Fig. 1. Free energy change of a microscopic system with a nucleus of new phase during nucleation and growth. The phase transformation generally dominates between the critical nuclei size corresponding to the critical nucleation work ΔG? (i.e., the point a) and the grain size corresponding to ΔG = 0 (i.e., the point b), and thereafter, pure growth controlled by the interface energy prevails.

Fig. 2. Thermo-kinetic calculations for the γ/α transformation of Fe-C alloys at different isothermally holding temperatures. (a) Kinetic activation energy as a function of thermodynamic driving force. (b) Ferrite fraction (f) as a function of transformed time (t). (c) Thermodynamic driving force and generalized stability evolving with f. The purple arrow indicates the evolution of thermodynamic driving force with f and the brown arrow indicates the evolution of generalized stability with f. Calculation details regarding the thermodynamic driving force, kinetic activation energy and generalized stability can be found in Supplementary note 2.

Fig. 3. Thermo-kinetic calculations for the γ/α transformation of Fe-0.04 at.% C alloy at isothermally holding temperatures of 1106.4 K and 1107.7 K. (a) Ferrite fraction (f) as a function of transformed time (t) extracted from Ref. [39]. (b) Generalized stability as a function of f. Calculation details regarding the thermodynamic driving force, kinetic activation energy and generalized stability can be found in Supplementary note 2.

Fig. 4. Thermo-kinetic calculations for α/γ transformation and grain growth of α phase in coarse-grained and nanostructured Fe alloys as depicted in Ref. [10]. (a) Thermodynamic driving force and kinetic activation energy for α/γ phase transformation calculated using Eq. (6). (b) Grain boundary energy and kinetic activation energy for grain growth of α phase calculated using Eq. (S3). It is due to the simultaneously increased ΔGchem or grain boundary energy, σ0, and Qb ? Qm (Eq. (6)) or Qb ? Q0 + ΔHseg (Eq. (S3)) that result in the simultaneously increased ΔG and Q for both the nano-scale α/γ transformation and the grain growth of α phase.

Fig. 5. Thermo-kinetic calculations for the martensitic transformation of Fe-0.2C-1Mn-1Si alloy after different austenitization treatments. (a) Calculated effective kinetic energy barrier as a function of thermodynamic driving force with the method adopted in Ref. [41] and Eq. (S25). (b) Relation between f and dQ/dΔG calculated using Eq. (17). (c) Martensite fraction (f) as a function of temperature (T) extracted from Ref. [41]. (d) Thermodynamic driving force and generalized stability evolving with f calculated using Eq. (2) and Eq. (18), respectively. The black and brown arrows indicate the evolution of thermodynamic driving force and generalized stability with f, respectively.

Fig. 6. Philosophy of materials design by the thermo-kinetic connectivity integrating the GS and the thermo-kinetic partition or the activation volume.