^{1}, Colleen M. Neal

^{1}, Baopeng Cao

^{1}, Martin F. Jarrold

^{1,a)}, Andrés Aguado

^{2,b)}and José M. López

^{2}

### Abstract

Heat capacities have been measured as a function of temperature for aluminum cluster anions with 35–70 atoms. Melting temperatures and latent heats are determined from peaks in the heat capacities; cohesive energies are obtained for solid clusters from the latent heats and dissociation energies determined for liquid clusters. The melting temperatures, latent heats, and cohesive energies for the aluminum cluster anions are compared to previous measurements for the corresponding cations. Density functional theory calculations have been performed to identify the global minimum energy geometries for the cluster anions. The lowest energy geometries fall into four main families: distorted decahedral fragments, fcc fragments, fcc fragments with stacking faults, and “disordered” roughly spherical structures. The comparison of the cohesive energies for the lowest energy geometries with the measured values allows us to interpret the size variation in the latent heats. Both geometric and electronic shell closings contribute to the variations in the cohesive energies (and latent heats), but structural changes appear to be mainly responsible for the large variations in the melting temperatures with cluster size. The significant charge dependence of the latent heats found for some cluster sizes indicates that the electronic structure can change substantially when the cluster melts.

We gratefully acknowledge the support of the National Science Foundation, Junta de Castilla y León, the Spanish MEC, and the European Regional Development Fund (Project Nos. FIS2008-02490/FIS and GR120).

I. INTRODUCTION

II. EXPERIMENTAL METHODS

III. EXPERIMENTAL RESULTS

IV. COMPUTATIONAL METHODS

V. COMPUTATIONAL RESULTS

VI. DISCUSSION

VII. CONCLUSIONS

### Key Topics

- Aluminium
- 87.0
- Latent heat
- 58.0
- Heat capacity
- 28.0
- Dissociation energies
- 21.0
- Ionization
- 21.0

## Figures

Heat capacities recorded as a function of temperature for aluminum cluster anions with 35–70 atoms. The heat capacities are plotted relative to the classical value , where and is the Boltzmann constant. The filled (blue) squares are the experimental measurements. The solid (blue) lines running through the points are spline fits. The thin black dashed line is the heat capacity derived from a modified Debye model (Ref. 29 ).

Heat capacities recorded as a function of temperature for aluminum cluster anions with 35–70 atoms. The heat capacities are plotted relative to the classical value , where and is the Boltzmann constant. The filled (blue) squares are the experimental measurements. The solid (blue) lines running through the points are spline fits. The thin black dashed line is the heat capacity derived from a modified Debye model (Ref. 29 ).

Comparison of the heat capacities recorded for aluminum cluster anions and cations with 37, 39, 42–44, 60, 61, and 51–57. The heat capacities are plotted relative to the classical value , where and is the Boltzmann constant. The filled (red) points are for the anions and the unfilled black points are for the cations. The solid lines running through the points are spline fits. The thin black dashed line is the heat capacity derived from a modified Debye model (Ref. 29 ).

Comparison of the heat capacities recorded for aluminum cluster anions and cations with 37, 39, 42–44, 60, 61, and 51–57. The heat capacities are plotted relative to the classical value , where and is the Boltzmann constant. The filled (red) points are for the anions and the unfilled black points are for the cations. The solid lines running through the points are spline fits. The thin black dashed line is the heat capacity derived from a modified Debye model (Ref. 29 ).

Examples of fits of the two and three state models to the heat capacities for aluminum cluster anions. The filled black points are the experimental results. The unfilled (blue) circles are simulations with the value of used to determine the heat capacities set to 50 K (the same value as used in the experiments). The solid (blue) line shows heat capacities calculated using . The dashed (blue) line shows the calculated heat capacity without the contribution from the latent heat. The results for and 66 are for the two state model. The lines at the bottom of these plots show the calculated relative abundances (using the scale on the right hand axes) of the solid (dark green) and liquid (light green) clusters as a function of temperature. The results for , 60, 61, and 65 were obtained with the three state model. The lines at the bottom of these plots show the calculated relative abundances of the solid (dark green), intermediate (red), and liquid (light green) clusters as a function of temperature.

Examples of fits of the two and three state models to the heat capacities for aluminum cluster anions. The filled black points are the experimental results. The unfilled (blue) circles are simulations with the value of used to determine the heat capacities set to 50 K (the same value as used in the experiments). The solid (blue) line shows heat capacities calculated using . The dashed (blue) line shows the calculated heat capacity without the contribution from the latent heat. The results for and 66 are for the two state model. The lines at the bottom of these plots show the calculated relative abundances (using the scale on the right hand axes) of the solid (dark green) and liquid (light green) clusters as a function of temperature. The results for , 60, 61, and 65 were obtained with the three state model. The lines at the bottom of these plots show the calculated relative abundances of the solid (dark green), intermediate (red), and liquid (light green) clusters as a function of temperature.

Plot of the melting temperatures determined for aluminum clusters. The filled (red) points show results for the anions and the unfilled points show results for the cations (the data for the cations are taken from Ref. 13 ). Where there are two well-resolved peaks in the heat capacities (i.e., for and ) we show values for both transitions.

Plot of the melting temperatures determined for aluminum clusters. The filled (red) points show results for the anions and the unfilled points show results for the cations (the data for the cations are taken from Ref. 13 ). Where there are two well-resolved peaks in the heat capacities (i.e., for and ) we show values for both transitions.

Plot of the latent heats determined from the heat capacities for aluminum clusters. The filled (red) points show results for the anions and the unfilled points show results for the cations (the data for the cations are taken from Ref. 13 ). Where there are two well-resolved peaks in the heat capacities (i.e., for and ) we show values for both transitions.

Plot of the latent heats determined from the heat capacities for aluminum clusters. The filled (red) points show results for the anions and the unfilled points show results for the cations (the data for the cations are taken from Ref. 13 ). Where there are two well-resolved peaks in the heat capacities (i.e., for and ) we show values for both transitions.

Cohesive energies for aluminum cluster anions compared to corresponding results for the cations (Ref. 18 ). The filled black points show results derived from experimental measurements. The unfilled (red) points show results obtained from calculations using DFT.

Cohesive energies for aluminum cluster anions compared to corresponding results for the cations (Ref. 18 ). The filled black points show results derived from experimental measurements. The unfilled (red) points show results obtained from calculations using DFT.

A selection of cluster anion structures. The number of atoms in the cluster is shown on top of each structure. The structural families identified here are the same as for the cluster cations (Ref. 18 ): ddf around 36 and 53 atoms, disordered isomers around 46 and 66 atoms, and fcc-like fragments (with or without sf) for the rest of the sizes. The disordered isomers for and 46 (which are not shown in the figure) are nearly degenerate with the ordered global minima.

A selection of cluster anion structures. The number of atoms in the cluster is shown on top of each structure. The structural families identified here are the same as for the cluster cations (Ref. 18 ): ddf around 36 and 53 atoms, disordered isomers around 46 and 66 atoms, and fcc-like fragments (with or without sf) for the rest of the sizes. The disordered isomers for and 46 (which are not shown in the figure) are nearly degenerate with the ordered global minima.

Cohesive energies of neutral aluminum clusters obtained from the DFT calculations. The corresponding experimental results cannot be obtained with the present experimental method.

Cohesive energies of neutral aluminum clusters obtained from the DFT calculations. The corresponding experimental results cannot be obtained with the present experimental method.

Calculated adiabatic ionization energies (filled black circles) and electron affinities [filled (red) squares] of neutral aluminum clusters with 13–70 atoms. The dashed lines are fits to a function , which represents the smooth size dependence expected from classical charged sphere models. In the case of the ionization energies, the fits give , in good agreement with the bulk work function . The results for are taken from Ref. 39 and are shown here because they are included in the fits.

Calculated adiabatic ionization energies (filled black circles) and electron affinities [filled (red) squares] of neutral aluminum clusters with 13–70 atoms. The dashed lines are fits to a function , which represents the smooth size dependence expected from classical charged sphere models. In the case of the ionization energies, the fits give , in good agreement with the bulk work function . The results for are taken from Ref. 39 and are shown here because they are included in the fits.

Plot showing the relationship between the melting temperatures of the aluminum cluster anions and cations and the structures of the aluminum clusters as determined from the DFT calculations. The dashed lines show approximately where transitions between different structural families occur for the anions. As noted above, the transition for the cations occurs at similar sizes (usually slightly larger). The labels above the plot identify the structural families described in the text: decahedral fragments, isomers, fragments, and fragments with stacking faults.

Plot showing the relationship between the melting temperatures of the aluminum cluster anions and cations and the structures of the aluminum clusters as determined from the DFT calculations. The dashed lines show approximately where transitions between different structural families occur for the anions. As noted above, the transition for the cations occurs at similar sizes (usually slightly larger). The labels above the plot identify the structural families described in the text: decahedral fragments, isomers, fragments, and fragments with stacking faults.

A schematic diagram showing the relationship between the caloric curves of cluster cations (top curve), neutrals (middle curve), and anions (bottom curve) for a fixed cluster size. Different melting temperatures and latent heats are shown for each charge state. Solid lines show the solid (low temperature) and liquid (high temperature) branches of the caloric curves and dashed lines are employed to represent the extensions of the solid and liquid branches of the caloric curves into the melting region. The latent heats are defined as the energy differences between the solid and liquid branch extensions evaluated at the middle point of the transition region. The slope is always larger in the liquid branches because the liquid has a larger heat capacity than the solid. We only show the temperature interval closely bracketing the melting temperatures, and hence we do not show explicitly that the slope of the caloric curves goes to zero at very low temperatures.

A schematic diagram showing the relationship between the caloric curves of cluster cations (top curve), neutrals (middle curve), and anions (bottom curve) for a fixed cluster size. Different melting temperatures and latent heats are shown for each charge state. Solid lines show the solid (low temperature) and liquid (high temperature) branches of the caloric curves and dashed lines are employed to represent the extensions of the solid and liquid branches of the caloric curves into the melting region. The latent heats are defined as the energy differences between the solid and liquid branch extensions evaluated at the middle point of the transition region. The slope is always larger in the liquid branches because the liquid has a larger heat capacity than the solid. We only show the temperature interval closely bracketing the melting temperatures, and hence we do not show explicitly that the slope of the caloric curves goes to zero at very low temperatures.

## Tables

The difference between the measured latent heats of the anions and cations, , for representative cluster sizes is compared to the values obtained from the calculations, . is the theoretical prediction for the difference between the latent heats of neutrals and cations.

The difference between the measured latent heats of the anions and cations, , for representative cluster sizes is compared to the values obtained from the calculations, . is the theoretical prediction for the difference between the latent heats of neutrals and cations.

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