Eutectic colony formation: a stability analysis

Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia

We investigate the role of auto-chemotaxis in the growth and motility of an epithelium advancing on a solid substrate. In this process, cells create their own chemoattractant allowing communications among neighbours, thus leading to a signaling pathway. As known, chemotaxis provokes the onset of cellular density gradients and spatial inhomogeneities mostly at the front, a phenomenon able to predict some features revealed in in vitro experiments. A continuous model is proposed where the coupling between the cellular proliferation, the friction on the substrate and chemotaxis is investigated. According to our results, the friction and proliferation stabilize the front whereas auto-chemotaxis is a factor of destabilization. This antagonist role induces a fingering pattern with a selected wavenumber k₀. However, in the planar front case, the translational invariance of the experimental set-up gives also a mode at k = 0 and the coupling between these two modes in the nonlinear regime is responsible for the onset of a Hopf-bifurcation. The time-dependent oscillations of patterns observed experimentally can be predicted simply in this continuous non-linear approach. Finally the effects of noise are also investigated below the instability threshold.

Ternary eutectic dendrites: Pattern formation and scaling properties

Extending previous work [Pusztai et al., Phys. Rev. E 87, 032401 (2013)], we have studied the formation of eutectic dendrites in a model ternary system within the framework of the phase-field theory. We have mapped out the domain in which two-phase dendritic structures grow. With increasing pulling velocity, the following sequence of growth morphologies is observed: flat front lamellae → eutectic colonies → eutectic dendrites → dendrites with target pattern → partitionless dendrites → partitionless flat front. We confirm that the two-phase and one-phase dendrites have similar forms and display a similar scaling of the dendrite tip radius with the interface free energy. It is also found that the possible eutectic patterns include the target pattern, and single- and multiarm spirals, of which the thermal fluctuations choose. The most probable number of spiral arms increases with increasing tip radius and with decreasing kinetic anisotropy. Our numerical simulations confirm that in agreement with the assumptions of a recent analysis of two-phase dendrites [Akamatsu et al., Phys. Rev. Lett. 112, 105502 (2014)], the Jackson-Hunt scaling of the eutectic wavelength with pulling velocity is obeyed in the parameter domain explored, and that the natural eutectic wavelength is proportional to the tip radius of the two-phase dendrites. Finally, we find that it is very difficult/virtually impossible to form spiraling two-phase dendrites without anisotropy, an observation that seems to contradict the expectations of Akamatsu et al. Yet, it cannot be excluded that in isotropic systems, two-phase dendrites are rare events difficult to observe in simulations.

Solidification of Bulk Lamellar Eutectics

Eutectic alloys from the ternary system Al-Cu-Ag are excellent model alloys for the investigation of coupled eutectic growth, not only because most materials properties are well known but also because the system offers a rich variety of crystal structures and crystal orientation relationships (ORs) being associated to distinct minima of the solid-solid interface energy. This paper describes three research topics specifically related to bulk lamellar Al-Al2Cu eutectics, e.g. the maze-to-lamellar transition during early growth, the role of fault lines during lamellar spacing selection close to the pinch-off limit and the onset of eutectic cell formation above the constitutional supercooling limit. These topics are central to the microgravity experiments SETA presently being prepared for the MSL / SQF.

Scaling theory of two-phase dendritic growth in undercooled ternary melts

Two-phase dendrites are needlelike crystals with a eutectic internal structure growing during solidification of ternary alloys. We present a scaling theory of these objects based on Ivantsov's theory of dendritic growth and the Jackson-Hunt theory of eutectic growth. The additional introduction of the relationship ρ∼λ (ρ: dendrite tip radius; λ: eutectic interphase spacing) suggested by recent experimental results [S. Akamatsu et al., Phys. Rev. Lett. 104, 056101 (2010)] leads to a complete solution of theselection problem and to the scaling rule ρ∼λ -1/2 (v: dendrite tip growth rate).

Spiral two-phase dendrites

We present real-time observations of a new growth pattern called a spiral two-phase dendrite, which we observed during univariant directional solidification of a ternary-eutectic alloy. Two different crystal phases grow from the apex of a parabolic finger, forming a spiral pattern that leaves behind in the solid a double helix microstructure. The parabola tip radius is nearly equal to the helix step. These lengths vary approximately as the reciprocal of the square root of the tip growth velocity. The direction of growth is not selected, but is initial-condition dependent.

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. 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 lamellae 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.

Eutectic growth under acoustic levitation conditions

Samples of Pb-Sn eutectic alloy with a high density of 8.5 x 10(3) kg/m(3) are levitated with a single-axis acoustic levitator, and containerlessly melted and then solidified in argon atmosphere. High undercoolings up to 38 K are obtained, which results in a microstructural transition of "lamellas-broken lamellas-dendrites." This transition is further investigated in the light of the coupled zone for eutectic growth and the effects of ultrasound. The breaking of regular eutectic lamellas and suppression of gravity-induced macrosegregation of (Pb) and (Sn) dendrites are explained by the complicated internal flow inside the levitated drop, which is jointly induced by the shape oscillation, bulk vibration and rotation of the levitated drop. The ultrasonic field is also found to drive forced surface vibration, which subsequently excites capillary ripples and catalyzes nucleation on the sample surface.

Traveling waves, two-phase fingers, and eutectic colonies in thin-sample directional solidification of a ternary eutectic alloy

We present an experimental investigation of the morphological transition of lamellar eutectic growth fronts called "formation of eutectic colonies" by the method of thin-sample directional solidification of a transparent model alloy, CBr4-C2Cl6. This morphological transition is due to the presence in the melt of traces of chemical components other than those of the base binary alloy (impurities). In this study, we use naphthalene as an impurity. The formation of eutectic colonies has generally been viewed as an impurity-driven Mullins-Sekerka instability of the envelope of the lamellar front. This traditional view neglects the strong interaction existing between the Mullins-Sekerka process and the dynamics of the lamellar pattern. This investigation brings to light several original features of the formation of eutectic colonies, in particular, the emission of long-wavelength traveling waves, and the appearance of dendritelike structures called two-phase fingers, which are connected with this interaction. We study the part played by these phenomena in the transition to eutectic colonies as a function of the impurity concentration. Recent theoretical results on the linear stability of ternary lamellar eutectic fronts [Plapp and Karma, Phys. Rev. E 60, 6865 (1999)] shed light on some aspects of the observed phenomena.