Exact solution of the Zwanzig-Lauritzen model of Polymer Crystallization under Tension

We solve a two dimensional model for polymer chain folding in the presence of mechanical pulling force ($f$) exactly using equilibrium statistical mechanics. Using analytically derived expression for the partition function we determine the phase diagram for the model in the $f$-temperature ($T$) plane. A square root singularity in the susceptibility indicates a second order phase transition from a folded to an unfolded state at a critical force ($f_c$) in the thermodynamic limit of infinitely long polymer chain. Surprisingly, the temperature dependence of $f_c$ shows a reentrant phase transition, which is reflected in an increase in $f_c$ as $T$ increases below a threshold value. For a range of $f$ values, the unfolded state is stable at both low and high temperatures. The high temperature unfolded state is stabilized by entropy whereas the low temperature unfolded state is dominated by favorable energy. The exact calculation could serve as a bench mark for testing approximate theories that are used in analyzing single molecule pulling experiments.