Phase field

Al—FeSi. Iron and shicon, present in primary aluminum, may also be added to produce enriched ahoys for specific purposes. The equhibrium phase fields in the Al—Fe—Si system are shown in Figure 20 and Table 17. The intermetahic phases have a limited range of composition when in equhibrium with the aluminum soHd solution. The amount of iron in soHd solution in the matrix is smah, so almost ah of the iron is in the intermetahic compounds. At low shicon contents the iron is present as Al Fe except for about 0.01% Fe in soHd solution. As shicon content increases, the ternary intermetahic... 

Increased pressures can lower the temperature at which crystallisation occurs. Experiments performed using Spectrosil (Thermal Syndicate Ltd.) and G.E. Type 204 (General Electric Company) fused siUcas (see Eig. 2) show that at pressures above 2.5 GPa (<25, 000 atm), devitrification occurs at temperatures as low as 500°C and that at 4 GPa (<40, 000 atm), it occurs at as low as 450°C (107). Although the temperatures and pressures were in the coesite-phase field, both coesite and quarts were observed. Both the devitrification rate and the formation of the stable phase (coesite) were enhanced by the presence of water. In the 1000—1700°C region at 500—4000 MPa (<5, 000-40,000 atm), a- and p-quarts were the primary phases. Crystal growth rates... 

Fig. 3.6. (a) The copper-nickel diagram is a good deal simpler than the lead-tin one, largely because copper and nickel are completely soluble in one another in the solid state. (b) The copper-zinc diagram is much more involved than the lead-tin one, largely because there are extra (intermediate) phases in between the end (terminal] phases. However, it is still an assembly of single-phase and two-phase fields. 

Pernod is a transparent yellow fluid consisting of water, alcohol and Evil Esters. The Evil Esters dissolve in strong water-alcohol solutions but precipitate out as tiny whitish droplets if the solution is diluted with more water. It is observed that Pernod turns cloudy at 60 wt% water at 0°C, at 70 wt% water at 20°C, and at 85 wt% water at 40°C. Using axes of T and concentration of water in wt%, sketch an approximate phase diagram (Fig. A1.3) for the Pernod-water system, indicating the single-phase and two-phase fields. 

The phase diagram for a binary alloy (Fig. A1.13) shows single-phase fields (e.g. liquid) and two-phase fields (e.g. liquid plus A). The fields are separated by phase boundaries. When a phase boundary is crossed, a phase change starts, or finishes, or both. 

DEF. When the constitution point lies in a single-phase region, the alloy consists of a single, homogeneous, phase. Its composition must (obviously) be that of the alloy. The phase composition and the alloy composition coincide in single-phase fields. 

DEF. When the constitution point for an alloy lies in the two-phase field the alloy breaks up into a mixture of two phases. The composition of each phase is obtained by constructing the tie line (the isotherm spanning the two-phase region, terminating at the nearest phase boundary on either side). The eomposition of eaeh phase is defined by the ends of the tie line. 

Figure A1.19 shows the phase diagram for the copper-zinc system. It is more complicated than you have seen so far, but all the same rules apply. The Greek letters (conventionally) identify the single-phase fields. 

DEF. A eutectic reaction is a three-phase reaction, by which, on cooling, a liquid transforms into two solid phases at the same time. It is a phase reaction, of course, but a special one. If the bottom of a liquid-phase field closes with a V, the bottom of the V is a eutectic point. 

The copper-zinc system shown in Fig. A1.38 has no fewer than five peritectic reactions. Locate them and ring the peritectic points. (Remember that when a single-phase field closes above at a point, the point is a peritectic point.)... 

Most pairs of homopolymers are mutually immiscible, so that phase diagrams are little used in polymer science... another major difference between polymers on the one hand, and metals and ceramics on the other. Two-phase fields can be at lower or higher temperatures than single-phase fields... another unique feature. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

It is not an easy task to develop computer codes which correctly treat the advancement of a folding interface as a boundary condition to a diffusion or flow field. In addition, the interface between a solid and a liquid, for example, is usually is not absolutely sharp on an atomic scale, but varies over a few lattice constants [32,33]. In these cases, it is sometimes convenient to treat the interface as having a finite non-zero thickness. An order parameter is then introduced, which for example varies from the value zero on one side of the interface to the value one on the other, representing a smooth transition from liquid to solid across the interface. This is called the phase-field... 

When the solid phase 0+ at x = -f oo coexists with the gas phase 0 at X = -oo, the stationary profile of the phase field is determined so as to minimize the free energy functional F (56). The functional derivative gives... 

The phase-field model and generalizations are now widely used for simulations of dendritic growth and solidification [71-76] and even hydro-dynamic flow with moving interfaces [78,79]. One can even use the phase-field model to treat the growth of faceting crystals [77]. More details will be given later. 

FIG. 8 Compact seaweed originating from the simulation of an isotropic phase by a phase-field model [120]. A doublon structure is just about to emerge from the chaotic background. 

The numerical solution of these equations is not trivial, since for reasonably low viscosities the flow becomes turbulent. A popular method of treating these equations (together with the equations of energy and mass conservation) is the MAC method [156,157]. For the case of immiscible fluids or moving internal interface a phase-field-type approach seems to be successful [78,158,159]. Because of the enormous requirements of computing ressources the development in this field is still relatively slow. We expect, however, an impact from the more widespread availability of massively parallel computers in the near future. 

Braun, S. Cornell, R. Sekerka. Phase field models for anisotropic interfaces. Phys Rev 48 10X6, 1993. 

A. Boesch, H. Miiller-Krumbhaar, O. Shochet. Phase field models for moving boundary problems Controlling metastability and anisotropy. Z Physik B 97 161, 1995. 

A. Karma, W.-J. Rappel. Phase field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E 55 R3017, 1996 A. Karma, W.-J. Rappel. Quantitative phase field modeling of dendritic growth in two and three dimensions. Phys Rev E 57 4111, 1998. 

R. Gonzalez-Cinca, L. Ramirez-Piscina, J. Casademunt, A. Hernandez-Machado, L. Kramer, T. Toth Katona, T. Borzsonyi, A. Buka. Phase field simulations and experiments of faceted growth in liquid crystals. Physica D 99 159, 1996. 

A. Karma. Phase field model of eutectic growth. Phys Rev E 49 2245, 1994. J. Burton, R. Prim, W. Slichter. J Chem Phys 27 1987, 1953. 

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