Modelling equilibrium grain boundary segregation, grain boundary energy and grain boundary segregation transition by the extended Butler equation

The Butler equation is extended to model equilibrium grain boundary (GB) energy and the equilibrium GB composition of a polycrystal, as a function of the following state parameters: bulk composition, temperature, pressure and the five degrees of freedom of the GB. In the simplest case of an ideal solution and equal atomic sizes of the components, the Butler equation reduces back to the well-known McLean equation of GB segregation. When the components repulse each other in the solid solution, grain boundary segregation transition (GBST) appears below the critical temperature of the bulk solid miscibility gap. The GBST line is a new equilibrium line in equilibrium phase diagrams. This new model is demonstrated for copper (Cu) segregation to the GBs in nickel (Ni) and for the phosphorous (P) segregation to the GBs in bcc iron (Fe). The GBST line appears in the Ni-rich (Fe-rich) corner of the Ni-Cu (Fe-P) phase diagram in coordinates of bulk Cu (P) mole fraction vs temperature at fixed pressure. The mole fraction of the solute (Cu or P), corresponding to the GBST line steadily increases with temperature. At a lower solute content (Cu or P), or at a higher temperature compared to the GBST line, the GB is composed mostly of the solvent atoms (Ni or Fe). Contrariwise, at a higher solute content (Cu or P), or at a lower temperature compared to the GBST line, the GB is composed mostly of the solute atoms (Cu or P). These low-segregation and high-segregation states of the GB are transformed into each other via a reversible first-order GBST. This latter process takes place when the GBST line is crossed by changing the bulk composition or the temperature. The results, theoretically estimated, are in agreement with earlier experimental results.