Inequivalence of fixed-force and fixed-extension statistical ensembles for a flexible polymer tethered to a planar substrate

Abstract

Recent advances in single macromolecule experiments have sparked interest in the ensemble dependence of force-extension relations. The thermodynamic limit may not be attainable for such systems, which leads to inequivalence of the fixed-force and the fixed-extension ensembles. We consider an ideal Gaussian chain described by the Edwards Hamiltonian with one end tethered to a rigid planar substrate. We analytically calculate the force-extension relation in the two ensembles and we show their inequivalence, which is caused by the confinement of the polymer to half space. The inequivalence is quite remarkable for strong compressional forces. We also perform Monte-Carlo simulations of a tethered wormlike chain with contour length 20 times its persistence length, which corresponds to experiments measuring the conformations of DNA tethered to a wall. The simulations confirm the ensemble inequivalence and qualitatively agree with the analytical predictions of the Gaussian model. Our analysis shows that confinement due to tethering causes ensemble inequivalence, irrespective of the polymer model.

Role of temperature in the decohesion of an elastic chain tethered to a substrate by onsite breakable links

“…Such investigations are coherent with our achievements and prove the possibility to have the nonequivalence between different (canonical) ensembles in statistical mechanics. In systems with strong interactions and low dimensionality, we always observe strong fluctuations, which are persisting also in the thermodynamic limit, thus inducing a different behavior for different statistical ensembles [116,117,119]. In our system the strong interactions, which are nonlocal and long-range, are generated by the ladder network geometry composed of linear and breakable springs.…”

“…In Ref. [116] a Gaussian chain is tethered on a rigid planar surface at one end and the ensemble nonequivalence comes from a pure confinement effect and does not involve any potential interaction, unlike in our case. Different rectangular, spherical and cylindrical geometries of confinement have been considered in Ref.…”

“…Although the equivalence of the Gibbs and Helmholtz statistical ensemble has been proved for a large class of systems (namely, single flexible polymer chains without confinement effects and with a continuous pairing interaction potential between neighboring monomers [86][87][88][89][90][91]), interestingly, it cannot be assumed for other structures. Indeed, other classes of problems, e.g., concerning the escape of a polymer confined between two surfaces and the desorption of a polymer initially tethered onto a surface, exhibit an unusual nonequivalence between the defined statistical ensembles [113][114][115][116][117][118]. In particular, in Refs.…”

“…As discussed in Refs. [116,117,119], the resulting fluctuations of the macroscopic observables cause a nonuniform convergence to the thermodynamic limit eventually producing the ensembles nonequivalence.…”

Elasticity of connected semiflexible quadrilaterals

“…The force-extension relation can be obtained in two different ensembles, namely the fixed-extension ensemble, and the fixed-force ensemble. 86 In the fixed-extension ensemble, the displacement of the tips is fixed and the external force fluctuates due to the thermal fluctuation of the polymer. In the fixed-force ensemble, the external force exerted on the tips of the polymer is fixed, while the displacement of the tips undergoes thermal fluctuations.…”