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The key to close the model is the determination of interfacial concentrations (C* l, i, C* s, i) under a given temperature T, i.e., the solidification path. For binary alloys, the solidification path is quite straightforward as indicated by the liquidus and solidus denoted in phase diagrams. When it comes to the multicomponent alloy system, the solidification paths become intricate as the information collections of liquidus and solidus are not curves in 2D any longer, e.g. the collections for ternary alloy system are curved surfaces in three dimensions. Thus it is necessary to impose other constraints on the liquidus and solidus information collections of multicomponent alloy phase diagrams to define the solidification paths.

A schematic of typical ternary eutectic alloy A-B-C system is shown in the left of Figure A1. Here the focus is set on the A rich corner. It should be noted that all the following derivations are not limited to the chosen ternary eutectic alloy system.

Fig. A1. Schematic of solidification paths for the ternary eutectic alloy system.

When temperature T1 lies in the range (TA, TeA-B), a primary solidification (L→α) occurs. As shown in the upper right of Fig A1, the temperature plane intersects with the liquidus surface and solidus surface of the phase diagram at curve mn and fg respectively. It can only be deduced that the interfacial liquid concentration point (C* l, B, C* l, C) lies on the curve mn while interfacial solid concentration point (C* s, B, C* s, C) lies on the curve fg. Unfortunately, the specific values of interfacial concentrations cannot be indicated from the phase diagram.

For this situation, the interface solid concentration can be expressed as,

/ [A1]
/ [A2]

and interfacial concentration constraint relation (Eq. (12)) is simplified as,

/ [A3]

Besides, the liquidus information can be expressed as

/ [A4]

Combining Equations [A1]-[A4], the interfacial concentrations can be ascertained under the given temperature.

When temperature T2 lies in the range (TeA-B, TE), a secondary solidification (L→ α + β) occurs. For this situation, the solid phase is composited of two solid phase (α + β). As shown in the right middle of Fig. A1, the temperature section intersects with phase diagram at three points. Thus, the specific value of each individual phase is obtained, and the interfacial liquid concentration (C* l, B, C* l, C)can be ascertained. However, the concentration of composite solid phase (C* s, B, C* s, C) cannot be determined directly from the phase diagram. One way is using the lever principles to three phase tie-triangles to calculate the substantial quantities of the liquid phase, α phase and β phase. Once the relative quantity of α phase in the composite solid is calculated, the concentration of composite solid phase (C* s, B, C* s, C) can be determined.

Here the interfacial concentration constraint relation is applied. It is assumed that the relative quantity of α phasein the mixed solid is x and x∈(0, 1). Then the concentrations of composite solid phase can be expressed as

/ [A5]
/ [A6]

Combining Equations [A5]-[A6] and interfacial concentration constraint relation (Eq. (12)), the solid interfacial concentrations (C* s, B, C* s, C) can be solved.

The last solidificationsituation in ternary A-B-C alloy is the ternary eutectic solidification (L→ α + β + γ)，as shown in the lower right of Fig. A1. According to the phase rule, the system freedom under this situation is zero (F = C-P+1 = 3-4+1 =0). Then the temperature is fixed at TE and all the concentrations are fixed as shown in Fig. A1. Considering the physical fact that all the liquid is transformed to the composite solid (α + β + γ), the interfacial concentrations are ascertained.

/ [A7]
/ [A8]

Substituting Equations [A7] and [A8] into Eq. [12], the constraint relation will always be true.

In conclusion, the possible solidification paths during current ternary alloy solidification can be calculated by the interfacial concentration constraint relations coupled with thermodynamic data of phase diagram. When it comes to different multicomponent alloy systems, e.g. ternary alloy system involving the peritecticreactions, similar derivations can be made.