Towards a quantitative phase-field model of two-phase solidification

We construct a diffuse-interface model of two-phase solidification that quantitatively reproduces the classic free boundary problem on solid-liquid interfaces in the thin-interface limit. Convergence tests and comparisons with boundary integral simulations of eutectic growth show good accuracy for steady-state lamellae, but the results for limit cycles depend on the interface thickness through the trijunction behavior. This raises the fundamental issue of diffuse multiple-junction dynamics.Comment: 4 pages, 2 figures. Better final discussion. 1 reference adde

The Effect of Bamboo Charcoal on pH and Hardware in Dailed Well Water

The problem that is often encountered in clean water is that the quality of the water, both groundwater and river water used by the community, does not meet the requirements of healthy clean water. This study aims to determine the effectiveness of activated charcoal from bamboo on pH and decrease the hardness level of dug well water in the village of Lampoh Keude, Aceh Besar. This study used a Quasi-experimental method, water obtained from residents' wells was processed using the filtration and adsorption method, the filtration media was made using media with a volume of 20 liters which had been modified using sand with a thickness of 10 cm, 10 cm of palm fiber, and 10 gravel cm. The highest pH increase test results occurred in TH III AB media where the resulting pH was 7.7 ± 0.01 compared to 4.5 before processing. The same thing also happened to a decrease in the hardness level of residents' well water, where the thickness of the charcoal and the length of contact time greatly affected the test results, from the results of the 3 existing media tests, the best results were obtained by TH III AB media with a contact time of 30 minutes where the level The hardness obtained was 279 ± 3.46 mg/L or a decrease of 47.1% from the initial hardness level of well waste. There is an increase in the pH value of well water after processing, and that the use of bamboo charcoal can also reduce the hardness level of well wate

Unsteady Crack Motion and Branching in a Phase-Field Model of Brittle Fracture

Crack propagation is studied numerically using a continuum phase-field approach to mode III brittle fracture. The results shed light on the physics that controls the speed of accelerating cracks and the characteristic branching instability at a fraction of the wave speed.Comment: 11 pages, 4 figure

Phase-Field Formulation for Quantitative Modeling of Alloy Solidification

A phase-field formulation is introduced to simulate quantitatively microstructural pattern formation in alloys. The thin-interface limit of this formulation yields a much less stringent restriction on the choice of interface thickness than previous formulations and permits to eliminate non-equilibrium effects at the interface. Dendrite growth simulations with vanishing solid diffusivity show that both the interface evolution and the solute profile in the solid are well resolved

Phase-field modeling of microstructural pattern formation during directional solidification of peritectic alloys without morphological instability

During the directional solidification of peritectic alloys, two stable solid phases (parent and peritectic) grow competitively into a metastable liquid phase of larger impurity content than either solid phase. When the parent or both solid phases are morphologically unstable, i.e., for a small temperature gradient/growth rate ratio ($G/v_p$), one solid phase usually outgrows and covers the other phase, leading to a cellular-dendritic array structure closely analogous to the one formed during monophase solidification of a dilute binary alloy. In contrast, when $G/v_p$ is large enough for both phases to be morphologically stable, the formation of the microstructurebecomes controlled by a subtle interplay between the nucleation and growth of the two solid phases. The structures that have been observed in this regime (in small samples where convection effect are suppressed) include alternate layers (bands) of the parent and peritectic phases perpendicular to the growth direction, which are formed by alternate nucleation and lateral spreading of one phase onto the other as proposed in a recent model [R. Trivedi, Metall. Mater. Trans. A 26, 1 (1995)], as well as partially filled bands (islands), where the peritectic phase does not fully cover the parent phase which grows continuously. We develop a phase-field model of peritectic solidification that incorporates nucleation processes in order to explore the formation of these structures. Simulations of this model shed light on the morphology transition from islands to bands, the dynamics of spreading of the peritectic phase on the parent phase following nucleation, which turns out to be characterized by a remarkably constant acceleration, and the types of growth morphology that one might expect to observe in large samples under purely diffusive growth conditions.Comment: Final version, minor revisions, 16 pages, 14 EPS figures, RevTe

Phase-Field Approach for Faceted Solidification

We extend the phase-field approach to model the solidification of faceted materials. Our approach consists of using an approximate gamma-plot with rounded cusps that can approach arbitrarily closely the true gamma-plot with sharp cusps that correspond to faceted orientations. The phase-field equations are solved in the thin-interface limit with local equilibrium at the solid-liquid interface [A. Karma and W.-J. Rappel, Phys. Rev. E53, R3017 (1996)]. The convergence of our approach is first demonstrated for equilibrium shapes. The growth of faceted needle crystals in an undercooled melt is then studied as a function of undercooling and the cusp amplitude delta for a gamma-plot of the form 1+delta(|sin(theta)|+|cos(theta)|). The phase-field results are consistent with the scaling law "Lambda inversely proportional to the square root of V" observed experimentally, where Lambda is the facet length and V is the growth rate. In addition, the variation of V and Lambda with delta is found to be reasonably well predicted by an approximate sharp-interface analytical theory that includes capillary effects and assumes circular and parabolic forms for the front and trailing rough parts of the needle crystal, respectively.Comment: 1O pages, 2 tables, 17 figure

Feedback control of unstable cellular solidification fronts

We present a numerical and experimental study of feedback control of unstable cellular patterns in directional solidification (DS). The sample, a dilute binary alloy, solidifies in a 2D geometry under a control scheme which applies local heating close to the cell tips which protrude ahead of the other. For the experiments, we use a real-time image processing algorithm to track cell tips, coupled with a movable laser spot array device, to heat locally. We show, numerically and experimentally, that spacings well below the threshold for a period-doubling instability can be stabilized. As predicted by the numerical calculations, cellular arrays become stable, and the spacing becomes uniform through feedback control which is maintained with minimal heating.Comment: 4 pages, 4 figures, 1 tabl

Eutectic Colony Formation: A Stability Analysis

Experiments have widely shown that a steady-state lamellar eutectic solidification front is destabilized on a scale much larger than the lamellar spacing by the rejection of a dilute ternary impurity and forms two-phase cells commonly referred to as `eutectic colonies'. We extend the stability analysis of Datye and Langer for a binary eutectic to include the effect of a ternary impurity. We find that the expressions for the critical onset velocity and morphological instability wavelength are analogous to those for the classic Mullins-Sekerka instability of a monophase planar interface, albeit with an effective surface tension that depends on the geometry of the lamellar interface and, non-trivially, on interlamellar diffusion. A qualitatively new aspect of this instability is the occurence of oscillatory modes due to the interplay between the destabilizing effect of the ternary impurity and the dynamical feedback of the local change in lamellar spacing on the front motion. In a transient regime, these modes lead to the formation of large scale oscillatory microstructures for which there is recent experimental evidence in a transparent organic system. Moreover, it is shown that the eutectic front dynamics on a scale larger than the lamellar spacing can be formulated as an effective monophase interface free boundary problem with a modified Gibbs-Thomson condition that is coupled to a slow evolution equation for the lamellar spacing. This formulation provides additional physical insights into the nature of the instability and a simple means to calculate an approximate stability spectrum. Finally, we investigate the influence of the ternary impurity on a short wavelength oscillatory instability that is already present at off-eutectic compositions in binary eutectics.Comment: 26 pages RevTex, 14 figures (28 EPS files); some minor changes; references adde