Entropic Elasticity of Weakly Perturbed Polymers

Taking into account the nonequivalence of fixed-force and fixed-length ensembles in the weak-force regime, equations of state are derived that describe the equilibrium extension or compression of an ideal Gaussian polymer chain in response to an applied force in such a manner that the calculated unstretched scalar end-to-end separation is the random-coil size rather than zero. The entropy-spring model for a polymer chain is thereby modified so that for calculational purposes, the spring is of finite rather than zero unstretched length. These force laws are shown to be consistent with observations from stretching experiments performed on single DNA molecules, wherein the measured extension approaches a non-zero limit as the external force is reduced. When used to describe single-chain dynamics, this approach yields a single exponential relaxation expression for a short Gaussian chain (bead-spring dumbbell), which when initially compressed or extended relaxes into a state having the random-coil end separation, in agreement with the Rouse-model result. An equation is derived that describes the elongational response of a charged, tethered chain to a weak electric field (as might occur in electrophoresis), a calculation not feasible with the traditional approach. Finally, an expression for the entropic work required to bring the ends of a chain together, starting from the random-coil configuration, is derived and compared with the Hookean result.