# Entropy and Phase Coexistence in Clusters: Metals vs. Nonmetals

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## Abstract

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## 1. Introduction

## 2. Dielectric Cluster Behavior

**Figure 1.**Configurational excitation of a cluster of 13 atoms bound by short-range pair interactions.

**Figure 2.**Structures of the lowest configurationally stable states for the 13-particle Lennard-Jones cluster at zero temperature [38].

**Figure 3.**The excitation energies and barrier energies for the lowest configuration excitations of the 13-atom Lennard-Jones cluster, according to [38].

**Figure 4.**A schematic representation of the single-particle excitations of a 13-atom cluster with short-range interparticle forces. When this schema is converted into an energy space surface structure, positions 1 and 1’ and also the positions 2 and 2’ are joined, and the cluster surface consists of 20 equilateral triangles.

**Figure 5.**(a) The dependence on the excitation energy of the anharmonic parameter (3)for an isolated Lennard-Jones cluster of 13 atoms [40,41] obtained from computer simulation [6]; (b), the temperature dependence of the anharmonic parameter (3) for the solid and liquid states [2,43] obtained on the basis of the results of computer simulation [6].

**Figure 7.**Evolution in time of the total potential energy of cluster atoms with short-time fluctuations averaged (a), for the 13-atom Lennard-Jones cluster under isothermal conditions [7] (b) and under adiabatic conditions [8] at the excitation energy of 10.8D (the melting point, i.e., the point of equal chemical potentials, corresponds to the excitation energy of 13.8D) (c). The data (b) and (c) results from computer simulations, time is measured in arbitrary units.

## 3. The Phase Transition in Metal Clusters

#### 3.1. Entropy of Melting for Small Metal Clusters

**Table 1.**Melting parameters for bulk argon and metals. The fourth column gives the melting point temperature per atomic binding energy in the cluster, ${\epsilon}_{0}$.

Bulk | ${\epsilon}_{o},\phantom{\rule{0.277778em}{0ex}}eV$ | ${T}_{m},\phantom{\rule{0.277778em}{0ex}}K$ | ${T}_{m}/{\epsilon}_{o}$ | $\Delta {H}_{fus},\phantom{\rule{0.277778em}{0ex}}eV$ | $\Delta {H}_{fus}/{\epsilon}_{o},\phantom{\rule{0.277778em}{0ex}}\%$ |
---|---|---|---|---|---|

Ar | 0.068 | 83.7 | 0.106 | 0.0123 | 18 |

Ni | 4.13 | 1728 | 0.036 | 0.181 | 4.4 |

Cu | 3.40 | 1358 | 0.034 | 0.138 | 4.1 |

Ag | 2.87 | 1235 | 0.037 | 0.120 | 4.2 |

Au | 3.65 | 1337 | 0.032 | 0.130 | 3.6 |

**Figure 9.**Schematic representation of the potential energy surface for configurational excitation of a dielectric cluster (a) and for a metal cluster (b) with its many intersecting PES’s.

**Figure 10.**The numbers of isomers for metal clusters of 13 atoms as functions of the excitation energy–in effect, the density of configurational states. a) $N{i}_{13}$, b) $A{g}_{13}$, c) $A{u}_{13}$ [54].

**Table 2.**The binding energy ${E}_{b}$ of cluster atoms and configurational excitation $\Delta \epsilon $ for 13-atom clusters of coinage metals. Here, the vibrational or thermal contributions $\delta {S}_{t}$ to the entropy change in the phase change and the configurational contributions $\Delta {S}_{con}$ are given separately, as well as their sum, $\Delta S$ and the percentage of that total due to configurational excitation.

Cluster | $N{i}_{13}$ | $A{g}_{13}$ | $A{u}_{13}$ |
---|---|---|---|

${E}_{b},\phantom{\rule{0.277778em}{0ex}}eV$ | 44.11 | 27.87 | 41.96 |

$\Delta \epsilon ,\phantom{\rule{0.277778em}{0ex}}eV$ | 0.73 | 0.66 | 0.11 |

$\Delta \epsilon /{E}_{b},\phantom{\rule{0.277778em}{0ex}}\%$ | 1.6 | 2.4 | 0.26 |

$\Delta E,\phantom{\rule{0.277778em}{0ex}}eV$ | 1.0 | 1.1 | 0.4 |

$dE/dn,\phantom{\rule{0.277778em}{0ex}}meV$ | 1.9 | 0.52 | 0.46 |

${T}_{m},\phantom{\rule{0.277778em}{0ex}}K$ | 860 | 820 | 440 |

$\delta {S}_{t}$ | 9.8 | 10.6 | 6.1 |

$\Delta {S}_{con}$ | 3.1 | 4.3 | 3.8 |

$\Delta S$ | 13 | 15 | 10 |

$\Delta {S}_{con}/\Delta S$, % | 24 | 29 | 38 |

**Figure 12.**The the root mean square of the bond length fluctuation for the cluster $A{g}_{13}$ [54]. 1 - the solid state, 2 - the melting range, 3 - the liquid cluster state.

#### 3.2. Phase Coexistence in Small Metal Clusters

**Figure 13.**The caloric curve for the $A{g}_{13}$ cluster [54]. a - the solid state, b - the melting range, c - the liquid cluster state.

## 4. Conclusions

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Berry, R.S.; Smirnov, B.M.
Entropy and Phase Coexistence in Clusters: Metals *vs*. Nonmetals. *Entropy* **2010**, *12*, 1303-1324.
https://doi.org/10.3390/e12051303

**AMA Style**

Berry RS, Smirnov BM.
Entropy and Phase Coexistence in Clusters: Metals *vs*. Nonmetals. *Entropy*. 2010; 12(5):1303-1324.
https://doi.org/10.3390/e12051303

**Chicago/Turabian Style**

Berry, Richard Stephen, and Boris Michailovich Smirnov.
2010. "Entropy and Phase Coexistence in Clusters: Metals *vs*. Nonmetals" *Entropy* 12, no. 5: 1303-1324.
https://doi.org/10.3390/e12051303