Elementary Thermodynamics of the Grain Size Dependence of Phase Transitions

Theory

The various examples described above show that the phase transitions, and more generally the phase diagrams of a material should be considered in a temperature–pressure–grain size space (or rather, reciprocal radius of curvature R for the latter degree of freedom). However, a general theory has not yet been put together. In this context, it is interesting to see what standard thermodynamics has to say about the matter:

Fig. 2.6. Temperature of the tetragonal–cubic phase transition in BaTiO₃ as a function of the grain size. Calculated result from and experimental result from

• The stable states of the system are no longer governed by the free enthalpy G at constant T and P. This role is fulfilled by a generalised free enthalpy function G∗ given by  G^∗ = G − 2γV/R .

• G^∗ is no longer a state function of the system. γ is the surface energy, V the molar volume of the material, and R the radius of the nanocrystal, assumed spherical.

• The equilibrium state of a system made from nanometric grains is no longer obtained by the condition dG = 0, but rather by dG∗ = 0.

• There is therefore equilibrium between the phase α and the phase β of the same body if  G^∗_α = G^∗_β.

• For given temperature and pressure, there is a critical radius Rc at which the phase transition occurs in nanometric grains.

• So the temperature Tc of the phase transition is, in particular, a function of the radius R of the nanocrystals:

Tc = T(P,R, γα, γβ) . This relation has been clearly demonstrated for the tetragonal–cubic transition in BaTiO₃.