Dear Kang,  the solution of this problem involve the formulation of the global Gibbs free energy of the system, which includes the complete solution of the elastic problem as such that the interfacial diffusion at the grain boundaries should be evaluated assuming that the bulk matrix is saturated with vacancies thereby the bulk diffusion can be overlooked.  Since you have isotropic system with respect to the interface diffusion the final morphology looks like honey comb grain structure with 120 dihedral angles. There would optimal grain size which is determined by the bulk to surface free energy ratio if the applied stress  is hydrostatic pressure up to 100MPa range. The surface of the bulk may have large undulations,which reduces the stored elastic energy at the coherent boundaries.  Otherwise you might have tangled dislocation forms  along the interfaces to relieve the strain energy there. 

We have large amount publications in this field using irreversible thermodynamic theory advocated by us and plus the computer simulation studies. BEST REGARDS

Hi again Mr. Dongsoo

I think the coherency of G' phase is not governed by elastic state between G and G' but by energy state.

According to Minoru Do, etal. (See p111), the energy state ETTL of a coherent particle in an infinite matrix is expressed by:

                                              ETTL=ESURF+ESTR+EINT

Here ESURF is the surface energy of the particle (i.e. the interfacial energy between the particle and the matrix), ESTR is the elastic strain energy due to the lattice mismatch between the particle and the matrix and EINT is the elastic interaction energy between particles. The energy state is important for interpreting the microstructure evolution of two-phase alloys.

According to Shobhan Paul (see p 56) "gamma prime phase exist at two main sizes, primary gamma prime and secondary gamma prime. The primary is much larger than the secondary in size and you can observe it at grain boundaries while the secondary gamma prime is uniformly dissipated within the gamma phase matrix.

Now, if the primary gamma prime volume fraction is reduced, the grain size is increased, even at temperature below the gamma prime solvus temperature.

This means that grain refining or coursing is strongly related to how much primary gamma prime exists at grain boundaries.

Now, in your question you assume grain boundary is refined by something, I think that no think effective to do this refining same as by primary gamma prime volume fraction which is, according to Shobhan Paul (see again p 56), "the primary gamma prime is retained throughout the manufacturing process to prevent the migration of grain boundaries and hence to control the grain size". Remember that migration of grain boundaries is the last step in contacting grains coursing process.

So, the higher volume fraction of  primary gamma prime affect in the interfacial energy for grain boundary which in turn prevent grain coarsening with no effect on the secondary gamma prime phase in grain matrix.

I hope this answer is helpful   

Minoru Do, etal. (See pp111) the link is:

https://www.researchgate.net/publication/264888748_GammaGamma-Prime_Microstructure_Formed_by_Phased_Separation_of_Gamma-Prime_Precipitates_in_a_Ni-Al-Ti_Alloy

Shobhan Paul (see pp 56) link on google is :

https://books.google.iq/books?id=_z23yiPChUUC&pg=PA56&lpg=PA56&dq=gamma+prime+grain+boundary&source=bl&ots=HsqV62E7Ds&sig=Ls5HyJSuhGRGDismZk7kEfYEzVY&hl=ar&sa=X&ved=0ahUKEwiAuOKLlbPMAhVHQpoKHb6KBuEQ6AEIYTAJ#v=onepage&q=gamma%20prime%20grain%20boundary&f=false

Article Gamma/Gamma-Prime Microstructure Formed by Phased Separation...