Thin-film Directional Solidification Research Articles

Role of interfacial energy and liquid diffusivities on pattern formation during thin-film three-phase eutectic growth

In this paper, we investigate three-phase eutectic growth during thin-film directional solidification of a model symmetric ternary eutectic alloy. In contrast to two-phase eutectics that have only a single possibility, ${\alpha}{\beta}{\alpha}{\beta}$..., as the growth pattern, during three-phase eutectic growth infinite possibilities exist. Here, we explore the possible existence of pattern selection influenced by the change in the solid-solid interfacial energies and the diffusivities. We begin the study by estimating the undercooling vs. spacing variation for the simplest possible configurations of pattern lengths 3 and 4, where phase-field simulations are utilized to quantify the influence of the solid-solid interfacial energy and the contrast in the component diffusivities. Subsequently, extended simulations consisting of multiple periods of ${\alpha}{\beta}{\delta}$... as well as ${\alpha}{\beta}{\delta}{\beta}$... are carried out for assessing the stability of the configurations to long-wavelength perturbations in spacing. Thereafter, growth competition among the simplest patterns is investigated through phase-field simulations of coupled growth of configurations of the type [${\alpha}{\beta}{\delta}]_m[{\alpha}{\beta}{\delta}{\beta}]_n$, (m,n) being the respective number of periods. Finally, pattern selection is studied by initializing with random initial configurations and classifying the emerging patterns based on the phase sequences. The principal finding is that, while there is no strong phase selection, between the solid-solid interfacial energies and the contrast in the component diffusivities, we find the latter to strongly influence pattern morphology.

Morphological Transitions of Shallow Cells during Thin-Film Solidification of an Alloy: Phase Field Modeling & Experimental Observation

The morphological transitions of shallow cells near the onset of Mullins-Sekerka instability during the thin-film directional solidification of a succinonitrile/acetone alloy have been simulated quantitatively by means of phase field modeling for the first time. Long time/length scale experiments at a constant pulling speed were performed for the purpose of comparison. The simulation results show good agreement with classic theories in the planar and λ(subscript c)↔λ(subscript c)/2 cellular bifurcation regions, where λ(subscript c) is the critical wavelength. At a higher pulling speed, V, an anti-trapping current was introduced to maintain the local equilibrium, and the region for Vλ^2~constant was found. Finally, multiplets appeared when λ was close to λ(subscript c)/4. In the experiments, due to slow solute accumulation in the molten zone, morphological transitions were also revealed. Some interesting nonlinear phenomena, such as tip splitting, coarsening, pinch-off, and asymmetric cells, were also confirmed through both simulations and experiments.

Quantitative phase field simulation of deep cells in directional solidification of an alloy

The formation of deep cells after the onset of Mullins–Sekerka instability during the thin-film directional solidification of a succinonitrile/acetone alloy has been simulated quantitatively by phase field modeling. The solute trapping introduced by the diffusive interface is corrected by a simple interface model, so that at the interface the equilibrium segregation is restored and the Gibbs–Thompson relation is satisfied. With the increasing pulling speed, the transitions from planar to λ c/2 shallow cells, smaller wavelength finite-depth cells, and deep cells are clearly illustrated. The formation of deep cells with change of overall morphologies is performed, and its wavelength transition is consistent with the reported experiments. Furthermore, during the development of a cellular pattern starting from a planar interface, the crossover wavelength under different solidification speeds, where the deformation is comparable to the wavelength, agrees reasonably well with the Warren–Langer theory.

Long-time scale morphological dynamics near the onset of instability during directional solidification of an alloy

The long-time scale nonlinear morphological dynamics of the interface during the thin-film directional solidification of an SCN/acetone alloy have been simulated by an efficient phase field simulation. In addition to the agreement with classic theories, the mode transition (λ c↔λ c/2) through periodic–chaotic–periodic solute pinch-off after a step change of the solidification speed is illustrated; λ c is the critical wavelength. The tip-splitting, coarsening, and overshooting during mode transition are consistent with experimental observations. However, the transition time is up to 5D L /kV 2 making a real experiment at lower solidification speed very difficult to perform, where D L is the acetone diffusivity, k the segregation coefficient, and V the solidification speed. The subcriticality of the primary bifurcation is further confirmed by simulation.

Similarity law for the tilt angle of dendrites in directional solidification of non-axially-oriented crystals

We report on an experimental and numerical study of tilted dendrites observed in thin-film directional solidification of a ${\mathrm{CBr}}_{4}{--8\mathrm{}\mathrm{mol}\mathrm{}%\mathrm{}\mathrm{C}}_{2}{\mathrm{Cl}}_{6}$ alloy for an arbitrary orientation of the crystal with respect to the plane of the sample and the pulling direction. The drift velocity ${V}_{d},$ or, equivalently, the tilt angle \ensuremath{\alpha} (defined as $\mathrm{tan}\ensuremath{\alpha}{=V}_{d}/V,$ where $V$ is the pulling velocity) of the dendrites is an increasing function of $V$ and the dendritic spacing \ensuremath{\lambda}. We show that \ensuremath{\alpha} scales like the P\'eclet number $P=\ensuremath{\lambda}V/D$ ($D$ is diffusion coefficient in the liquid). In the limit of large $P$ values $(P>7),$ \ensuremath{\alpha} tends to a constant, crystallographically determined value. We show that the \ensuremath{\alpha} versus $P$ master curve is insensitive to the temperature gradient as long as $V$ is not too close to the cellular-threshold velocity. The tilt dynamics of dendrites in directional solidification is thus essentially determined by the interactions between neighboring dendrites within the approximately periodic pattern. We also show that the shape of the master curve only slightly depends on the value of the interfacial anisotropy, as long as it is not too small.

Symmetry-broken double fingers and seaweed patterns in thin-film directional solidification of a nonfaceted cubic crystal.

We present a detailed experimental and numerical investigation of the directional-solidification growth patterns in thin films of the ${\mathrm{CBr}}_{4}$--8 mol % ${\mathrm{C}}_{2}$${\mathrm{Cl}}_{6}$ alloy, as a function of the orientation of the (fcc) crystal with respect to the solidification setup. Most experiments are performed with single-crystal samples about 10 mm wide and 15 \ensuremath{\mu}m thick. The crystal sometimes contains small faceted gas inclusions, the shape of which gives us direct information about the orientation of the crystal. Numerical simulations by a fully dynamical method are carried out with parameters corresponding to the experimental system. We find experimentally that, in crystals with a {111} plane (nearly) parallel to the plane of the thin film, the growth pattern is nondendritic and unsteady over the explored velocity range (5${\mathit{V}}_{\mathit{c}}$--50${\mathit{V}}_{\mathit{c}}$; ${\mathit{V}}_{\mathit{c}}$\ensuremath{\approxeq}1.9 \ensuremath{\mu}m ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ is the cellular threshold velocity). By studying the time evolution of this pattern, we establish that it is essentially similar to the ``seaweed pattern'' characteristic of vanishingly small capillary and kinetic anisotropies of the solid-liquid interface, recently studied numerically [T. Ihle and H. M\uller-Krumbhaar, Phys. Rev. E 49, 2972 (1994)]. The building blocks of this pattern are local structures---pairs of symmetry-broken (SB) fingers called ``SB double fingers'' or ``doublons,'' and more complex structures called ``multiplets''---whose lifetime is long but finite.We show experimentally that, in agreement with numerical findings, doublons obey selection rules, but do not have a preferential growth direction. We furthermore find that ``dendritic'' doublons also appear in crystals with a 〈100〉 axis close to the pulling direction (thus having a strong two-dimensional anisotropy) above a critical velocity (\ensuremath{\approxeq}20${\mathit{V}}_{\mathit{c}}$). The existence and stability of dendritic doublons in directional solidification at high velocity are confirmed by the simulations. Another crystal orientation of interest is that in which two 〈100〉 axes are symmetrically tilted at \ifmmode\pm\else\textpm\fi{}45\ifmmode^\circ\else\textdegree\fi{} with respect to the pulling axis. We show experimentally and numerically that the nondendritic unsteady ``degenerate'' pattern observed in this case, and previously noticed by F. Heslot and A. Libchaber [Phys. Scr. T9, 126 (1985)], is qualitatively different from the seaweed pattern. In crystals close to this orientation, we find experimentally transitions between the degenerate and the two possible tilted dendritic states. The small amplitude of the sidebranches of dendrites in our system and the fact that the simulations with a purely capillary anisotropy do not reproduce these transitions lead us to attribute them to the increasing effect of kinetic anisotropy as the pulling velocity increases.

Long-time-scale dynamics observed in directional solidification of a binary alloy.

Asymptotic analysis and finite-element simulations are used to explore the nonlinear dynamics present in cellular interface patterns that arise in thin-film directional solidification, as caused by the presence of a codimension-two bifurcation at a specific value of the spatial wavelength for the cells. For certain parameter sets, the quadratic nonlinearities that dominate the dynamics because of 2:1 spatial resonance give rise to secondary steady-state bifurcation and to time-periodic states with periods that scale as ${\mathrm{\ensuremath{\delta}}}^{\mathrm{\ensuremath{-}}3/2}$, where \ensuremath{\delta} measures the distance from the parameter values for the codimension-two state. The structure of this dynamics is explored in closed form for a modified Swift-Hohenberg equation and for the two-sided, solutal model for directional solidification. The asymptotic results for directional solidification are verified by finite-element numerical simulations for small amplitude cells.

Observation expérimentale de la morphologie "en algue" et d'états asymétriques appariés en solidification directionnelle de films minces d'alliages CBr4-C2Cl6

The morphology of the growth in the thin-film directional solidification of a CBr4 - 8 mol% C2Cl6 alloy is studied for a particular crystalline orientation, namely, where a axis is perpendicular to the plane of the film. In this case, the effective surface tension as well as attachment kinetics anisotropies are very weak, and a disordered, seaweed structure is observed. In good agreement with recent numerical studies by Ihle and Muller-Krumbhaar (Phys. Rev. E 49 2972 (1994)), we show that this disordered structure is mainly composed of pairs of asymmetric fingers. This pairing phenomenon is in fact generic of all crystal orientations and is as well observed in standard, highly anisotropic grains, for higher velocities and thus higher undercoolings, as predicted numerically. In addition to the pulling velocity and the effective anisotropy in the plane of the experiment, we point out the relevance of the cellular spacing as a determinant parameter for the generation of symmetric or paired asymmetric structures. We also discuss the differences between the periodic patterns found numerically and the disordered morphologies observed experimentally.

Quantitative determination of the physical parameters relevant to the thin-film directional solidification of the CBr 4-C 2Cl 6 eutectic alloy

A relatively large number of physical parameters are involved in the formation of lamellar eutectic patterns in thin-film directional solidification: parameters pertaining to the phase diagram of the alloy, diffusion coefficient in the liquid, solid-liquid and solid-solid surface energies. We present a quantitative determination of all these quantities for the CBr 4-C 2Cl 6 eutectic alloy, employing only the thin-film directional-solidification setup. By a study of the experimental steady-state shapes of the growth front, we obtain, in particular, the value of the characteristic length known as the Jackson-Hunt minimum-undercooling wavelength.

Experimental study of the planar-to-cellular transition during thin-film directional solidification: Observations of the long-time-scale dynamics of microstructure formation.

The morphological evolution of the crystal-melt interface formed during the thin-film directional solidification of the succinonitrile-acetone alloy is studied using an experimental system that permits extremely-long-time-scale experiments under conditions close to those for the onset of cellular growth from the planar interface. Long exposure times of the alloy to high temperatures result in thermal decomposition of the succinonitrile, which increases the total concentration of solute or impurity in the sample. This effect alone may be responsible for the apparent hysteresis observed by others in the measurement of the critical solidification rate ${\mathit{V}}_{\mathit{c}}$ for the onset of cellular growth and may explain the differences between those experiments and numerical calculations. We show that the evolution of the planar interface into cellular structures occurs first in packets of more rapidly growing undulations separated by regions of slower growing cells. These cellular structures propagate over the entire interface and at very long times the interface exhibits shallow cells without a selected wavelength. The interface dynamics appears to be spatiotemporally chaotic. The wavelength distribution is dispersed about a mean that is almost a factor of 4 below the critical value expected from linear stability theory. Deeper cells exhibit stronger wavelength selection behavior and are observed at solidification rates only slightly above ${\mathit{V}}_{\mathit{c}}$; the mean wavelength increases with increasing V for deep cells.

Dynamical wavelength selection by tilt domains in thin-film lamellar eutectic growth.

Tilt domains are dynamical defects of cellular growth fronts consisting of a group of asymmetric cells traveling laterally along the front. We study the dynamics of these defects in thin-film directional solidification of the CBr 4 -C 2 Cl 6 eutectic alloy. We show that the sweeping of the growth front by tilt domains, at a given growth velocity V, brings it into a well-defined «dynamically selected» state. Once this state is reached, the tilt domains travel with a constant width. A sudden increase (decrease) of the growth velocity transforms constant-width tilt domains to growing (shrinking) ones

Tilt bifurcation and dynamical selection by tilt domains in thin-film lamellar eutectic growth: Experimental evidence of a tilt bifurcation.

In thin-film directional solidification of the ${\mathrm{CBr}}_{4}$-${\mathrm{C}}_{2}$${\mathrm{Cl}}_{6}$ eutectic alloy, we observe a quasi-instantaneous transition from symmetric states to drifting parity-broken (``tilted'') states as a result of a sudden increase of the pulling velocity. We show that (i) the observed transition is essentially the forward-tilt bifurcation that Kassner and Misbah [Phys. Rev. Lett. 65, 1458 (1990); 66, 522 (1991)] have recently found, numerically, in a model without anisotropy of the solid phases; (ii) crystal anisotropy, in the eutectic system, makes the bifurcation slightly imperfect. In lamellar eutectics, phase diffusion is extremely slow, and permanent spatial modulations of the pattern wavelength are always present, generated at eutectic grain boundaries. We show that the tilt angle, i.e., the degree of parity breaking of the pattern, adjusts itself locally to the wavelength in a very short time, in contrast with the slow phase diffusion.

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.