# Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
## 3 Synthesis of Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
### Nonequilibrium Solidification  
The melting and solidification of matrix alloys, including Sn-Bi in SHMMCs, rarely achieve equilibrium conditions because of the time-dependent kinetic processes such as diffusion [95; 99; 100]. Micro-segregation occurs across dendrites because of the nonequilibrium solidification of the matrix alloy. During solidification, the resulting liquid becomes enriched in solute, leading to constitutional undercooling ahead of the solidification front. Initially, a spherical solid nucleus forms in the undercooled region of the melt, but as it grows, the spherical morphology becomes unstable, resulting in non-planar (dendritic, columnar) solidification fronts caused by perturbations. As dendrites continue to grow, the inter-dendritic liquid becomes progressively enriched in solute, approaching the eutectic composition. Assuming limited diffusion in the solid, complete mixing in the liquid phase, and equilibrium at the solid-liquid interface, [PERSON]'s equation (Equation (2)) can be employed to determine the solute distribution within a solidified bar [95; 99; 100; 101]. [PERSON]'s equation, also known as the nonequilibrium lever rule, is given by:  
\[\mathrm{Cs}^{*}=\mathrm{k}\;\mathrm{C}_{\mathrm{O}}\left(1-\mathrm{f}_{ \mathrm{S}}\right)\mathrm{{}^{k}}^{-1} \tag{2}\]  
where \(\mathrm{Cs}^{*}\) is the concentration of the solute in the solid, \(\mathrm{C}_{\mathrm{O}}\) the initial concentration of the liquid, \(\mathrm{k}\) the partition coefficient, and \(\mathrm{f}_{\mathrm{S}}\) the fractional distance (solidified) along the bar.  
A nonequilibrium solidification microstructure is beneficial in designing the matrix alloy of SHMMCs. This is because, within the matrix alloy, there will always be inter-dendritic eutectic phases that possess a lower melting point compared to the surrounding matrix. These eutectic phases are capable of melting first, thereby providing a healing liquid that acts as a \"welding\" agent during the healing process of an off-eutectic composition of matrix alloy. This unique characteristic facilitates the effective repair of cracks or defects in the SHMMCs, enhancing their structural integrity.  
Figure 1: Tin–bismuth binary phase diagram. Reprinted/adapted with permission from Ref. [50]. 2013, [PERSON].  
(c) Thermodynamics of Healing Self-healing is a nonequilibrium process in which the restoring thermodynamic force (e.g., diffusion) causes healing by shifting the system away from thermodynamic equilibrium [1]. The healing of the SMA-reinforced SHMMCs can be viewed as a self-organization process that results in increased orderliness or decreased entropy of the material. Placing the system in a metastable state causes it to deviate from thermodynamic equilibrium. When degradation occurs in a system, it breaks the fragile metastable equilibrium and drives the system to its new, most stable state. Metastability is achieved in an SMA-reinforced SHMMC by prestraining the SMA, by heating that causes the phase transformation, and by designing the matrix to partially liquefy at the healing temperature. When heat is applied to the system, the martensitic-austenitic phase transformation of the SMA recovers the strain, resulting in compressive force to the matrix and crack clamping. Furthermore, partial liquefaction of the matrix welds the crack/void via wetting and capillary interactions between the eutectic liquid and the crack/void. These two phenomena result in a more stable configuration for SMA-reinforced SHMMCs by decreasing the entropy (increasing the orderliness) at higher scales. During healing, the entropy production at a particular scale is compensated for by entropy production at another level. Though the orderliness of the SMA-reinforced SHMMC as observed during healing grows (entropy decreases), the excess entropy should be produced at the lower scale. Hence, the strain of the reinforced SMA and volume fraction of liquid during healing can be considered as healing parameters necessary for driving self-healing.