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7. Configurational entropy of polydisperse systems can never reach zero

Author

Baranau, Vasili and Tallarek, Ulrich

Subjects
Condensed Matter - Statistical Mechanics, Condensed Matter - Soft Condensed Matter
Abstract

We present examples of systems whose configurational entropy $S_{\text{conf}}$ can never reach zero and is instead limited from below by the entropy of mixing $S_{\text{mix}}$ of the corresponding ideal gas. We use $S_{\text{conf}}$ defined through the local minima of the potential energy landscape, $S_{\text{conf}}^{\text{PEL}}$. We show that this happens in mean-field models, in collections of hard spheres with infinitesimal polydispersity, and for one-dimensional hard rods. We demonstrate that these results match recent advances in understanding the configurational entropy defined in the free energy landscape, $S_{\text{conf}}^{\text{FEL}}$. We demonstrate that if $\min( S_{\text{conf}}^{\text{FEL}} ) = 0$, then for an arbitrary system $\min( S_{\text{conf}}^{\text{PEL}} ) = A N + S_{\text{mix}}$, where $N$ is the number of particles and $A$ is some constant determined by the interaction potential. We discuss which implications these results have on the Adam--Gibbs (AG) and RFOT relations and show that the latter retain a physically meaningful shape for both configurational entropies, $S_{\text{conf}}^{\text{FEL}}$ and $S_{\text{conf}}^{\text{PEL}}$., Comment: The paper is not scientifically sound and makes several unjustified assumptions. The examples that are sound are unfortunately almost trivial. We did not find a reasonable way to fix or alleviate these problems. These deficiencies were pointed out by several independent reviewers during a peer-review process

Published
2018

9. Another resolution of the configurational entropy paradox as applied to hard spheres

Author

Baranau, Vasili and Tallarek, Ulrich

Subjects
Condensed Matter - Statistical Mechanics
Abstract

Recently, Ozawa and Berthier [J. Chem. Phys., 2017, 146, 014502] studied the configurational and vibrational entropies $S_c$ and $S_v$ from the relation $S_{tot}=S_c+S_v$ for polydisperse mixtures of spheres. They noticed that because $S_{tot}/N$ shall contain the mixing entropy per particle $k_B s_m$ and $S_v/N$ shall not, the configurational entropy per particle $S_c/N$ shall diverge in the thermodynamic limit for continuous polydispersity due to the diverging $s_m$. They also provided a resolution for this paradox and related problems-it relies on a careful redefining of $S_c$ and $S_v$. Here, we note that the relation $S_{tot}=S_c+S_v$ is essentially a geometric relation in the phase space and shall hold without redefining $S_c$ and $S_v$. We also note that the total entropy per particle $S_{tot}/N$ diverges with $N \to \infty$ with continuous polydispersity as well. The usual way to avoid this and other difficulties with $S_{tot}/N$ is to work with the excess entropy $\Delta S_{tot}$ (relative to the ideal gas of the same polydispersity). Speedy [Mol. Phys., 1998, 95, 169] applied this approach to the relation above and wrote this relation as $\Delta S_{tot}=S_c+\Delta S_v$. This form has flows as well, because $S_v/N$ does not contain the $k_B s_m$ term and the latter is introduced into $\Delta S_v/N$ instead. Here, we suggest that this relation shall actually be written as $\Delta S_{tot}=\Delta_c S_c+\Delta_v S_v$, where $\Delta=\Delta_c+\Delta_v$ while $\Delta_c S_c=S_c-k_B N s_m$ and $\Delta_v S_v=S_v-k_B N[1+\ln(V/\Lambda^d N)+U/N k_B T]$ with $\Lambda$ standing for the de Broglie wavelength. In this form, all the terms per particle are always finite for $N \to \infty$ and continuous when introducing a small polydispersity to a monodisperse system. We also suggest that the Adam-Gibbs and related relations shall in fact contain $\Delta_c S_c/N$ instead of $S_c/N$.

Published
2017
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23. On Microstructure-Property Relationships Derived by Virtual Materials Testing with an Emphasis on Effective Conductivity

Author

Neumann, Matthias, Furat, Orkun, Hlushkou, Dzmitry, Tallarek, Ulrich, Holzer, Lorenz, Schmidt, Volker, Barbosa, Simone Diniz Junqueira, Series Editor, Filipe, Joaquim, Series Editor, Kotenko, Igor, Series Editor, Sivalingam, Krishna M., Series Editor, Washio, Takashi, Series Editor, Yuan, Junsong, Series Editor, Zhou, Lizhu, Series Editor, Baum, Marcus, editor, Brenner, Gunther, editor, Grabowski, Jens, editor, Hanschke, Thomas, editor, Hartmann, Stefan, editor, and Schöbel, Anita, editor

Published
2018
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