Different statistical mechanical ensembles for a stretched polymer.

Imposing to a single polymer chain of N monomers either a fixed pair of forces +/-f acting at the chain ends (stress ensemble) or a fixed end-to-end vector R (strain ensemble) does correspond to the use of different statistical mechanical ensembles. In particular, the two elasticity laws, R(f)=g(f) and f(R)=h(R), where R(f) is the length of the average end-to-end vector <R>(f) in the stress ensemble and f(R) is the intensity of the average internal force <f>(R) in the strain ensemble, are not equivalent. For these conjugated ensembles, the quantity Delta(f)=f-h(g(f)) and more generally Delta(O)=<O>(f)-<O>(R) where O is an arbitrary observable, is studied systematically in this paper for a wide class of polymer models corresponding to chains at temperatures equal or above the theta point. The leading term Delta((2))(O) of an expansion of Delta(O) in terms of the successive moments of the end-to-end vector fluctuations in the stress ensemble can be used to analyze the scaling properties of Delta(f). For the Gaussian and the freely jointed chain models, Delta(O) proportional to 1/N for large N with the particularity that, for the elasticity law, Delta(f) strictly vanishes for the Gaussian chain at any finite N. For chains in good solvent, the usual result Delta(f) proportional to 1/N at fixed f is only valid in the highly stretched chain regime (Pincus regime). N independent large ensemble differences of the order of 20% on Delta(f) are noticed when the chain is stretched over a distance of the order of the unstretched chain average end-to-end distance R0. These effects decrease to the 1% level for R(f)>3R(0). Monte Carlo calculations for a chain model containing both excluded volume and finite extensibility features illustrate the distinction between the elasticity laws in the two ensembles over all stretching regimes. Our study suggests that the nature of the constraints used in single chain micromanipulations could be relevant to the interpretation of experimental elasticity law data.