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 Al2SiO5 Polymorphs
We can apply the ideas discussed above to the Al2SiO5 polymorphs. The three Al2SiO5 polymorphs minerals are very important in metamorphic petrology as they are indicators of metamorphic grade (i.e., metamorphic facies or isograds).
Structures of the Al2SiO5 Polymorphs
We can understand the relative stabilities of the Al2SiO5 polymorphs in terms of their structures.
Andalusite, VIAlVAlO[SiO4] has Al in 6 and 5-fold coordination giving the largest molar volume (V= 51.3 cm3/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, VIAl2O[SiO4] has all of the Al in 6-fold coordination giving the most dense structure (V = 44.09 cm3/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, VIAl[IVAlSiO4] 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 cm3/mole. The unit cell is orthorhombic.