Phase Diagrams: The Beginning of Wisdom

Pure Iron

The most important distinction in this field is the one between property diagram and phase diagram. Figure 1 shows the enthalpy of pure iron as function of temperature at constant pressure of 1 bar. It is a property diagram; the areas in that diagram have no meaning. Only the curve has a meaning. It is labeled with the different phases associated with the three different continuous parts of the enthalpy curve, BCC (α), FCC (γ), and BCC (δ), where BCC and FCC are a shorthand for the different crystal structures, body centered cubic and face centered cubic, respectively. Figure 1 is not a phase diagram.

Open image in new window

Figure 2 shows the phase diagram of pure iron. This is a unary system, build from a single component. The axes are the state variables, temperature T and pressure P, the pressure is shown in log-scale to reveal the gas phase region together with the high-pressure part. All of the areas in that diagram have a meaning as seen from the following example. Consider the state point at 1000 K and 10³ bar, indicated by the crosshairs in Fig. 2. That is, we put a certain amount of pure Fe in a closed box and fix the temperature and pressure at these values. Now the Fe atoms may arrange themselves in various phases, such as Gas, Liquid, BCC, FCC, or HCP. Each phase has a unique value of Gibbs energy, G ^phase (T, P), at that given state point. The phase with the lowest value of Gibbs energy G is the “winner”; it is the stable phase under the given condition. In this case, it is the BCC (α) phase. In fact, the outcome of competition among phases in the entire region bounded by the boundary lines between BCC (α)/FCC (γ) and BCC (α)/HCP is the same, i.e., BCC (α) wins. In other words, this is the stable single-phase region of BCC (α) phase, and at any state point in this region the Gibbs energy value of any competing phase is higher than that of the BCC (α) phase. The same competition rule applies to all state points in the entire phase diagram in Fig. 2 which composes of the single phase regions of Gas, Liquid, BCC (δ), FCC (γ), BCC (α), and HCP. If one chooses to use other state variables different from T and P, other thermodynamic functions instead of Gibbs energy must be minimized to find the stable field of each phase. This aspect will not be further discussed here.

Open image in new window

The boundary line between the BCC (α) and FCC regions is defined by the equality of the Gibbs energy values, G ^BCC (T, P) = G ^FCC (T, P). That line not only reveals the pressure dependence of the BCC (α)/FCC phase transformation temperature, but also forms the two-phase region BCC (α) + FCC in the phase diagram. All these two-phase regions in Fig. 2 are one-dimensional lines because of the choice of state variables, T and P, for plotting this diagram. The three-phase region BCC (α) + FCC + HCP occurs only at a unique state point, 757 K and 10^5.019 bar, in the phase diagram. Therefore, this is an invariant equilibrium, also called a triple point. The degrees of freedom are zero at an invariant equilibrium according to the Gibbs phase rule. That means one cannot change any state variable (T or P) without loosing a phase from the equilibrium. By contrast, the two-phase equilibrium BCC (α) + FCC in Fig. 2 is monovariant, it has one degree of freedom. One variable may be changed, at least infinitesimally, and by adjusting the other one along the T–P relating line the BCC (α) + FCC equilibrium may be maintained. When plotting the phase diagrams with the choice of different state variables as axes (e.g. enthalpy instead of temperature) the topology of the two- and three-phase regions may change so that they are no longer one- or zero-dimensional, as will be discussed later.

The geological relevance of the phase diagram of iron is that it helps explaining why the inner core of our earth is solid. That is peculiar because the temperature increases all the way down to the earth’s center so that the outer core, composed mainly of an iron-nickel alloy, is liquid. However, going further down, the alloy solidifies at the boundary to the inner core, even though the temperature still increases. That is because of the monotonous pressure increase, resulting in a dramatic increase of the melting temperature, shown for pure Fe in Fig. 2. The pressure at the boundary is about 330 GPa, or 3.3 × 10⁶ bar, just outside the frame of Fig. 2, and the melting temperature of the Fe-rich alloy at this pressure is estimated to be 5600 K.[12] The additional pressure increase subsequently solidifies the alloy.

The industrial importance of steel basically originates from a peculiarity of the phase diagram in Fig. 2, the occurrence of two separate regions of BCC, one at high temperature BCC (δ), and one at low temperature, BCC (α). Normally the more densely packed FCC crystals should be stable down to room temperature. However, the less densely packed BCC iron crystals undergo magnetic ordering, which significantly decreases the values of G ^BCC (T), making the BCC phase stable again at lower temperature. That results in the occurrence of the γ/α phase transition of iron, which may be modified and tailored by addition of carbon and other alloying elements in steel. Properly designing this γ/α solid-state phase transition forms the basis for controlling the variety of microstructures in steel, from the as-cast structure through heat treatment or thermo-mechanical processing. Without the FCC (γ)/BCC (α) transition in Fig. 2 at ambient pressure man could not produce steel.

H₂O and SiC

All other unary, or one-component, P-T phase diagrams can be understood following the example of iron. The rules and the resulting topology are identical if the component is an element or a pure substance, defined as a compound that cannot change composition. A prominent example is the phase diagram of pure H₂O, shown in many textbooks with the phase regions of vapor, water, ice and, at high pressure, the various crystalline ice phases. However, it is quite a special case because all the phase transitions of H₂O (e.g. melting, evaporation) are congruent ones, where at any two-phase equilibrium the compositions of both phases are identical. That is clearly fulfilled for all phases of H₂O.

There are many other solid stoichiometric compounds with negligible solid solution range, or negligible stoichiometry deviation, forming a pure solid substance. Upon melting, however, the composition is often not maintained. That is for instance the case for silicon carbide, SiC, an important compound semiconductor and also an important hard ceramic material. Heating SiC at 1 bar above 2823 °C[13] produces a silicon-rich liquid phase with only 17 at.% C and in addition a certain amount of graphite, thus balancing the original 50 at.% C of SiC. That is, the single phase SiC crystal decomposes into a liquid and graphite, also called incongruent melting. That is a three-phase equilibrium, not simple two-phase melting. In the P-T phase diagram of SiC at 1 bar the single-phase region of solid SiC ends at 2823 °C, beyond a gap forms, the two-phase region liquid + graphite. Calculation with the database PanMg[9] shows that a temperature of 3600 °C (and pressure of at least 8 bar to avoid evaporation) is necessary to obtain single-phase liquid “SiC” with 50 at.% C. This 777 K wide gap, liquid + graphite, in the “P-T phase diagram of SiC” occurs because, technically speaking, it is a section in the P-T-x phase diagram of the binary Si-C system at constant 50 at.% C. Such phase diagram sections may exhibit phase compositions which are off the section plane as will be discussed in more detail later; they are also known as isopleths.

Therefore, the compound SiC does not form a unary P-T phase diagram, in contrast to the compound H₂O. In order for the simple topology of Fig. 1, observed for any element, to prevail also for a compound it is necessary that this compound exists in all phases under all conditions (at all state points) as stable single phase, making all phase transitions congruent. Otherwise the topology of the P-T phase diagram will be more complex. The important distinction between property and phase diagram, however, is generally valid and must be kept in mind also for multicomponent systems.

Cu-Ni Phase Diagram

Figure 3 shows the phase diagram of the binary system Cu-Ni. The entire diagram is valid for a constant total pressure of 1 bar. Consider the state point indicated by the crosshairs in Fig. 3 at 1200 °C and composition x _Ni = 80 at.% Ni. The mole fraction of Ni in the system, x _Ni, may be written as x _Ni = 0.80 or x _Ni = 80 % in the “atomic percent of Ni”, at.% Ni, composition scale. That is, we put 0.20 mol of Cu atoms and 0.80 mol of Ni atoms in a closed box. One may choose any number of moles of atoms of that fixed composition instead of 1 mol, phase diagrams are independent of total amount. In addition we fix T = 1200 °C and P = 1 bar in the box. At that particular state point (T, P, x _Ni) the Cu and Ni atoms may arrange themselves in various phases, such as Liquid, FCC, Gas, or in a combination of phases. Just like in the unary example the thermodynamic equilibrium requires G ^system (T, P, x _Ni) = minimum. At that state point the minimum is attained if all atoms are in the FCC phase, thus G ^system (T, P, x _Ni) = G ^FCC (T, P, x _Ni). In the following the pressure condition P = 1 bar is not shown explicitly, also because the equilibrium temperatures among condensed phases (Liquid, Solid), in contrast to the gas phase, are almost independent of pressure up to about 100 bar as exemplified for iron in Fig. 2.

Open image in new window

The FCC phase is a substitutional solid solution, the Cu and Ni atoms may substitute each other completely in the crystal structure. That is a necessary requirement for the formation of a complete solid solution range between the pure components. Because the absolute minimum of G ^system (T = 1200 °C, x _Ni = 0.8) is found if all atoms are in the FCC phase, the state point at the crosshairs in Fig. 3 presents the stable single-phase constitution of FCC. The same result, G ^system (T, x _Ni) = G ^FCC (T, x _Ni) = minimum (with different numerical values), is obtained at any state point within the contiguous region labeled as FCC. Therefore, in the case of Cu and Ni, a complete single-phase region of the solid solution phase FCC is formed below the solidus line, and contiguous liquid forms above the liquidus line as shown in Fig. 3. In that single-phase Liquid region the absolute minimum of G ^system (T, x _Ni) = G ^Liquid (T, x _Ni) is attained by putting all atoms in the liquid solution phase.

For any state point within the contiguous region labeled as Liquid + FCC the same result, G ^system (T, x _Ni) = G ^Liquid+FCC = minimum (with different numerical values), is obtained by smart distribution of atoms on the two phases. The resulting phase compositions are plotted as the phase boundaries Liquid/(Liquid + FCC), the liquidus line, and FCC/(Liquid + FCC), the solidus line, in Fig. 3. At any state point in the Liquid + FCC region they can be read from the tie line joining the two equilibrium phase compositions, as plotted for the example of the state point at 1320 °C and 60 at.% Ni with Liquid at 52.6 at.% Ni and FCC at 65.6 at.% Ni. The tie line is horizontal because the two phases must of course be at the same temperature to be in equilibrium. Any two-phase region in a binary temperature-composition phase diagram follows these same rules.

Phase Diagram Applications Exemplified with Cu-Ni

For a simple application to solidification under strict equilibrium conditions we may consider an alloy with fixed composition of 52.6 at.% Ni which is completely molten at 1400 °C and then slowly cooled. At 1320 °C the liquidus point of this alloy is encountered and the first small crystal of composition 65.6 at.% Ni may form. Upon further cooling both the Liquid and FCC compositions become more Cu-rich. That may be envisaged by a sequence of tie lines while the state point (at constant 52.6 at.% Ni) moves down from 1320 to 1271 °C where it hits the phase boundary FCC/(Liquid + FCC) at the solidus point. At this last tie line the last droplet of liquid has attained 37.9 at.% Ni and the composition of FCC equals the 52.6 at.% Ni of the alloy. The equilibrium solidification is thus terminated and further cooling occurs in the single-phase FCC region. It is emphasized that, generally, there is no such thing as a “melting point” of an alloy but a melting interval between solidus and liquidus with ΔT = 49 K in this example. The significant composition variation of 52.6-37.9 at.% Ni in Liquid and 65.6-52.6 at.% Ni in solid FCC during “equilibrium” solidification requires sufficiently fast diffusion to attain complete equilibrium in all parts of the alloy, especially in the solid. It will be discussed later how to relieve this constraint using the Scheil simulation.

Another very useful application of the phase diagram is materials compatibility and materials bonding. Assume we drop a piece of solid nickel, preheated to 1320 °C, into pure liquid copper kept in a furnace at constant 1320 °C. In a first step we plot the initial status of materials into Fig. 3, one point at pure Cu the other at pure Ni, both at 1320 °C. The phase diagram tells us that there is no tie line between these points, thus, there is no equilibrium and therefore a reaction is expected. In a second step we calculate or estimate the overall composition of this (closed) system. Assume the piece of nickel is small, amounting to only 10 at.% Ni in our Cu-Ni experiment. This defines the state point at 1320 °C and 10 at.% Ni, which is in the single-phase liquid region in Fig. 3. That is the direction into which the reaction between the initial materials is expected to go and also the final state of the system; the solid piece of Ni will completely dissolve in the melt.

If our experiment is performed the other way around, pouring liquid copper in a nickel crucible at 1320 °C, the first two steps in applying the phase diagram are done in the same way. If we have a thin Ni crucible, amounting to a total of 10 at.% Ni in the system (as before), the dissolution will produce a hole in the crucible and a mess in the furnace. With a thick Ni crucible and, say, total 85 at.% Ni of the system, that state point is in the single-phase FCC region in Fig. 3. Now, in a third step, we use this knowledge to assess a realistic reaction path. Ni from the crucible will dissolve in the liquid Cu, shifting its composition towards the liquidus point. Concurrently, some Cu will dissolve in the FCC-Ni, shifting its composition in the direction of the solidus point, in a much slower solid state diffusion process. During that process the amount of liquid decreases because Cu is taken out of the liquid. A composition gradient builds up in the crucible, and at the Liquid/FCC interface a local equilibrium is attained with the composition of the two phases given by the tie line in Fig. 3. Eventually, the amount of liquid goes to zero, with the last drop of liquid at 52.6 at.% Ni and the FCC at 65.6 at.% Ni at the last contact point. Now the system is isothermally solidified. The composition gradient in the single-phase (not yet homogeneous!) FCC crucible is eventually leveled out by solid state diffusion until the equilibrium dictated by the state point at 85 at.% Ni and 1320 °C is attained in the homogeneous FCC.

Exactly that process is widely used in transient liquid phase bonding (TLP) or diffusion brazing.[4,14] In our example a thin Cu foil would be clamped between two relatively thicker Ni plates. This assembly is heated above the melting point of Cu but much below that of Ni so that the state point of the assembly is in the single-phase FCC region. At constant bonding temperature, just above 1085 °C, a solid bond is formed, which is even stable at much higher temperature, up to 1397 °C, the solidus point in our example of 85 at.% Ni system composition. That process is applied in many material systems where at least the part around the substrate is similar to the Ni-rich region in Fig. 3 with significant solid solubility, a low melting filler material and no intermetallic phases occurring at the bonding temperature. No intermetallic phases are formed in the bond during the diffusion brazing process in contrast to soldering or diffusion soldering[15, 16, 17] which is ruled by different phase diagrams.

Mg-Al Phase Diagram

The Mg-Al phase diagram is shown in Fig. 4 as an example for a system with formation of compounds, which are also called intermetallic phases if the components are metals as in this example. For the state point indicated by the crosshairs at 400 °C and composition 95 wt.% Al the equilibrium constitution is obviously the single-phase FCC solid solution. The state point at 63.59 wt.% Al and 100 °C is also in a single-phase “region”, namely the β phase region. However, this region appears degenerate to a line in the plot at constant 63.59 wt.% Al, thus, the β phase is also called a line compound. By contrast the single phase region of γ is so large it is easily discerned. Another name often used for the γ phase is Al₁₂Mg₁₇, however, the stoichiometry suggested by this name is only approximate and the phase diagram elucidates how the stoichiometry variation, or the stable solid solution range of γ, depends on temperature. Thermodynamics require that all compounds must have a finite solid solution range, but it may not be resolved on the plotted phase diagram as is the case for the phases β and ε in Fig. 4. All single-phase regions in Fig. 4 are marked by color shading, leaving all the two-phase regions blank.

Open image in new window

The region where the three phases HCP + L + γ are in equilibrium at 436.3 °C is marked by the line, with dots indicating the phase compositions. This is the intersection of three two-phase equilibria, HCP + L, HCP + γ, and L + γ, which may also be seen as the overlap of these three tie lines. This three-phase equilibrium region is a true line (1D), while all the two-phase and single-phase regions are areas (2D), remembering that even the β and ε regions will extend under sufficiently high magnification. That is why the three-phase equilibrium is an invariant equilibrium, temperature and all phase compositions are fixed. All other three-phase equilibria in Fig. 4 follow the same rules.

This topological degeneracy of the invariant equilibrium to a phase region with lower dimension in the phase diagram is general. Look at the congruent melting point of γ at 463.4 °C and 48.8 wt.% Al, which is also an invariant equilibrium, L + γ, at a true point (0D) where the liquidus and solidus lines of γ touch at the maximum. On either side of that point the L + γ two-phase equilibrium extends into a normal 2D region and a melting interval. This is also seen in systems with more (or less) than two components. Consider for example pure Mg in Fig. 4. The single phase regions HCP and L are lines (1D) along the temperature axis, whereas the invariant two-phase region HCP + L is a point (0D), the melting point of magnesium.

If a state point is chosen in a three-phase region, for example at 20 wt.% Al and 436.3 °C in Fig. 4 we can read from the phase diagram that the three phases HCP, L, and γ occur in equilibrium with compositions at 12.7, 33.3, and 42.5 wt.% Al, respectively. However, the phase fractions are not fixed and the lever rule cannot be applied because this system composition is not on a unique tie line. This is also the case at the melting point of pure Mg where the phase fractions of HCP and L are not determined by just fixing the temperature because we cannot know if we are at the begin or the end of the melting process of that invariant equilibrium. That is generally the case for all invariant equilibria in this temperature-composition phase diagram: one can read only the type of phases and their composition but not the phase fractions. For any state point outside the invariant equilibrium regions one can read the complete constitution information: type of phases, composition of phases, and fraction of phases.

All invariant equilibria, or invariants, can be written as invariant reactions, for example L = HCP + γ. This is the so-called eutectic reaction, where L decomposes into the solid phases HCP and γ upon extraction of heat. In other words it is attempted to decrease the temperature of the alloy which is only possible in equilibrium once the liquid phase completely disappeared and the subsequent two-phase region HCP + γ is entered. The accepted notation is to always write the reaction in this direction of decreasing temperature.

Such a decomposition reaction is also observed for the three-phase equilibrium γ + ε + β at 250.1 °C, ε = γ + β. This is called a eutectoid reaction; the ending “oid” indicates that solid phases only are involved. It would be a eutectic reaction if ε is replaced by a liquid. The only other possible reaction type is the formation reaction, where two phases react to form a third one. This is seen for example at the other γ + ε + β three-phase equilibrium at 409.8 °C, γ + β = ε. This is called a peritectoid reaction. If one of the reactant phases were a liquid it would be called a peritectic reaction, not occurring in the Mg-Al system. It occurs in the Mg-Zr system where L + (Mg) + (Zr) form the invariant equilibrium at 653.6 °C with the peritectic reaction L + (Zr) = (Mg). The notation (Mg) is used here as an alternative to denote the HCP solid solution phase, similarly for (Zr). Mg and Zr (without the brackets) denote the components, not the solid phases. The peritectic reaction L + (Zr) = (Mg) occurs above the melting point of pure Mg (649.8 °C). That is because the solubility of Zr in solid (Mg) is larger than in the liquid, thus, the melting interval (Mg) + L develops to higher temperature with Zr addition as opposed to the case of Al addition in Fig. 4.

An important distinction between the decomposition and formation reaction type is in the kinetics. Generally, the formation reaction (peritectic/peritectoid), λ + φ = μ, slows itself down because the solid product phase μ produces a growing diffusion barrier between the reactant phases λ and φ. Therefore this reaction type is prone to not finish under real alloy cooling conditions and to leave unreacted, non-equilibrium remains. As opposed to that, the decomposition reaction (eutectic/eutectoid), λ = φ + μ, may occur much closer to the equilibrium phase diagram because there is no barrier phase formed. With higher cooling rates the microstructure of the growing φ + μ phases simply adjusts to a finer size with shorter diffusion distances.

When reading a phase diagram of the type in Fig. 4 a simple rule is helpful. If we cross a phase boundary we either gain a phase or loose a phase. Consider the path indicated by the dashed arrow for the Mg90Al10 (wt.%) alloy between 650 and 350 °C. We start in the single-phase liquid phase and the phase sequence is L/HCP + L/HCP/HCP + γ and the rule obviously holds. For the Mg80Al20 (wt.%) alloy between 600 and 350 °C the phase sequence is L/HCP + L/HCP + L + γ/HCP + γ. Highlighted by bold font is the eutectic three-phase region and the rule holds. This clarifies that the horizontal eutectic line is a phase region, not just a phase boundary. Otherwise we would read the wrong phase sequence L/HCP + L/HCP + γ, where the “gain a phase or loose a phase” rule is violated.

Phase Diagram Applications Exemplified with Mg-Al

One important application of thermodynamic software packages is the calculation of expected equilibrium phase fractions as function of temperature for a given alloy composition. That is shown for the Mg90Al10 (wt.%) example alloy in Fig. 5. In principle one could calculate that manually using the lever rule, see Eq 7-8, after plotting the tie lines at various temperatures in the phase diagram in Fig. 4. That is not only tedious but becomes difficult or impossible if the alloy contains more than two components. On the other hand the thermodynamic calculation is done rapidly for any number of components along the line of state points with fixed composition and variable temperature and an easy to read diagram similar to Fig. 5 is obtained. That is why this type of calculation is also called a line calculation, or a 1D (scan) calculation, or lever rule calculation. The result is not a phase diagram but a property diagram, such as in Fig. 5.

Open image in new window

Reading in the direction of decreasing temperature the liquid fraction shrinks from 1 (liquidus point, 598 °C) to zero (solidus point, 472 °C), thus defining the freezing range, 126 K, of this alloy. Note that this is the equilibrium freezing range, which is a limiting case requiring complete equilibration in all parts of the alloy at each temperature step, as discussed above for the Cu47.4Ni52.6 alloy. Another useful limiting case, the Scheil simulation, will predict a wider freezing range for this alloy as discussed later. Concurrently with the shrinking amount of liquid the fraction of solid phase, only HCP for this case, grows from 0 to 1. Between 472 and 380 °C the single-phase region of solid HCP exists and this defines a potential solution heat treatment window for this alloy. This information is important because during non-equilibrium solidification, discussed later, some secondary phase may have formed in a coarse microstructure that should be dissolved during a solution heat treatment of that alloy. Subsequently an aging heat treatment may be performed to produce the second phase as fine precipitates in the HCP matrix during a solid state process. This is in principle possible for this alloy because below 380 °C the secondary phase γ forms at the expense of the HCP phase. That is also seen in the phase diagram from the retrograde solid solubility line (solvus line), the HCP/(HCP + γ) phase boundary. It is also seen that the solution heat treatment window widens for lower Al content of the alloy. For our Mg90Al10 example alloy the solution treatment must be performed below 472 °C to avoid partial melting and above 380 °C to form the single-phase HCP. Thereafter a relatively fast cooling down to room temperature will form a supersaturated solid solution (SSS) once the state point moves into the HCP + γ region, however, the nucleation and growth kinetics is too slow to produce the γ phase. In the subsequent aging treatment the SSS is heated to a point in the HCP + γ region with favorable kinetics, and held for example at 200 °C. Eventually the γ phase will precipitate and from tie line and lever rule at that state point a very large equilibrium mass fraction of f ^γ = 0.19 should finally form in the HCP matrix. These generic phase diagram features are essential for such processing steps and widely applied for microstructure design of alloys, even though our Mg90Al10 example alloy is not among the industrially used alloys.

The calculated molar phase fractions for the more Al-rich Mg80Al20 (wt.%) example alloy is shown in Fig. 6. The primary crystallizing phase is again HCP, starting to grow at the liquidus temperature of 539 °C, but then the secondary phase γ is formed in the eutectic reaction L = HCP + γ at 436.3 °C which terminates the solidification. The equilibrium freezing range of this alloy is 103 K. At the fixed eutectic temperature the liquid fraction drops from 0.35 to zero and simultaneously the solid fractions increase, from 0 to 0.24 for γ and from 0.65 to 0.76 for HCP. At this point the primary crystallized HCP and the eutectic HCP are exactly the same phase, both with 12.7 wt.% Al, however, they are distinguished in the microstructure of the solidified alloy. The primary HCP has grown freely in the melt, probably forming dendrites, whereas the eutectic HCP is found in finer microstructure, often lamellar, jointly with the γ phase. The fraction of eutectic HCP, 0.11 as dictated by the phase diagram, is therefore an important information. After completion of the invariant eutectic three-phase reaction the two-phase region HCP + γ is entered and Fig. 6 shows that the fraction of γ continues to grow at the expense of HCP. That is also seen in Fig. 4 because the solid solubility limits, the solvus lines, vary with temperature and for the given Mg80Al20 alloy the lever rule shows the same increasing fraction of γ. For a quantitative comparison one may calculate the mass fractions because the phase diagram is given in wt.%. However, in a micrograph the observed phase fraction are closer related to the volume fraction and quite often the molar phase fraction is a better approximate to that as compared to the mass phase fraction.

Open image in new window

Mg-Al-Zn Phase Diagram

Isothermal and Vertical Sections

In a ternary system, composed of three components, we select the state variables T, P, and composition. We assume constant total pressure, high enough to suppress any gas phase formation in all of the following. At constant temperature, say T = 500 °C, only two variables are left to fix the state point of the system. In the Mg-Al-Zn system we may select the contents of Al and Zn, thus fixing [wt.% Mg] = 100 − [wt.% Al] − [wt.% Zn]. The natural way to plot these two variables is in rectangular coordinates as shown in Fig. 7(a). Only the right-angle triangle area highlighted by the colored phase regions corresponds to real alloys; outside that range at least one composition becomes negative. The compositions on the straight line [wt.% Mg] = 0 are in the binary edge system Al-Zn. In order to obtain symmetry among the components the y-axis is tilted to get the equilateral or Gibbs triangle in Fig. 7(b), generally used if the complete composition range is covered. For enlargements of small composition ranges the rectangular diagram is much more useful, the tilted axis is disapproved for partial diagrams. Both figures are true phase diagrams because any point in the triangles corresponds to a fixed state point with unique constitution. That is shown for the intersection of the dotted lines for 30 wt.% Al and 20 wt.% Zn, where the single phase liquid is stable. At 500 °C in the Mg-Al-Zn system five single-phase regions are observed, L, HCP, FCC, τ, and MgZn₂, the latter developing as a line compound due to significant solubility of Al in binary MgZn₂. The term “line compound” refers to the one-dimensional extension of the single-phase region in the phase diagram section under consideration; it is also used for the phases β and ε in Fig. 4. A compound with no solubility range for any component is denoted as “stoichiometric compound”; the phase ε is an example that will be discussed in Fig. 8.

Open image in new window

Open image in new window

Coming back to the isothermal section of the Mg-Al-Zn phase diagram at 500 °C in Fig. 7, a number of two-phase regions, such as L + FCC, span between the solubility limits of single phases, marked by series of selected tie lines. For example, the tie line passing through the state point of the Mg10Al70Zn20 (wt.%) alloy indicates the compositions of the equilibrium phases L (50.2 wt.% Al, 33.4 wt.% Zn) and FCC (86.7 wt.% Al, 8.6 wt.% Zn). The lever rule can also be applied here, providing the mass fractions of the two phases, f ^Liquid = 0.458 and f ^FCC = 0.542. This is easily checked by the materials balance for Al, 50.2 × 0.458 + 86.7 × 0.542 = 70.0 wt.% Al, and analogously for Zn or Mg. Finally, two three-phase regions occur, marked by the red tie triangles. That completes the entire area of the phase diagram with unique constitution at each state point.

The isothermal section at 400 °C is shown in Fig. 8. At this lower temperature the extension of the single-phase liquid region shrinks into two separate patches and three more solid phases become stable, γ, ε, β, and Mg₂Zn₃. Note that the solid phases at 400 °C may be categorized as follows: HCP and FCC are terminal solid solutions, extending from the pure components; the Zn-rich HCP phase region is very small because we are just below the melting point of Zn at 419.5 °C. Another useful notation for these terminal solid phases is (Mg) and (Zn) for HCP and (Al) for FCC. We have two separate stable patches of the same phase HCP, just as we have two separate patches of L. Next we have γ, β, MgZn₂, and Mg₂Zn₃, binary intermetallic phases with significant ternary solubility; only γ shows also a significant binary solution range, the other three appear as line compounds in the ternary. Next there is ε, a binary intermetallic phase that remains stoichiometric, visible only as a point on the Mg-Al binary edge. Finally there is τ, the only truly ternary solid phase since it does not connect in a continuous single-phase range to any of the binary edges, not even at different temperature. Another ternary solid phase, φ, will form below 387 °C.

Let us have a closer look at the three-phase region FCC + τ + β, marked by one of the red tie triangles in Fig. 8 with one apex at each of these single-phase boundaries. The state point of an example alloy Mg30Al50Zn20 (wt.%), marked by the crosshairs, is located inside this region, indicating that this combination of three phases is the most stable configuration of distribution of atoms on available phases with the absolute minimum of the Gibbs energy of the system. The composition of each phase is fixed and may be read at the corners of the tie triangle, FCC (88.6 wt.% Al, 1.6 wt.% Zn), τ (42.3 wt.% Al, 25.9 wt.% Zn), and β (55.9 wt.% Al, 10.2 wt.% Zn). The mass fractions are f ^FCC = 0.104, f ^τ = 0.681, and f ^β = 0.215, again determined by the lever rule. For any other alloy composition located inside this three-phase region FCC + τ + β the phase compositions remain fixed, only the phase fractions change. The closer the state point moves to a corner of this tie triangle the larger the fraction of this phase becomes.

One may envisage the complete ternary Mg-Al-Zn phase diagram (at constant pressure) as a prismatic 3D model, spanned by the triangular composition base and a vertical temperature axis. Each point inside that prism is a fixed state point with unique constitution: Type(s), composition(s), and fraction(s) of phase(s) are given by the equilibrium condition of Gibbs energy minimum. The isothermal sections in Fig. 7 and 8 form one way to produce a 2D section through the prism for quantitative display of the phase relations. The other way is the vertical section, also called T-x section or isopleth. If the interest is in phase relations along a series of alloys at various temperatures, the 3D prism will be sectioned parallel to the temperature axis along the composition line defined by the series of alloys. The result is a 2D section, such as in Fig. 9 for the example of ternary alloys at constant 20 wt.% Zn. The selected composition line along the alloy range Mg80Zn20-Al80Zn20 (wt.%) is also indicated by the dotted lines in Fig. 7 and 8. Along this line at 500 °C the same phase relations appear in Fig. 7 and 9. The stable regions of the liquid phase, or the FCC phase, are easily discerned in both diagrams.

Open image in new window

The important distinction occurs in the two-phase region, such as L + FCC. Figure 7 clearly shows that for the alloy Mg10Al70Zn20 the phase compositions of L and FCC are off the section plane at constant 20 wt.% Zn, thus, the phase compositions cannot be read in Fig. 9. That is generally true for all two- or three-phase regions in a vertical section: Only the type of phases, not the composition of phases can be read from the vertical section. Therefore, the lever rule cannot be applied to determine the phase fractions. The complete constitution information can only be read from the isothermal section because all the tie lines are in the plane of that 2D section, and that is always the case for ternary systems.

In very rare and special cases all the tie lines may also happen to lie inside the plane of the vertical 2D section. In that case it is also necessary that the end points of the composition section are in a single-phase region, α or β, for example of a stoichiometric melting compound or a pure component. Not even that condition is met for the end point Mg80Zn20, that alloy is two-phase HCP + L. If all the tie lines are inside the plane this vertical phase diagram section between the phases α and β is called a pseudobinary system. It may be read like a binary system α-β, and it will provide the full constitution information on type, composition, and fraction of phases. Such pseudobinary systems are sometimes found in systems between stoichiometric oxides, such as Bi₂O₃-Fe₂O₃ in the Bi-Fe-O system,[18] or if a complete series of solid solutions exist between congruent melting compounds, such as GdA1₂-NdA1₂.[19] In the Mg-Al-Zn system any composition section will cut at least one tie line or tie triangle, thus, no pseudobinary section exists.

For the isothermal or vertical 2D sections, such as in Fig. 8 and 9, the simple rule outlined above is again helpful: If we cross a phase boundary we either gain a phase or loose a phase. Start by reading the phase regions at 400 °C and Al80Zn20 in the single-phase FCC region in Fig. 9. With decreasing Al-content, at constant 20 wt.% Zn, we cross the phase boundary at 79.5 wt.% Al and must enter a two-phase region. From Fig. 8 it is obvious that this is the FCC + MgZn₂ region, so we have gained the phase MgZn₂, as correctly labeled in Fig. 9. The next boundary is at 75.9 wt.% Al and, according to the rule, it might be single- or three-phase. From Fig. 8 we see that this is the narrow three-phase region FCC + MgZn₂ + τ, so we have gained the phase τ. At 75.3 wt.% Al we cross the boundary to the wide region labeled as FCC + τ in Fig. 9, so we have lost the phase MgZn₂, in consistency with Fig. 8. Now it becomes obvious that the next phase boundary at 53.7 wt.% Al in Fig. 9, where we gain the phase β, reflects just the cut through the FCC + τ edge of the tie triangle FCC + τ + β in Fig. 8. All phase compositions of FCC, τ, and β, are way off the vertical section. For the example alloy Mg30Al50Zn20 compositions can only be obtained from the isothermal section, as detailed above. One may continue reading the 400 °C line in Fig. 9 and add labels to all phase regions in consistency with Fig. 8 and the “gain/loose-a-phase-rule”.

Liquidus Projection

For melting and solidification processes the equilibria of the liquid with solid phases are especially important. The extensions of the liquidus lines from the binary edges form the liquidus surface in the 3D prism and its projection to the composition triangle forms the liquidus projection, shown in Fig. 10. It immediately answers the question which phase crystallizes primary. For any alloy composition in the region marked “HCP” this is the primary phase that may grow freely in the melt, thus forming a typical dendritic or globulitic microstructure. In the adjacent primary γ region this intermetallic will form first from the melt. At the intersection line the liquid phase compositions are in equilibrium with both HCP and γ; the liquid is double saturated. This intersection line displays the projection of the monovariant three-phase equilibrium L + HCP + γ. It emerges from the Mg-Al edge were it starts as the binary invariant eutectic reaction L = HCP + γ at 436.3 °C, proceeding to lower temperature into the ternary. At 400 °C it is seen as the tie triangle L + HCP + γ in Fig. 8. The triangular shape tells that it is monovariant, at a given temperature all three phase compositions are fixed. Only the trace of the apex at phase L is plotted in the projection in Fig. 10, the solid compositions would make that graph too busy. Each primary phase region is confined by such monovariant lines of double saturated liquid, or by the binary edges.

Open image in new window

At an intersection point of different monovariant lines we have a contact point of three primary phase regions, such as FCC, τ, and β. That unique liquid composition is (triple) saturated with all three phases, thus forming the four-phase equilibrium L + FCC + τ+β. That is exactly the invariant reaction of Eq 9, associated with the unique temperature of 447.07 °C in the projection. Therefore, Fig. 10 reveals all the liquid compositions involved in invariant equilibria. One simply reads the types of the three adjoining primary regions, e.g. τ, γ, and φ to see the composition of L in the invariant L + τ + γ = φ at (34.7 wt.% Zn, 14.7 wt.% Al) and 387.37 °C, discussed in Eq 11.

In total there are 12 different invariant four-phase reactions in the Mg-Al-Zn system corresponding to the intersection points in Fig. 10. The closing of the tiny primary field of Mg5Zn2, at 52 wt.% Zn, 0.1 wt.% Al and 338.9 °C, produces the 12th point, which cannot be discerned on the graph. These invariant reactions are connected by the network of monovariant three-phase equilibria, some of them ending in the binary edges, some just occur in the ternary, such as L + τ + MgZn₂. This line offers a special case, the maximum at 530.1 °C with liquid composition (13.4 wt.% Al, 64.4 wt.% Zn). The maximum occurs because the tie triangle L + τ + MgZn₂ degenerates to a line and at exactly this point the three-phase equilibrium becomes a unique and invariant three-phase reaction,

$$ {\text{L }} = \tau + {\text{ MgZn}}_{ 2} \quad {\text{at 53}}0. 1\,^\circ {\text{C}}. $$

(12)

This reaction is of the eutectic type because the liquid composition is located exactly in between the τ + MgZn₂ tie line. Therefore, the liquid may decompose into these two solid phases without any composition shift. That explains why this reaction is invariant, similar to the binary eutectic. Other double saturated lines show a monotonous temperature variation only, such as L + MgZn₂ + FCC from 475.9 °C down to 355.4 °C at the Zn-rich end. One should be careful in assigning a reaction type because it may change within the monovariant range and the transition from eutectic to peritectic type may be hard to detect especially if significant solid solubilities are involved.[2] The safest way, also for multicomponent alloys, is to calculate the phase fractions with a small decreasing temperature step for a given alloy composition and temperature. The shrinking phases go to the left hand side and the growing phases to the right hand side of the reaction equation, valid only at that state point. It can be shown that even within a given tie triangle the transition from eutectic (α = β + γ) to peritectic (α + β = γ) type may occur by just changing the alloy composition at fixed temperature. Therefore, one should generally denote just the three-phase equilibrium (α + β + γ) unless it degenerates to an invariant reaction at a minimum or maximum temperature, such as in Eq 12.

A 2D graphical display of the network of monovariant three-phase equilibria, connecting the invariant reactions of the binary edge system with those in the ternary system, can be given by the “Scheil Reaction Scheme”.[20] It may be used to prove the consistency of the phase diagram; for example the number of three-phase equilibria meeting at a four-phase reaction must be four, and so on. An established notation for invariant equilibria and liquidus projections is developed[21] that covers also more complex cases, such as liquid miscibility gaps intersecting primary crystallization fields in a liquidus surface.

A particularity of the primary regions of φ and τ in Fig. 10 is that they do not touch the binary edges. That may be seen as another indication that φ and τ are true ternary solid phases, however, the decisive distinction is that their solid solution ranges do not touch the binary edges at any temperature. As additional information the projections of selected isothermal liquidus lines are plotted in Fig. 10. These contour lines give a better impression on the shape of the liquidus surface in the 3D prism phase diagram. Moreover they are used to read, or interpolate, the liquidus temperature of a given alloy in addition to the type of primary phase that starts crystallizing at that temperature.

Phase Diagram Applications Exemplified with Mg-Al-Zn

For melting processes the completely molten (single-phase liquid) region needs to be identified and that is obviously done from Fig. 7 to 10. For solution heat treatment the single-phase solid regions must be known, given in Fig. 7 to 9. Similarly the constitution of a ternary multiphase material is read from these diagrams to answer the question if the observed phase assembly is a stable one, or which assembly may be expected after equilibration. For solidification applications the important information on liquidus temperatures and primary crystallizing phase has been discussed above. The equilibrium melting/freezing range may be read from vertical sections, such as in Fig. 9.

A very powerful tool is the calculation of phase fractions (and compositions) of a fixed alloy composition of interest, such as shown in Fig. 5 and 6 for the binary example. That will be demonstrated in detail in the section on Scheil and equilibrium solidification simulation. In principle the equilibrium phase fractions could be read from a series of isothermal sections using the lever rule. That is only simple if the solid phases are all stoichiometric, in that special case even the liquidus projection and knowledge of the stoichiometries is sufficient. In a real world alloy system, with significant solid solubilities, the tie lines and their directions change with temperature, making this a very tedious manual task even if many isothermal sections in small temperature steps are available. The thermodynamic calculation using a software package and a reliable database is highly recommended for that application.

Another very important application is materials compatibility, applied in interface reactions, joining, durability of refractory crucibles for alloy melting, attack of slag, and so on. Initially two materials A and B are brought in contact and heated. For example, if the alloy plate composed of Mg80Zn20 is clamped to another one, Al80Zn20 (wt.%), the phase diagram in Fig. 9 reveals that in the temperature range 300-400 °C various product phases may form between the materials, and also partial melting may occur. At 400 °C more details are seen from Fig. 8, the dotted line at 20 wt.% Zn indicates all possible overall compositions of the clamped material system. All the phase regions crossed by that line indicate potential temporary product phases and the state point, calculated from the overall composition of the two plates, gives the final equilibrium state.

As another example, consider a thin film layer of an alloy Mg60Zn40 (wt.%) deposited on a disk of pure Al and then heating the coated disk at 400 °C. We apply the same technique as in section 3.2, the Cu-Ni example. In a first step we plot the initial composition of materials into Fig. 8, one point at pure Al the other at alloy Mg60Zn40. The phase diagram tells us that there is no tie line between these points, thus, there is no equilibrium and therefore a reaction is expected. In a second step we calculate or estimate the overall composition of this (closed) system, which must be on the straight line between the starting points. For a very thin film our state point will be in the single-phase FCC region, thus, after equilibration all the Mg and Zn atoms from the film will be dissolved in the (Al) disk. The disk converts to a solid solution that is eventually homogenized by solid state diffusion. Temporarily a partial melting of the film may occur, because that point is in the L + HCP region. After some Al from the disk went into the layer even a temporary complete liquid layer might form on the disk. Subsequently a number of reactions may occur, involving the phases τ, γ, and β, until the equilibrium state, dictated by the state point, is reached. With growing film thickness the state point may be located beyond the (Al) solvus in the adjacent two-phase region FCC + τ, and the τ phase is expected as final secondary phase on the saturated (Al) solution phase. With even larger film thickness, say total mass 5 g film on a thin Al disk of 5 g, the state point is located at Mg30Al50Zn20 (wt.%) and the three-phase equilibrium phase assembly FCC + τ + β, well discussed above, constitutes the final state of the reaction.

The Scheil Approximation in Context of Multicomponent Multiphase Solidification

Equilibrium solidification simulation, discussed above, often does not reflect the phases actually observed in the as-cast state of alloys. That discrepancy is most pronounced if the crystallizing phases exhibit wide ranges of solid solubility and if solid-state diffusion is relatively slow compared to the solidification rate. Under these conditions the so-called “Scheil approximation” is widely applied as a much better description.

However, it cannot be applied to real-world scenarios of multicomponent alloys with different types of solid phases precipitating. A most significant progress has been made by combining Calphad-type generated thermodynamic databases with the basic Scheil approximation in assumptions (i)-(iii), but releasing assumption (iv). The precipitation of sequential (infinitesimal) layers of any assembly of solid phases is quantitatively calculated from the current local equilibrium at the liquid/solid(s) interface. That is solved for each “layer step” by numerical procedures which are available in software packages such as Pandat,[5] Thermocalc[6] or Factsage.[7] In this modern approach the concept of a “distribution coefficient k ₀” is not used at all. Rather the calculation of the multicomponent equilibrium tie line (or tie triangle etc. for multiphase solidification) at each temperature at the liquid/solid(s) interface is embedded in the numerical procedure. That is the basis for the most successful application of the Scheil approximation to real-world multicomponent multiphase alloys. It allows to quantitatively calculate a limiting case for the various solid phase fractions and phase composition gradients (segregation) developing in the Scheil-freezing-range of the alloy by using thermodynamic data only without requiring any kinetic material parameter.

This Scheil approximation is often much more realistic compared to the other limiting case, the “equilibrium” or “lever rule” approximation, where assumption (iii) is replaced by assuming infinitely fast solid-state diffusion and, thus, complete equilibration also in all solid parts of the alloy at any temperature. This “equilibrium” condition requires complete diffusion back into the regions crystallized at an early stage, which may be far remote from the liquid/solid interface at a late stage of solidification. Results of such line calculations under equilibrium conditions, also done numerically, were shown in Fig. 5 and 6. Again, the concept of “k ₀” is not used at all.

Therefore, the calculation of both the “Scheil approximation” and the “equilibrium approximation” often provides useful bounds to assess the solidification of a multicomponent alloy and to gain insight in heat treatment conditions. Another limitation in both approximations is that any nucleation or growth undercooling is assumed to be zero. The two main limitations in the Scheil approximation are: (i) The diffusion layer in the liquid in front of the moving interface is neglected; it will be build up by concentration accumulation if the solubility of the pertinent component is larger in the liquid compared to the solid (or depletion in the other case). (ii) Back diffusion is completely excluded, even if the concentration gradient in a solid phase may be very steep at the final stage of solidification or if the component is known as fast diffusing. However, to quantitatively address these two limitations kinetic data are required, such as diffusion coefficients in all phases, or better the mobilities because diffusion coefficients can then be calculated using the thermodynamic factors obtained from the already known thermodynamic description. That is done in advanced solidification models that may also take the convection in the liquid into account as another transport mechanism. However, kinetic data are often not known for real multicomponent systems, leaving the Scheil approximation as the best alternative.

For special cases, if one component is known to diffuse very fast in the solid phases it could be set as diffusing infinitely fast, while the diffusion of all other components is kept completely blocked. In that case partial back diffusion is simulated in a partial equilibrium of the homogeneous residual liquid with the solid(s).[32,33] This is related to the paraequilibrium conditions of ternary (or higher order) systems with two sluggishly diffusing components, such as substitutional Fe and Mn, and the fast diffusing interstitial C.[2] That approach is especially useful if the interstitial components occur in phases with wide solid solution range, such as carbon in the austenite (FCC) phase of steel.

If all solid phases exhibit negligible solid solution range and only eutectic (no peritectic or transition-type) reactions occur, the Scheil simulation will give exactly the same results as the equilibrium solidification simulation. The reason is simply that the condition to block the diffusion in the solid phases will not make a difference because it is impossible to build up a concentration gradient in a stoichiometric solid phase, requiring solid-state diffusion to be leveled out. That also explains why the equilibrium (or lever rule) solidification calculation works nicely in simple eutectic systems.

Solidification Simulation Exemplified with Mg-Al-Zn

The different features of these two limiting cases, the equilibrium and Scheil approximation, respectively, will be exemplified for the solidification simulation of a ternary Mg90Al9Zn1 (wt.%) alloy. That is the simplified nominal alloy composition of the important magnesium casting alloy AZ91. Its overall composition is plotted as the starting point in Fig. 20, the Mg-rich part of the Mg-Al-Zn liquidus projection.

Open image in new window

The blue curve is the solidification path under equilibrium conditions, showing the composition variation of the residual liquid phase while its fraction shrinks from 1 to zero. The liquid composition moves from Mg-Al9Zn1 at the liquidus temperature of 600 °C to Mg-Al23.6Zn10.3 at the solidus temperature of 446 °C. Simultaneously, the liquid phase fraction moves from 1 to zero as shown in Fig. 21, where a logarithmic scale is used. Concurrently, the solid phase (Mg) crystallizes and its fraction grows from zero to 1. Thus, solidification terminates without precipitation of any secondary phase. The reason is simply that all the alloying components can be dissolved in the (Mg) solid solution, attaining exactly Mg-Al9Zn1 at the solidus point, where a tie line stretches out to the last droplet of liquid at Mg-Al23.6Zn10.3. That is also seen in Fig. 8 – the solid solution range of HCP, or (Mg), is wide enough to cover the alloy composition Mg90Al9Zn1 even at 400 °C. This single-phase (Mg) region exists from 446 to 381 °C for this alloy, as seen in Fig. 21. Below that (solvus) temperature the secondary phase γ precipitates in a solid-state reaction from the (Mg) matrix phase. The solidification path in Fig. 20 is strongly curved to the Zn-rich side in the final stage. That is because the L + (Mg) tie lines, passing through the fixed Mg90Al9Zn1 point, rotate significantly with decreasing temperature. For practical applications the solution heat treatment window for this alloy from 446 to 381 °C in the single-phase (Mg) region is important.

Open image in new window

The red curve in Fig. 20 is the solidification path under Scheil conditions. It starts at the same liquidus point as in the equilibrium case but then differs significantly. While crystallizing primary (Mg) the liquid composition moves in an almost straight line from Mg-Al9Zn1 to Mg-Al29.7Zn5.0 where it hits the monovariant line of double saturation, L + (Mg) + γ at 429 °C. This significant distinction to the equilibrium solidification arises because much less of the alloying components can be dissolved in the (Mg) phase due to the blocked back diffusion. The core of the first crystal, formed at 600 °C from Mg-Al9Zn1, remains frozen at the low composition of Mg-Al2.6Zn0.06. A remarkable composition gradient is formed in the growing (Mg) phase from that core crystal to the last layer solidified with composition Mg-Al11.7Zn0.5 at 429° C. At that point f ^(Mg) = 0.834 and f ^Liquid = 0.166, seen at the bend in the liquid fraction curve in Fig. 22. The solidification path in Fig. 20 also shows a break at that point, after which it continues with decreasing Al-content. Subsequently, the secondary phase γ crystallizes jointly with (Mg) from the melt in the monovariant reaction L → (Mg) + γ from 429 to 365.34 °C.

Open image in new window

The next break point occurs at 365.34 °C (liquid composition Mg-Al1.7Zn34.3) where only a small amount of residual liquid is left, f ^Liquid = 0.004, and the accumulated solid fractions are f ^(Mg) = 0.886, and f ^γ = 0.110. At that point the invariant reaction L + γ = (Mg) + φ is encountered at 365.34 °C. This is a ternary transition-type reaction, and, as explained in Eq 10, the formation of the product phases (Mg) + φ would form a solid-state diffusion barrier. Therefore, this type of reaction cannot proceed under Scheil conditions, it is overrun. Solidification simply proceeds with the residual liquid, now saturated with (Mg) + φ. The amount of γ remains constant, that phase is frozen-in with f ^γ = 0.110. It will be overgrown by the following crystallizing phases, thus, loosing contact to the moving liquid/solid phase boundary. That is typical for such unreacted remains, or peritectic phases, often producing a characteristic microstructure.

The subsequent solidification L → (Mg) + φ occurs from 365.34 to 337.67 °C, producing the small fraction of φ, f ^φ = 0.002. Now the solidification path in Fig. 20 hits the next invariant reaction, L + φ = (Mg) + τ, at 337.67 °C. This is again a transition-type reaction that will be overrun in the Scheil simulation, the phase fractions do not change at this invariant point. Solidification proceeds with the residual liquid, f ^Liquid = 8.7 × 10⁻⁴, along the short line of L → (Mg) + τ in Fig. 20. Finally, the ternary eutectic L = (Mg) + τ + MgZn is encountered at 336.77 °C. This decomposition-type reaction proceeds fully, thus terminating solidification. The residual ternary eutectic liquid, f ^Liquid = 7.9 × 10⁻⁴, decomposes completely to produce some more (Mg) and τ and the additional phase MgZn. At this point, and at T < 336.77 °C, the alloy Mg90Al9Zn1 is composed of f ^(Mg) = 0.887, f ^γ = 0.110, f ^φ = 0.002, f ^τ = 1.1 × 10⁻⁴, and f ^MgZn = 3.4 × 10⁻⁴. These values are shown in Fig. 22 at the end of the various phase fraction curves and the point for MgZn.

This Scheil simulation of the as-cast constitution is complemented by the information about the sequence of phase precipitation, providing important clues on the schematic alloy microstructure with primary, secondary etc. phase formation. Moreover, the compositions of all phases are also obtained and could be plotted together with the phase fractions as function of the local phase formation temperature, revealing the expected segregation within the solid solution phases. In addition, the “Scheil solidus” at 336.77 °C is a good approximation of the incipient melting temperature of the as-cast alloy. That is an important temperature limit during extrusion or other hot forming processes. The assembly of five solid phases is of course a non-equilibrium constitution. Heating at 336 °C will eventually produce the equilibrium constitution (Mg) + γ with f ^(Mg) = 0.95 and f ^γ = 0.05, see Fig. 21, by dissolution of the non-equilibrium phases φ, τ, MgZn and leveling out the segregation in the solution phases. This also suggests a safe two-step heat treatment process, a first step slightly below 336 °C to avoid incipient melting and removing the low melting ternary eutectic, and a second step inside the solution heat treatment window, shown in Fig. 21, to produce a fully single-phase (Mg) alloy.

This concept of the incipient melting was applied as reverse “Scheil melting” to the frozen-in microstructure from a Scheil simulation and used to model the partial re-melting of feedstock in Thixomolding.[34] Care should be taken if the residual liquid fraction becomes very small. The example in Fig. 22 has been discussed until the termination of solidification in the ternary eutectic with f ^Liquid = 0 to highlight all aspects of the different reaction types in relation to the solidification path. This sequence of transition-type and eutectic reaction is also seen in many other ternary or multicomponent alloys and may be considered a generic example.

For practical applications, however, the calculation may be cut off at an arbitrary limit of f ^Liquid = 0.01 to obtain a “realistic Scheil solidus at 1% liquid “, in our example at 399 °C in Fig. 22. That corresponds to a “last liquid” composition of Mg-Al18.8Zn21.9 in Fig. 20, in the middle of the L → (Mg) + γ path. The next phase, φ, would only appear if the arbitrary cut-off limit is lowered to f ^Liquid = 0.001 (at 341 °C). The small phase fractions are not observed experimentally as shown in Fig. 23; the solidified microstructure of Mg90Al9Zn1 alloy only shows the phases (Mg) and γ. More importantly, it demonstrates that the coarse secondary phase γ crystallized jointly with (Mg) from the melt, thus proving the reaction L → (Mg) + γ predicted by the Scheil simulation as opposed to the equilibrium solidification in Fig. 21, where γ is formed by solid-state precipitation only. It is emphasized that the as-cast microstructure in Fig. 23 is obtained by a cooling rate of only 1 K/min.[35] Even this slow rate is fast enough to enable a good description by the Scheil approximation in this alloy system.

Open image in new window

In other material systems it is even more important to set a cut-off limit to the Scheil solidus. If the Scheil simulation is not stopped by a eutectic it may fade out down to unrealistically low temperature with f ^Liquid ≪ 10⁻⁴, suggesting a much too wide freezing range and a too low incipient melting temperature. After all, the Scheil simulation is based on the approximations detailed above and may be prone to some artifacts if the liquid fraction becomes very small in the final stage.

For the correlation with actual casting defects other useful data may be obtained from Scheil simulations and phase diagrams. Generally, a narrow freezing range is beneficial. More precisely, the temperature range at the final stage of solidification is relevant. Quantitatively, the partial freezing range near termination of solidification, the terminal freezing range (TFR)[36] may be obtained from Scheil simulation. For example, this value may be taken from 88 to 98% fraction solid, indicating the TFR of the “almost” last 10% of solidifying liquid. This value is more relevant than the one for the “last 10%” because of the limitations of the Scheil approximations in the last 1 or 2% of liquid fraction. Secondly, a small TFR should be beneficial in order to avoid hot tearing. It is less useful to account for a limit of 0% liquid in that application instead of a small but not harmful residual limit. The experimentally observed hot-tearing susceptibility (HTS) has been successfully correlated to the phase diagram features of Mg-Al-Ca alloy castings: wide freezing range and low eutectic content result in higher HTS.[37] Scheil simulations have been used to reveal that the HTS increases with increasing fraction solid at the end of primary solidification of Mg-Al-Sr alloys.[38] This powerful tool was also used to predict and reduce liquation-cracking susceptibility during welding of Al alloys with different filler alloys by Scheil simulation of the respective solidified fractions.[39]

The superposition of the solidification path on the liquidus projection in Fig. 21 is very instructive for ternary alloys but hard to read and not recommended for more than three components. For quaternary or even higher multicomponent alloys the diagram types shown in Fig. 21 and 22 are widely used and simple to read, they may just show more phases. In addition, the Scheil simulation provides the compositions of all phases. It is instructive to plot these together with the phase fractions as function of the local phase formation temperature, revealing the expected segregation within the solid solution phases. For the quaternary Mg-Al-Zn-Mn alloy system such Scheil simulations have also been used to obtain the isotherms for the non-equilibrium solidus temperature (NEST), defined at the cut-off limit at 98% fraction solid, or f ^Liquid = 0.02. These predicted NEST isotherms are well validated by experimental data on incipient melting for the group of alloys with high alloying contents.[40]