Isometric and Isotensional Force‐Length Profiles in Polymethylene Chains

Abstract

Summary: The force‐length curves and related thermodynamic quantities of single polymethylene (PM) chains in the high‐force region were calculated by the statistical mechanics. Two statistical mechanics ensembles (isometric and isotensional) were used to represent the chain stretching under the conditions, respectively, of the fixed length L or of the fixed force F in single‐chain experiments by AFM and related techniques. The input deformation potentials of highly extended conformations of PM chains were obtained from the molecular‐mechanics calculations. Variations of the energy, entropy, and Helmholtz energy with the end‐to‐end length L of chains were computed under the condition of fixed length. The ensuing isometric profile of the mean force 〈F〉(L) is non‐monotonic, featuring a sawtooth‐like pattern of ascending peaks. In contrast, the mechanical equation under isotensional conditions, given by the variation of the mean length 〈L〉(F), shows a conventional monotonous shape with a distinct plateau region at low temperatures. The considerable difference in the shapes of 〈F〉(L) and 〈L〉(F) curves arises from the large fluctuations of mean values due to the small number of conformers present in molecules at high strains. The computed data should be relevant to the AFM stretching experiments on short polyethylenes or on other soft matter materials involving linear paraffinic chains. It is argued that the dual character of elastic response described by the conjugated force profiles is a universal feature of mechanochemistry of chain molecules whenever the chain length discontinuously increases at a transition.Force‐length curves of short polymethylene chains at 300 K.magnified imageForce‐length curves of short polymethylene chains at 300 K.

Free Energy of Deformation of the Radius of Gyration in Semiflexible Chains

“…In recent years it had been realized that elasticity laws of single macromolecules critically depend on which variables are held fixed in experiments or calculations and which are allowed to fluctuate 2–7. The vector R connecting the ends of a chain and the vector force f acting on the chain ends and their scalar equivalents R and f are the state variables.…”

Stretching a self-interacting semiflexible polymer

“…It is known that the two ensembles -fixed distance and fixed force -are equivalent only in the thermodynamical limit [12,22]. It will either be obvious from the context or specified in each case which ensemble we use in the following.…”

“…The multi-peak behaviour is similar to the one observed in refs. [12,22] for semi-flexible polymers and polymethylene chains. In all these cases the peak pattern was usually ascending, whereas in the present one it displays a more irregular behaviour.…”

Theoretical Approaches for Understanding the Interplay Between Stress and Chemical Reactivity

“…Controlled separation corresponds to fixing the distance, R, between the groups associated with the degree of freedom to which F is applied and measuring the F required to maintain R. Such conditions apply, for example, to forceextension curves obtained in AFM experiments in which the end-to-end distance of a molecule is fixed and the F required to keep the distance fixed is measured. Measuring F for multiple values of R yields a force-extension curve, which can be useful in determining the values of F needed to induce processes such as the unfolding proteins [35,56,57] or the rupture of polymers [58,59]. At the other end of the spectrum, isotensional conditions involve subjecting a molecule to a constant tensile force, F, and allowing the affected degrees of freedom to change as needed to accommodate F. Such conditions have been achieved in AFM experiments of stretched macromolecules [7,60] and the ultrasonic cleavage of metalligand coordination complexes [61][62][63], for example.…”