A lattice model of semiflexible linear chains (with equilibrium polydispersity) containing free volume is solved exactly on a Husimi cactus. A metastable liquid (ML) is discovered to exist only at low temperatures and is distinct (and may be disjoint) from the supercooled liquid (SCL) that exists only at high temperatures. The free volume plays a significant role in that the spinodals of the ML and SCL merge and then disappear as the free volume is reduced. The Kauzmann temperature $T_{\mathrm{K}}$ occurs in the ML without any singularity. At $T_{\mathrm{MC}}>T_{\mathrm{K}},$ the ML specific heat has a peak. For infinitely long polymers, the peak height diverges and the free volume vanishes at $T_{\mathrm{MC}},$ resulting in a continuous liquid-liquid transition. Contrary to the conventional wisdom, both $T_{\mathrm{K}}$ and $T_{\mathrm{MC}}$ occur in the ML and not in the SCL.

• Received 17 July 2002

DOI:https://doi.org/10.1103/PhysRevE.67.052501