Pattern Stability and Trijunction Motion in Eutectic Solidification

We demonstrate by both experiments and phase-field simulations that lamellar eutectic growth can be stable for a wide range of spacings below the point of minimum undercooling at low velocity, contrary to what is predicted by existing stability analyses. This overstabilization can be explained by relaxing Cahn's assumption that lamellae grow locally normal to the eutectic interface.Comment: 4 pages, 5 eps figure

Pattern formation in directional solidification under shear flow. I: Linear stability analysis and basic patterns

An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near the onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies, described in a companion paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.Comment: 20 pages, Latex, accepted for Physical Review

Eutectic colony formation: A phase field study

Eutectic two-phase cells, also known as eutectic colonies, are commonly observed during the solidification of ternary alloys when the composition is close to a binary eutectic valley. In analogy with the solidification cells formed in dilute binary alloys, colony formation is triggered by a morphological instability of a macroscopically planar eutectic solidification front due to the rejection by both solid phases of a ternary impurity that diffuses in the liquid. Here we develop a phase-field model of a binary eutectic with a dilute ternary impurity and we investigate by dynamical simulations both the initial linear regime of this instability, and the subsequent highly nonlinear evolution of the interface that leads to fully developed two-phase cells with a spacing much larger than the lamellar spacing. We find a good overall agreement with our recent linear stability analysis [M. Plapp and A. Karma, Phys. Rev. E 60, 6865 (1999)], which predicts a destabilization of the front by long-wavelength modes that may be stationary or oscillatory. A fine comparison, however, reveals that the assumption commonly attributed to Cahn that lamella grow perpendicular to the envelope of the solidification front is weakly violated in the phase-field simulations. We show that, even though weak, this violation has an important quantitative effect on the stability properties of the eutectic front. We also investigate the dynamics of fully developed colonies and find that the large-scale envelope of the composite eutectic front does not converge to a steady state, but exhibits cell elimination and tip-splitting events up to the largest times simulated.Comment: 18 pages, 18 EPS figures, RevTeX twocolumn, submitted to Phys. Rev.

Capillary instabilities in deep cells during directional solidification

Motivated by recent experiments, we discuss a capillary instability that arises in funnel-shaped, deep-cell roots during directional solidification. A linear stability analysis of thin, axisymmetric roots reveals that due to surface energy, the roots are always unstable. The eventual outcome of this instability in experiments is a break-up of the root into droplets. The distance between successive droplet centers is in reasonable agreement with the wavelength of the fastest growing mode.Nous discutons, à la lumière d'expériences récentes, la stabilité des sillons cellulaires profonds au cours de la solidification directionnelle. Une analyse linéaire montre que les sillons tridimensionnels axisymétriques étroits présentent toujours une instabilité d'origine capillaire, qui peut expliquer l'émission par les sillons de gouttes liquides observée dans les expériences. La distance mesurée entre gouttelettes successives est de l'ordre de la longueur d'onde la plus dangereuse

Lamellar eutectic growth at large thermal gradient: I. Stationary patterns

We study stationary front profiles of directionally solidified lamellar eutectics. We show that, in the limit of large thermal gradients, front shapes can be calculated analytically without resorting to the Jackson-Hunt ansatz [7] of equal average front undercooling of both phases. We find that the difference between them is comparable with the global average one Δ T, but that the variation of ΔT with the lamellar period λ exhibits a minimum for a value of λ of the same order of magnitude as that predicted by J. H. We look for stationary tilted solutions. These can occur in general for isolated values of the tilt angle, but are absent in the large gradient limit, in agreement with the experimental observation of a velocity threshold for « tilt waves ».Nous étudions les profils de fronts stationnaires des eutectiques lamellaires en solidification directionnelle. Nous montrons que, dans la limite des grands gradients thermiques, on peut calculer analytiquement ces formes de front sans avoir recours à l'hypothèse d'égalité des surrefroidissements de front moyens des deux phases faite par Jackson et Hunt [7]. Nous trouvons que la différence entre ces quantités est comparable au surrefroidissement moyen global Δ T, mais que ΔT présente un minimum à une valeur de la longueur d'onde lamellaire du même ordre de grandeur que celle prédite par J. H. Nous cherchons s'il existe des solutions à lamelles inclinées, et montrons qu'elles ne peuvent, en général, exister que pour des valeurs isolées de l'angle d'inclinaison, mais qu'elles sont absentes à grand gradient, en accord avec l'observation expérimentale d'un seuil de vitesse pour les « ondes d'inclinaison »