign particle that is suspended in the melt with a planar solidifying interface may induce the onset of morphological instabilities provided that its distance from the interface falls below a critical value. This distance, which is of the order of the particle’s radius, depends on the governing processing and physical parameters. When the particle is in nearcontact with the solid-liquid interface, the disjoining pressure in the melt film that separates the particle from the interface influences the interaction. We derive an expression for the film thickness at which rupture occurs. The critical film thickness, which depicts the competition between the stabilizing influence of surface tension and thermal gradients and the destabilizing influence of the intermolecular forces, varies as (Sh)1/4, where Sh is the Scheludko number that is modifed by the imposed thermal gradients. We note the existence of a critical value for the particle’s radius below which the stabilizing effects are primarily due to surface tension and above which they are due to the thermal gradients.

Phase field modeling of directional solidification of a binary alloy at high velocities

Some computational results on directional solidification of a binary alloy obtained by using the phase field model are reported. Frozen temperature approximatio n is adopted to simplify the modeling. The onset of morphological instability, t he pattern evolution and the solute redistribution are shown. The cell to planar interface transition is modeled near the high velocity absolute stability condi tion. The cell spacings, tip temperatures and solute partitioning coefficients a t different velocities are calculated. The solute trapping is also observed.

The growth and the fluid dynamics of protein crystals and soft organic tissues: models and simulations, similarities and differences

The fluid-dynamic environment within typical growth reactors as well as the interaction of such flow with the intrinsic kinetics of the growth process are investigated in the frame of the new fields of protein crystal and tissue engineering. The paper uses available data to introduce a set of novel growth models. The surface conditions are coupled to the exchange mass flux at the specimen/culture-medium interface and lead to the introduction of a group of differential equations for the nutrient concentration around the sample and for the evolution of the construct mass displacement. These models take into account the sensitivity of the construct/liquid interface to the level of supersaturation in the case of macromolecular crystal growth and to the “direct” effect of the fluid-dynamic shear stress in the case of biological tissue growth. They then are used to show how the proposed surface kinetic laws can predict (through sophisticated numerical simulations) many of the known characteristics of protein crystals and biological tissues produced using well-known and widely used reactors. This procedure provides validation of the models and associated numerical method and at the same time gives insights into the mechanisms of the phenomena. The onset of morphological instabilities is discussed and investigated in detail. The interplay between the increasing size of the sample and the structure of the convective field established inside the reactor is analysed. It is shown that this interaction is essential in determining the time evolution of the specimen shape. Analogies about growing macromolecular crystals and growing biological tissues are pointed out in terms of behaviours and cause-and-effect relationships. These aspects lead to a common source (in terms of original mathematical models, ideas and results) made available for the scientific community under the optimistic idea that the contacts established between the “two fields of engineering” will develop into an ongoing, mutually beneficial dialogue.

Stress-driven morphological instability and catastrophic failure of microdevices

In microdevices, the competition between surface energy and elastic energy could lead at the phenomenon known as stress-driven morphological instability (MI), causing an increase of surface roughness with time. Several different mass transport mechanisms can trigger such a morphological alteration and operate simultaneously: surface and bulk diffusion, evaporation and condensation, chemical reactions. Unstable solids could eventually evolve towards crack-like surfaces thus altering mechanical, electrical and optical properties of the devices or even leading to catastrophic failures by supercritical crack propagation. In this work, a more general kinetic law is employed to estimate the onset of MI, considering the effect of the stress field on the atomic mobility. A more intuitive and straightforward approach is used to determine the stability conditions, where the rate of atomic mass motion is introduced as a stability parameter. The critical loads and wavelengths for the onset of MI, determined as a function of material parameters α and β, are compared with the limiting conditions for the supercritical crack propagation (SC) of a crack-like surface in order to asses if and under which situations catastrophic failures by SC can be observed. Two practical cases are investigated: fixed wavelength (Case I) and arbitrary rough surface with a fixed remote load (Case II). In Case I, absolute and relative threshold loads are found below which MI could never occur and a transitional wavelength over which MI would always lead to SC is introduced. In Case II, it is shown that dominant perturbation for MI would always lead to SC given enough time for the surface to evolve towards a crack-like profile. The influence of the material properties α and β on the critical parameters is also addressed.

Phase field modeling of shallow cells during directional solidification of a binary alloy

We report some computational results obtained by using the phase field model for binary alloy solidification. We study directional solidification and concentrate on the transition between a planar interface and steady shallow cells near the onset of morphological instability at low growth speeds. The model is formulated on the principle of local positive entropy production. This gives rise to a set of partial differential equations for the temperature, composition and the phase fields. We make a frozen temperature approximation and compute steady-state nearly periodic cellular crystal–melt interfaces in two dimensions. The cell shapes can be characterized by a small number of Fourier coefficients and become nearly sinusoidal as stability is approached. Wavelength selection by tip splitting or coarsening is also observed.

Critical Role of Crystalline Anisotropy in the Stability of Cellular Array Structures in Directional Solidification.

We calculate numerically the full Floquet-Bloch stability spectrum of cellular array structures in a symmetric model of directional solidification. Our results demonstrate that crystalline anisotropy critically influences the stability of these structures. Without anisotropy, the stability balloon of cells in the plane of wave number and velocity closes near the onset of morphological instability. With a finite, but even small, amount of anisotropy this balloon remains open and a band of stable solutions persists for higher velocities into a deep cell regime. The width of the balloon depends critically on the anisotropy strength. {copyright} {ital 1996 The American Physical Society.}

Fluctuations in solidification.

We present an analytical treatment of (i) the incorporation of thermal noise in the basic continuum models of solidification, (ii) fluctuations about nonequilibrium steady states, and (iii) the amplification of noise near the onset of morphological instability. In (i), we find that the proper Langevin formalism, consistent with both bulk and interfacial equilibrium fluctuations, consists of the usual bulk forces and an extra stochastic force on the interface associated with its local kinetics. At sufficiently large solidification rate, this force affects interfacial fluctuations on scales where they are macroscopically amplified and, thus, becomes relevant. An estimate of this rate is given. In (ii), we extend the Langevin formalism outside of equilibrium to characterize the fluctuations of a stationary and a directionally solidified planar interface in a temperature gradient. Finally, in (iii), we derive an analytic expression for the linear growth of the mean-square amplitude of fluctuations slightly above the onset of morphological instability. The amplitude of the noise is found to be determined by the small parameter ${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{E}}$${\mathit{d}}_{0}^{\mathit{c}}$${\mathit{l}}_{\mathit{T}}$/\ensuremath{\gamma}${\ensuremath{\lambda}}_{\mathit{c}}^{4}$ where \ensuremath{\gamma} is the surface energy, ${\mathit{d}}_{0}^{\mathit{c}}$ is the chemical capillary length, ${\mathit{l}}_{\mathit{T}}$ is the thermal length, and ${\ensuremath{\lambda}}_{\mathit{c}}$ is the critical wavelength. Possible applications to experiment are discussed.

Morphological instability and Si diffusion in nanoscale cobalt silicide films formed on heavily phosphorus doped polycrystalline silicon

In this letter we examine the interdiffusion and reaction of deposited cobalt layers during the formation of nanoscale (<35 nm) silicide films on heavily phosphorus doped polycrystalline silicon films. The onset of morphological instability is indicated by an increase of layer resistivity. Cross-section transmission electron micrographs of high resistance films, which were formed by rapid thermal annealing at 700 °C for 30 s, show a series of disconnected ‘‘islands’’ of CoSi2 with or without a highly disordered (amorphous) Si-rich top surface layer. A continuous band of voids, attributed to the Kirkendall effect and to phase transformation induced volume changes, initially appear at the CoSi/CoSi2 interface. In the highly agglomerated films, these voids separate the buried CoSi2 islands and buried polycrystalline Si from the disordered surface layer. The sequence of events is analyzed in terms of grain boundary diffusion, grain boundary grooving, and the impact of phosphorus concentration (∼1020 cm−3) on Si diffusivity.

The Planar to Cellular Transition and the Long Time Scale Dynamics of Pattern Formation During Thin-Film Directional Solidification

AbstractExperimental observations of the melt/crystal interface in the directional solidification of a thin sample of a binary alloy are reported for conditions only slightly beyond those for the onset of morphological instability. Spectral analysis for long times shows dynamics leading to a band of most probable wavelengths with sharply selected peaks at wavelengths significantly below the most dangerous wavelength predicted by linear stability theory. These large-amplitude cells develop first in packets that spread to fill the interface.

Buoyancy effects on morphological instability during directional solidification

The onset of morphological instability during the directional solidification of a single-phase binary alloy at constant velocity vertically upwards is treated by a linear stability analysis. We consider the case in which a heavier solute is rejected at the solidifying interface, and study the effect of natural convection on the critical concentration for the onset of instability. For tin containing lead we find a small destabilization of the system at low growth velocities, and a large increase in the wavelength of the instability at the onset. Calculations show that the destabilization is enhanced as the variation of density with solute concentration is reduced, and in the limit of neutrally-dense solute, there is a long wavelength instability for which the critical solute concentration is several orders of magnitude lower than that predicted by the Mullins and Sekerka analysis in the absence of convection. For the neutrally-dense solute, a simplified analysis indicates the roles played by the interface deformation and thermal convection in promoting the instability. In particular, the destabilization is very sensitive to the ratio of crystal and melt thermal conductivities.

Effect of a forced Couette flow on coupled convective and morphological instabilities during unidirectional solidification

Abstract The effect of a forced Couette flow, parallel to a horizontal crystal-melt interface during directional solidification of an alloy of lead containing tin, on the onset of convective and morphological instabilities, is calculated numerically via a linear stability analysis. Such a flow does not affect perturbations with wave vectors perpendicular to the flow. For perturbations with wave vectors parallel to the flow, the onset of morphological instability is somewhat suppressed and thermosolutal convection is greatly suppressed. When instabilities occur, they are oscillatory and correspond to travelling waves. For values of the crystal growth velocity for which mixed morphological and convective modes occur, the presence of a forced flow produces sufficient decoupling to allow otherwise degenerate branches to be identified.

Morphological instability of vapour-grown ice crystals: SEM investigations

The morphological stability of ice crystals freely grown from the vapour was investigated by means of scanning electron microscopy. The replica method was used. The onset of morphological instability was observed on different faces of plate-like and column-like ice crystals obtained at temperatures from -1 to -12°C. The smaller ice crystals were growing stably. It was established that the instability of the bigger crystals begins to develop from the face with the largest growth rate. For plate-like crystals this is the prismatic face, and in the case of column-like crystals this is the basal face. This experimental result is in agreement with the theoretical prediction of Chernov and with the numerical calculations of Irisawa, Kuroda and Ookawa.