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# Atomic rheology of gold nanojunctions

## Abstract

Despite extensive investigations of dissipation and deformation processes in micro- and nano-sized metallic samples1,2,3,4,5,6,7, the mechanisms at play during the deformation of systems with ultimate (molecular) size remain unknown. Although metallic nanojunctions, which are obtained by stretching metallic wires down to the atomic level, are typically used to explore atomic-scale contacts5,8,9,10,11, it has not been possible until now to determine the full equilibrium and non-equilibrium rheological flow properties of matter at such scales. Here, by using an atomic-force microscope equipped with a quartz tuning fork, we combine electrical and rheological measurements on ångström-size gold junctions to study the non-linear rheology of this model atomic system. By subjecting the junction to increasing subnanometric deformations we observe a transition from a purely elastic regime to a plastic one, and eventually to a viscous-like fluidized regime, similar to the rheology of soft yielding materials12,13,14, although orders of magnitude different in length scale. The fluidized state furthermore exhibits capillary attraction, as expected for liquid capillary bridges. This shear fluidization cannot be captured by classical models of friction between atomic planes15,16 and points to an unexpected dissipative behaviour of defect-free metallic junctions at ultimate scales. Atomic rheology is therefore a powerful tool that can be used to probe the structural reorganization of atomic contacts.

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## Data availability

The data shown in the plots and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

A.S. acknowledges funding from the European Union’s H2020 Framework Programme/ERC Starting Grant agreement number 637748 – NanoSOFT. L.B. acknowledges funding from the European Union’s H2020 Framework Programme/ERC Advanced Grant – Shadoks.

### Reviewer information

Nature thanks Erio Tosatti and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

A.N., L.B. and A.S. conceived and supervised the project. A.S. and A.N. designed the experiments. J.C. developed the experimental setup. J.C. and A.L. performed the experiments. J.C. wrote the manuscript with input from all the authors.

### Competing interests

The authors declare no competing interests.

Correspondence to Alessandro Siria.

## Extended data figures and tables

### Extended Data Fig. 1 Raw data.

a, b, Raw data showing the frequency shift δf = δω/2π (a) and the excitation voltage E (b) far from the substrate (red curves) and during a typical rheological experiment on a metallic gold contact (N = 11) (blue).

### Extended Data Fig. 2 Effect of excitation frequency on the plastic transition.

Mechanical impedance Z′ and Z″ of the gold junction as a function of oscillation amplitude as of the substrate with fs = 200 kHz. Entry in the plastic regime occurs at the critical substrate oscillation amplitude $${a}_{{\rm{s}}}^{{\rm{c}}}$$. Cross-sectional atom number is = 15.

### Extended Data Fig. 3 Force spectroscopy at large oscillation amplitude.

a, b, Force spectroscopy showing the approach (black) and retraction (blue) of the gold tip to the gold substrate at large oscillation amplitude (1 nm) for the conservative mechanical impedance Z′ (a) and the dimensionless conductance G/G0 (b).

### Extended Data Fig. 4 Energy balance for the shear-induced fluidization of the junction.

Dissipated energy in the junction over one oscillation cycle as a function of melted volume (see Methods for details). The dashed line has a slope of 1.

### Extended Data Fig. 5 Typical rheological curves.

Typical rheological curves for various contact conductances, showing the conservative modulus Z′ (black), the dissipative modulus Z″ (red) and the mean current (blue). The general trends of the rheological curves are maintained, with: (1) a plastic transition at a critical oscillation amplitude aY, corresponding to a decrease in Z′, an increase in Z″ and an increase in current fluctuations; (2) a plateau in the dissipative impedance at large oscillation amplitude; and (3) a decrease in the conservative impedance to negative values Z′ < 0, corresponding to a capillary-like attraction at a critical oscillation amplitude aL.

### Extended Data Fig. 6 Reversibility of the plastic transition.

Rheological curve obtained for increasing (dots) and decreasing (cross) oscillation amplitude (here for N = 20). The plastic transition is found to be completely reversible, with negligible hysteresis.

### Extended Data Fig. 7 Prandtl–Tomlinson model.

a, b, Schematic of the model, with a mass-spring oscillator of mass M and stiffness K (a) interacting with a corrugated potential of N potential wells of corrugation energy U0 and corrugation period b (b). c, Simulation results, showing the dimensionless dissipative and conservative impedance $$\widetilde{Z}=Z/\left(N{U}_{0}/b\right)$$ as a function of dimensionless oscillation amplitude x/b. Inset, dimensionless dissipative force, FD/(NU0/b).

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• #### DOI

https://doi.org/10.1038/s41586-019-1178-3