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A Legendre-Fenchel Transform for Molecular Stretching Energies
Eivind Bering1, Dick Bedeaux2, Signe Kjelstrup2
1PoreLab, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway.
Nanomaterials (Basel, Switzerland)
|December 2, 2020
Summary
Thermodynamics in small systems is explored. Single-molecule polymer stretching reveals Helmholtz and Gibbs energies relate via Legendre-Fenchel transform, applicable to other small systems.
Area of Science:
- Thermodynamics
- Polymer Physics
- Statistical Mechanics
Background:
- Thermodynamics typically applies to large systems, but its applicability to nanoscale systems like single-molecule polymers is an active area of research.
- Previous work showed that isometric and isotensional stretching of small polymers yield Helmholtz and Gibbs energies, differing from the Legendre transform observed in long polymers.
- Experimental evidence supports this thermodynamic disparity in small polymer systems.
Purpose of the Study:
- To investigate the thermodynamic relationship between Helmholtz and Gibbs stretching energies in single-molecule polymers.
- To determine if a Legendre-Fenchel transform can bridge the observed disparity in thermodynamic behavior for small polymer systems.
- To explore the potential application of these findings to other small systems in the context of Hill's thermodynamics.
Main Methods:
- Molecular dynamics simulations were employed to model the stretching of single-molecule polymers.
- Specific simulations focused on polyethylene-oxide to analyze stretching energies under different conditions.
- Thermodynamic relationships, including the Legendre-Fenchel transform, were analyzed between Helmholtz and Gibbs energies.
Main Results:
- For the first time, it was demonstrated that Helmholtz and Gibbs stretching energies in single-molecule polymers can be related by a Legendre-Fenchel transform.
- This finding establishes a connection between thermodynamic potentials previously thought to be unrelated in small systems.
- The simulations confirmed the theoretical predictions regarding stretching energies in isometric and isotensional conditions.
Conclusions:
- The Legendre-Fenchel transform provides a valid method to relate Helmholtz and Gibbs energies in small single-molecule polymer systems.
- This research extends the applicability of thermodynamic transforms to systems that are small in the sense defined by Hill.
- The findings open new avenues for applying advanced thermodynamic concepts to nanoscale phenomena.
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