# A Legendre–Fenchel Transform for Molecular Stretching Energies

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Simulation Details

_{3}−[O−CH

_{2}−CH

_{2}]

_{n}−O−CH

_{3}in molecular dynamics simulations. It is a united-atom model with each bead representing either a methyl group, a methylene group or an oxygen atom. This model is based on a common model documented in the literature [18,19,20], and has all the standard contributions to the potential energy from bond stretching, bending, and torsion, and includes also the breaking of bonds. It therefore lends itself well to a testing of the stretching energies. In this particular force-field, the standard harmonic bond stretching potential is replaced by a Morse potential

#### 2.2. Energy Transforms

## 3. Simulation Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Force as a function of length per bond from isometric and isotensional simulations for chains of poly-ethylene oxide (PEO) composed of $N=12$, 24 and 51 united atoms. The ensemble inequivalence is most pronounced for the smallest systems.

**Figure 2.**Energy as a function of length per bond for chains of PEO composed of $N=12$, 24 and 51 united atoms. While the Legendre transform of the Helmholtz energy F is different from minus the Gibbs energy G, we see that the Legendre–Fenchel transform ${G}_{\mathrm{LF}}$ is an excellent approximation in all three cases.

**Figure 3.**Energy as a function of force for chains of PEO of composed of $N=12$ and 24 united atoms. The smallest system displays multiple singularities, one of which is emphasized in the insert. Although less pronounced, singularities can be seen also in the system with $N=24$.

**Figure 4.**The force-elongation curve ${x}_{\mathrm{LF}}$ computed from the Legendre–Fenchel transform cf. Equation (17) is compared to the force-elongation curves from Figure 1. We recognize the singular points in ${G}_{\mathrm{LF}}$ as jumps in ${x}_{\mathrm{LF}}\left(f\right)$, particularly visible in the smallest system with $N=12$.

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**MDPI and ACS Style**

Bering, E.; Bedeaux, D.; Kjelstrup, S.; de Wijn, A.S.; Latella, I.; Rubi, J.M.
A Legendre–Fenchel Transform for Molecular Stretching Energies. *Nanomaterials* **2020**, *10*, 2355.
https://doi.org/10.3390/nano10122355

**AMA Style**

Bering E, Bedeaux D, Kjelstrup S, de Wijn AS, Latella I, Rubi JM.
A Legendre–Fenchel Transform for Molecular Stretching Energies. *Nanomaterials*. 2020; 10(12):2355.
https://doi.org/10.3390/nano10122355

**Chicago/Turabian Style**

Bering, Eivind, Dick Bedeaux, Signe Kjelstrup, Astrid S. de Wijn, Ivan Latella, and J. Miguel Rubi.
2020. "A Legendre–Fenchel Transform for Molecular Stretching Energies" *Nanomaterials* 10, no. 12: 2355.
https://doi.org/10.3390/nano10122355