Calculation of phase diagrams

CALPHAD METHOD
CALPHAD method which means calculation of phase diagrams is used to calculate phase diagrams of metallic ,ceramics, and polymer materials.The CALPHAD method is based on minimization of the free energy of the system and is, thus not only completely general and extensible, but also theoretically meaningful.
There is some thermodynamic consideration for phase equilbiria and transformations:
The subject of thermodynamic stability in materials is of fundamental importance , underpinning considerations of phase equilibiria and phase transformations. Considering a system at constant temperature and pressure , the relative stability is dependent on the Gibbs energy, G defined as: G=H-TS (1)
Where H is the enthalpy,T the absolute temperature, and S the entropy of the system .
The enthalpy relates to the heat content of the system and is expressed as : H=E+PV (2)
Where E is the internal energy of the system,P the pressure and V the volume.
The dependence of the Gibbs free energy on composition at a given temperature of binary solution of metal A and metal B can be given by the following equation:-

XA and XB mole fractions of metal A&B,GA &GB free energy of metal A&B,?HXS the excess enthalpy ,T absolute temperature, and ?S_m the entropy of mixing.
* Gibbs Free Energy of a Mechanical Mixture and Solid Solution:-There is a difference between Gibbs free energy of a mechanical mixture and that of a solid solution. For hypothetical phase diagram of two metals A and B , the possible condensed phases are the pure component A and B, and the solutions of components in each other. If each component has only one crystalline form and the two have the same crystal structure , complete intersolubility will be possible in solid state ,as well as in the liquid state. Such a system may exist at any given temperature, pressure, and overall composition, in any one of the following state:-
-a mechanical mixture of the pure components(mechanical mixture being defined as an intimate association of the two pure components in a state of subdivision large enough so that the percent of atoms within the interphase boundaries is negligibly small).
-a single solution( a solution being defined as a mixture of the components on an atomic scale).
-a mechanical mixture of solutions of different compositions.
Gibbs free energy of a mechanical mixture in any binary system is :-
GM=XAHA+XBHB-T(XASA+XBSB) (4)
XA and XB mole fractions of A and B in the alloy, HA,HB,SA and SB are enthalpies and entropies of the pure components.
For the same alloy existing as a single solution Gibbs free energy is:-


H ?_B-H ?_A: partial molar enthalpy of the component when it is in solution.
(H_A not the same as (H_A ) ? because the environment of A atoms in pure A not the same as that of A atoms in solution of B and A).
the The entropy of mixing: The change in entropy of mixing is purely statistical in nature; it is due to the fact that there are two kinds of atoms mixed together. Before mixing the atoms are in a condition of relatively high order, after mixing the atoms are in much more random arrangement, since both kinds of atoms can move throughout the whole system. The extra randomness is equivalent to an increase in the entropy of mixing. Since this entropy increment is not associated with either component alone or with individual atoms but rather with the solution as a whole .The entropy of mixing is given by:


The enthalpies and entropies in the eqs.(4)& (5) all vary with temperature, pressure, and composition.