  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.
