The physical basis for the relative stabilities of the phases at P and T.
Thermodynamics and Phase stabilities
The stability of a phase at a given pressure P and temperature T is conveniently expressed in terms of its Free Energy:
G = U - TS + PV
Here, U is the internal energy, S is the entropy and V is the volume. From the geochemistry module, you might be more familiar with the enthalpy H rather than U. The two quantities are related by H = U + PV. When atoms come together to make a crystal, they want to make G a large negative number. They can do this by making U (or H) a large negative number. At high temperature, however, they might find that going to a new structure with a slightly higher U but an even larger S may give a decrease in the free energy. At high pressure, they will try to make V as small as possible. The structure which is adopted reflects the compromise between U, TS and PV.
Stabilities of the Al_{2}SiO_{5} Polymorphs
We can apply the ideas discussed above to the Al_{2}SiO_{5} polymorphs. The three Al_{2}SiO_{5} polymorphs minerals are very important in metamorphic petrology as they are indicators of metamorphic grade (i.e., metamorphic facies or isograds).
Structures of the Al_{2}SiO_{5} Polymorphs
We can understand the relative stabilities of the Al_{2}SiO_{5} polymorphs in terms of their structures.
Andalusite
Andalusite, ^{VI}Al^{V}AlO[SiO_{4}] has Al in 6 and 5-fold coordination giving the largest molar volume (V= 51.3 cm^{3}/mole) of the three polymorphs; It is stable at the lowest pressures. The entropy (S(298K)= 93.22 J/mol-K) is intermediate between that of kyanite and sillimanite. The unit cell is orthorhombic.
Kyanite
Kyanite, ^{VI}Al_{2}O[SiO_{4}] has all of the Al in 6-fold coordination giving the most dense structure (V = 44.09 cm^{3}/mol). Kyanite is stable at high pressure. However, the high density and low symmetry (triclinic) means it has the lowest entropy S(298 K) = 83.22 J/mol-K. If you look carefully at the unit cell, you can see that it is triclinic.
Sillimanite
Sillimanite, ^{VI}Al[^{IV}AlSiO_{4}] has the greatest entropy (S(298K) = 96.11 J/mol-K) because it has a high symmetry (orthorhombic). The molar volume is V =49.9 cm^{3}/mole. The unit cell is orthorhombic.