# Mechanism of Spontaneous Surface Modifications on Polycrystalline Cu Due to Electric Fields

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Classification of Surface Atoms

**.**We can see the existence of small number of wrongly identified surface atoms in the bulk. The number of such atoms decreased further at lower temperatures and in the case of undeformed surfaces.

#### 2.2. Generation and Preparation of Simulated Polycrystal

#### 2.3. Simulations with Electric Fields

#### 2.3.1. Simulations with Ramped Electric Fields

#### 2.3.2. Simulations with Constant Applied Electric Fields

#### 2.4. Analysis of Surface Diffusion

#### 2.4.1. Temperature Dependence of Critical Mechanical Stress

#### 2.4.2. Dependence of the Deformation Time with the Force Acting on the Surface Atoms

## 3. Results and Discussion

#### 3.1. Linearly Increasing Stress

#### 3.2. Constant Mechanical Stress

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Pohjonen, A.S.; Parviainen, S.; Muranaka, T.; Djurabekova, F. Dislocation nucleation on a near surface void leading to surface protrusion growth under an external electric field. J. Appl. Phys.
**2013**, 114, 33519. [Google Scholar] [CrossRef] - Calatroni, S.; Kovermann, J.; Taborelli, M.; Timko, H.; Wuensch, W.; Descoeudres, A.; Djurabekova, F.; Kuronen, A.; Nordlund, K.; Pohjonen, A. Breakdown studies for the CLIC accelerating structures. In Proceedings of the LINAC2010, Tsukuba, Japan, 12–17 September 2010; pp. 217–219. Available online: http://epaper.kek.jp/LINAC2010/papers/mop070.pdf (accessed on 6 June 2014).
- Braun, H.H.; Döbert, S.; Wilson, I.; Wuensch, W. Frequency and Temperature Dependence of Electrical Breakdown at 21, 30, and 39 GHz. Phys. Rev. Lett.
**2003**, 90, 224801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Descoeudres, A.; Ramsvik, T.; Calatroni, S.; Taborelli, M.; Wuensch, W. DC breakdown conditioning and breakdown rate of metals and metallic alloys under ultrahigh vacuum. Phys. Rev. Spéc. Top. Accel. Beams
**2009**, 12, 032001. [Google Scholar] [CrossRef] [Green Version] - Descoeudres, A.; Levinsen, Y.; Calatroni, S.; Taborelli, M.; Wuensch, W. Investigation of the dc vacuum breakdown mechanism. Phys. Rev. Spéc. Top. Accel. Beams
**2009**, 12, 092001. [Google Scholar] [CrossRef] [Green Version] - Latham, R.V. High Voltage Vacuum Insulation: Basic Concepts and Technological Practice, 1st ed.; Academic Press: London, UK, 1995. [Google Scholar]
- Miller, R.; Lau, Y.Y.; Booske, J.H. Schottky’s conjecture on multiplication of field enhancement factors. J. Appl. Phys.
**2009**, 106, 104903. [Google Scholar] [CrossRef] - Jimenez, M.; Noer, R.J.; Jouve, G.; Jodet, J.; Bonin, B. Electron field emission from large-area cathodes: Evidence for the projection model. J. Phys. D Appl. Phys.
**1994**, 27, 1038–1045. [Google Scholar] [CrossRef] - Pohjonen, A.S.; Djurabekova, F.; Kuronen, A.; Fitzgerald, S.P.; Nordlund, K. Analytical model of dislocation nucleation on a near-surface void under tensile surface stress. Philos. Mag.
**2012**, 92, 3994–4010. [Google Scholar] [CrossRef] - Kyritsakis, A.; Veske, M.; Eimre, K.; Zadin, V.; Djurabekova, F. Thermal runaway of metal nano-tips during intense electron emission. J. Phys. D Appl. Phys.
**2018**, 51, 225203. [Google Scholar] [CrossRef] [Green Version] - Cicu, P.; Demontis, P.; Spanu, S.; Suffritti, G.B.; Tilocca, A. Electric-field-dependent empirical potentials for molecules and crystals: A first application to flexible water molecule adsorbed in zeolites. J. Chem. Phys.
**2000**, 112, 8267–8278. [Google Scholar] [CrossRef] [Green Version] - Parviainen, S.; Djurabekova, F.; Timko, H.; Nordlund, K. Electronic processes in molecular dynamics simulations of nanoscale metal tips under electric fields. Comput. Mater. Sci.
**2011**, 50, 2075–2079. [Google Scholar] [CrossRef] - Vigonski, S.; Djurabekova, F.; Veske, M.; Aabloo, A.; Zadin, V. Molecular dynamics simulations of near-surface Fe precipitates in Cu under high electric fields. Model. Simul. Mater. Sci. Eng.
**2015**, 23, 25009. [Google Scholar] [CrossRef] - Schiøtz, J. A Maximum in the Strength of Nanocrystalline Copper. Science
**2003**, 301, 1357–1359. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bolesta, A.V.; Fomin, V.M. Molecular dynamics simulation of polycrystalline copper. J. Appl. Mech. Tech. Phys.
**2014**, 55, 800–811. [Google Scholar] [CrossRef] - Cretegny, L.; Saxena, A. AFM characterization of the evolution of surface deformation during fatigue in polycrystalline copper. Acta Mater.
**2001**, 49, 3755–3765. [Google Scholar] [CrossRef] - Jansson, V.; Baibuz, E.; Kyritsakis, A.; Vigonski, S.; Zadin, V.; Parviainen, S.; Aabloo, A.; Djurabekova, F. Growth mechanism for nanotips in high electric fields. Nanotechnology
**2020**, 31, 355301. [Google Scholar] [CrossRef] [PubMed] - Tsong, T.T.; Kellogg, G. Direct observation of the directional walk of single adatoms and the adatom polarizability. Phys. Rev. B
**1975**, 12, 1343–1353. [Google Scholar] [CrossRef] - Kyritsakis, A.; Baibuz, E.; Jansson, V.; Djurabekova, F. Atomistic behavior of metal surfaces under high electric fields. Phys. Rev. B
**2019**, 99, 205418. [Google Scholar] [CrossRef] [Green Version] - Kaur, I.; Mishin, Y.; Gust, W. Fundamentals of Grain and Interphase Boundary Diffusion, 3rd ed.; John Wiley: Chichester, UK; New York, NY, USA, 1995. [Google Scholar]
- Xydou, A.; Parviainen, S.; Aicheler, M.; Djurabekova, F. Thermal stability of interface voids in Cu grain boundaries with molecular dynamic simulations. J. Phys. D Appl. Phys.
**2016**, 49, 355303. [Google Scholar] [CrossRef] - Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Daw, M.S.; Baskes, M.I. Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals. Phys. Rev. Lett.
**1983**, 50, 1285–1288. [Google Scholar] [CrossRef] - Mishin, Y.; Mehl, M.J.; Papaconstantopoulos, D.A.; Voter, A.F.; Kress, J.D. Structural stability and lattice defects in copper:Ab initio, tight-binding, and embedded-atom calculations. Phys. Rev. B
**2001**, 63, 224106. [Google Scholar] [CrossRef] [Green Version] - Djurabekova, F.; Parviainen, S.; Pohjonen, A.; Nordlund, K. Atomistic modeling of metal surfaces under electric fields: Direct coupling of electric fields to a molecular dynamics algorithm. Phys. Rev. E
**2011**, 83, 026704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Veske, M.; Kyritsakis, A.; Eimre, K.; Zadin, V.; Aabloo, A.; Djurabekova, F. Dynamic coupling of a finite element solver to large-scale atomistic simulations. J. Comput. Phys.
**2018**, 367, 279–294. [Google Scholar] [CrossRef] - Veske, M.; Kyritsakis, A.; Djurabekova, F.; Sjobak, K.N.; Aabloo, A.; Zadin, V. Dynamic coupling between particle-in-cell and atomistic simulations. Phys. Rev. E
**2020**, 101, 053307. [Google Scholar] [CrossRef] [PubMed] - Jackson, J.D. Classical Electrodynamics, 3rd ed.; Wiley: New York, NY, USA, 1999. [Google Scholar]
- Hirel, P. Atomsk: A tool for manipulating and converting atomic data files. Comput. Phys. Commun.
**2015**, 197, 212–219. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool. Model. Simul. Mater. Sci. Eng.
**2010**, 18, 015012. [Google Scholar] [CrossRef] - Polak, E. Optimization: Algorithms and Consistent Approximations; Springer: New York, NY, USA, 1997. [Google Scholar]
- Shinoda, W.; Shiga, M.; Mikami, M. Rapid estimation of elastic constants by molecular dynamics simulation under constant stress. Phys. Rev. B
**2004**, 69, 134103. [Google Scholar] [CrossRef] - Schneider, T.; Stoll, E. Molecular-dynamics study of a three-dimensional one-component model for distortive phase transitions. Phys. Rev. B
**1978**, 17, 1302–1322. [Google Scholar] [CrossRef] - Binh, V.T.; Garcia, N.; Purcell, S. Electron Field Emission from Atom-Sources: Fabrication, Properties, and Applications of Nanotips. Adv. Imaging Electron Phys.
**1996**, 95, 63–153. [Google Scholar] [CrossRef] - Nordlund, K.; Djurabekova, F. Defect model for the dependence of breakdown rate on external electric fields. Phys. Rev. Spéc. Top. Accel. Beams
**2012**, 15, 071002. [Google Scholar] [CrossRef] - Larsen, P.M.; Schmidt, S.; Schiøtz, J. Robust structural identification via polyhedral template matching. Model. Simul. Mater. Sci. Eng.
**2016**, 24, 055007. [Google Scholar] [CrossRef] - Honeycutt, J.D.; Andersen, H.C. Molecular dynamics study of melting and freezing of small Lennard-Jones clusters. J. Phys. Chem.
**1987**, 91, 4950–4963. [Google Scholar] [CrossRef] - Antoine, C.Z.; Peauger, F.; le Pimpec, F. Electromigration occurences and its effects on metallic surfaces submitted to high electromagnetic field: A novel approach to breakdown in accelerators. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2011**, 665, 54–69. [Google Scholar] [CrossRef] [Green Version] - Frantz, J.; Rusanen, M.; Nordlund, K.; Koponen, I.T. Evolution of Cu nanoclusters on Cu(100). J. Phys. Condens. Matter
**2004**, 16, 2995. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Perspective and 2D view of the resulting surface in the case of deformed geometry at 900 K. Atoms are color coded by the z-coordinate. A small number of low coordination false positives can be seen in the rightmost graph.

**Figure 2.**Schematic workflow of the simulation process. 20 polycrystalline structures were obtained after the preparation process. Increasing applied field simulations were conducted for each structure at 7 different temperatures. Constant applied field simulations were conducted for a specific structure at 900 K for 10 fractional values of the critical stress. Each of those were repeated twice to obtain adequate statistics.

**Figure 3.**(

**A**) Grain structure underlying the surface. Atoms are colored by PTM structure type—green: fcc atoms, grey—grain boundary atoms, red: hcp atoms (atoms part of a stacking fault). (

**B**) Height map of the surface in the case of critical surface deformation. Atoms are colored by the z-coordinate. Surface protrusion has formed above a quadruple junction.

**Figure 4.**The cross section (

**A**) and a perspective view (

**B**) of a surface protrusion. Atoms are colored by the PTM parameter. Video S1 of the protrusion formation is given in Supplementary Materials.

**Figure 6.**Height mapping (

**left**) and the underlying grain structure (

**right**) of the system subject to constant tensile stress. A clear tendency of self-roughening is seen under the constant stress, which is lower than critical one in the left image.

**Figure 8.**Logarithmic graph of the protrusion formation time under different stress conditions. Each Data point was.averaged over 3 simulations. Least squares fit used to estimate atomic parameters is given by the blue line.

**Figure 9.**(

**A**) Total displacement of atoms until critical surface deformation. (

**B**) Height map of the surface and uderlying grain structure. It can be seen that atoms are most mobile on surface and grain boundary intersections. Only a small surface slide is shown for clarity.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kuppart, K.; Vigonski, S.; Aabloo, A.; Wang, Y.; Djurabekova, F.; Kyritsakis, A.; Zadin, V.
Mechanism of Spontaneous Surface Modifications on Polycrystalline Cu Due to Electric Fields. *Micromachines* **2021**, *12*, 1178.
https://doi.org/10.3390/mi12101178

**AMA Style**

Kuppart K, Vigonski S, Aabloo A, Wang Y, Djurabekova F, Kyritsakis A, Zadin V.
Mechanism of Spontaneous Surface Modifications on Polycrystalline Cu Due to Electric Fields. *Micromachines*. 2021; 12(10):1178.
https://doi.org/10.3390/mi12101178

**Chicago/Turabian Style**

Kuppart, Kristian, Simon Vigonski, Alvo Aabloo, Ye Wang, Flyura Djurabekova, Andreas Kyritsakis, and Veronika Zadin.
2021. "Mechanism of Spontaneous Surface Modifications on Polycrystalline Cu Due to Electric Fields" *Micromachines* 12, no. 10: 1178.
https://doi.org/10.3390/mi12101178