A Legendre–Fenchel Transform for Molecular Stretching Energies

Abstract

Single-molecular polymers can be used to analyze to what extent thermodynamics applies when the size of the system is drastically reduced. We have recently verified using molecular-dynamics simulations that isometric and isotensional stretching of a small polymer result in Helmholtz and Gibbs stretching energies, which are not related to a Legendre transform, as they are for sufficiently long polymers. This disparity has also been observed experimentally. Using molecular dynamics simulations of polyethylene-oxide, we document for the first time that the Helmholtz and Gibbs stretching energies can be related by a Legendre–Fenchel transform. This opens up a possibility to apply this transform to other systems which are small in Hill’s sense.

Legendre-Fenchel transforms capture layering transitions in porous media

“…For low surface number densities Γ, the system is small, and the Helmholtz energy is non-convex and the integral in equation 10 does not reduce to the Legendre transform. If the distribution of the normal pressure is sharply peaked it can however be calculated with a saddlepoint approximation [17,20,21],…”

“…We have recently reported the changes in free energy during polymer stretching. The free energy depends on the conditions used, whether the polymer is stretched at controlled length or force [16,17]. For sufficiently short polymers, the Helmholtz energy is a non-convex function of the controlled length of the polymer.…”

“…For small systems, this is not always the case. However, in the case of polymer stretching, we have recently found that the Legendre-Fenchel transform can be used [17]. This experience has led us to wonder whether Legendre-Fenchel transforms can be used also for fluids confined to the slit pore, thereby motivating this paper.…”

Special Issue on Nanoscale Thermodynamics

“…The issue contains three extensions of Hill's theory, see papers [6][7][8]. Hill stated that small systems do not obey Legendre transforms, a clear disadvantage.…”

“…Hill stated that small systems do not obey Legendre transforms, a clear disadvantage. Beering et al [6] have been able to show for the first time in their article "A Legendre-Fenchel Transform for Molecular Stretching Energies" that an alternative for small systems lies in the Legendre-Fenchel transform. This is an extension of Hill's theory that may prove useful in practice.…”