An alloy is cooled from a temperature at which its equilibrium structure is a single phase a to a temperature at which two phases, a and are in equilibrium. The phase nucleates heterogeneously at grain boundaries in form of lens-shaped precipitates. a) Assuming a strain free nucleation, derive an expression for the critical radius of nuclei and also show that the critical free energy for heterogeneous nucleation is less than that for homogeneous nucleation. In your derivation you may assume the following: Volume of precipitates=(2+ cose) (1 - cose) Surface area of the precipitate= 4r (1-cose) Where r is the radius of a spherical cap and is the contact angle between the nucleus and [16 marks] the grain boundary. b) Briefly describe how the effect of volumetric strain can be incorporated into the above model.