# Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
## 3 Synthesis of Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
### Thermodynamics of Solidification and Healing  
Understanding the thermodynamics of solidification and healing is crucial for the development and optimization of SMA-reinforced SHMMCs. Solidification transforms the molten metal matrix into a solid-state during cooling, facilitating the healing mechanism. By heating the composite above the SMA's phase transformation temperature, partial melting of the matrix alloy occurs, allowing it to infiltrate cracks and initiate healing. It is crucial to maintain a specific volume ratio of dendrites to eutectic in the matrix alloy, which ensures that the crack cavities are adequately infiltrated by a liquid phase after heating and healing, thereby providing sufficient capillary pressure [1]. As the SMA returns to its original shape and clamps the crack, the partially molten matrix alloy solidifies, sealing the crack and enabling self-healing. In this section, we will delve into the thermodynamic principles governing the solidification and healing of a NiTi-reinforced Sn-Bi matrix. Partial Melting and Solidification Equilibrium phase diagrams for multicomponent material mixtures are valuable tools for predicting the formation of thermodynamically stable phases across various temperature, pressure, and composition ranges. These diagrams provide insights into the phase behavior of the mixtures and aid in understanding the stability of different phases under different conditions. The renowned Gibbs phase rule [92; 93; 94; 95], formulated by American physicist [PERSON] in his seminal work \"On the Equilibrium of Heterogeneous Substances\", published in parts between 1875 and 1878 [96], relates the number of phases present (P), the degrees of freedom (F), and the number of components (C) under constant pressure as:  
\[\mathrm{P+F=C+2} \tag{1}\]  
The Gibbs phase rule (given as Equation (1)) allows for the prediction of the degrees of freedom and the number of phases present in a binary system. The rule predicts that in the Sn-Bi system (C = 2, P + F = 4), two-phase regions (P = 2) with two degrees of freedom (F = 2) can exist. Figure 1 is an equilibrium phase diagram of Sn-Bi, which shows that a two-phase region exists between the liquid solution phase (L) and the solid solution of Sn and Bi. Consequently, the alloy experiences partial melting within the two-phase region for a given composition and temperature. For a fixed composition in a binary eutectic phase diagram, upon increasing the temperature, melting in the alloy will initiate beyond the solidus temperature and completely melt just above the liquidus temperature. At a specific composition in the phase diagram, the liquidus and solidus temperatures of the alloy coincide and reach their minimum, resulting in congruent melting akin to pure metal. This composition is referred to as the eutectic composition, and the temperature is known as the eutectic melting point. In the Sn-Bi eutectic system, the eutectic composition is approximately Sn-57 wt.%Bi, and the eutectic melting point is around 139 \({}^{\circ}\)C [97; 98]. A binary eutectic alloy has a lower melting point in its eutectic composition than either of its constituent elements. Choosing an alloy with a eutectic phase is essential for designing the matrix of SMA-reinforced SHMMCs. Crystals of the primary phase initiate nucleation and growth when the matrix alloy is cooled from the liquid into a liquid-solid two-phase region, except at eutectic compositions. The lever rule [92; 94; 95] allows for the design of the matrix alloy for SHMMCs with a predictable liquid fraction required for the healing of the composite at a given temperature. Conversely, the temperature corresponding to a fixed liquid fraction of the matrix alloy for a given composition can also be calculated by using the equations of the solidus and liquidus lines and the lever rule. It is worth noting that designing the matrix alloy for SHMMCs using the equilibrium phase diagram assumes ideal equilibrium conditions that are rarely found in real systems.