The coil–stretch transition of polymers in external fields

Abstract

We consider the exact statistical mechanical properties of a simple cubic lattice chain with one end anchored at the origin and with all other segments experiencing an external potential of one of two forms: −c‖x‖α (even Hamiltonian) or −c sgn(x)‖x‖α (odd Hamiltonian), for α an arbitrary exponent greater than zero and for c an arbitrary field strength. The problem is exactly soluble numerically for N not too large and for arbitrary α by transfer matrix techniques. In addition, the odd Hamiltonian with α=1 is especially simple to solve, yielding closed form expressions for a number of properties. Both Hamiltonians exhibit a first-order phase transition at c=0 in the limit of large N. The even Hamiltonian exhibits a coil (c<0) to stretch (c>0) transition. The odd Hamiltonian exhibits a left-stretched (c<0) to right-stretched (c>0) transition. For N sufficiently large and for α>1, the entire chain participates in the transition, becoming completely stretched for c only slightly greater than zero. When α<1, the transition is concentrated in one end of the chain. This transition is related to the coil–stretch transition of polymers in elongational flow. This model (unlike real polymers) does not exhibit hysteresis in the position of the transition, in agreement with the generally held belief that such hysteresis is due to hydrodynamic screening in the coil.