# Grain Size-Dependent Thermal Expansion of Nanocrystalline Metals

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Classical Molecular Dynamics Modelling

#### 2.2. Ab Initio Modelling

## 3. Results

#### 3.1. Thermal Expansion of Amorphous and Crystalline Morphologies

#### 3.2. Thermal Expansion of Nanocrystalline Microstructures

#### 3.3. Electronic Scale Analysis

## 4. Discussion

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Linear Thermal Expansion

## References

- Kolahalam, L.A.; Viswanath, I.V.K.; Diwakar, B.S.; Govindh, B.; Reddy, V.; Murthy, Y.L.N. Review on nanomaterials: Synthesis and applications. Mater. Today Proc.
**2019**, 18, 2182. [Google Scholar] [CrossRef] - Petford-Long, A.K.; Chiaramonti, A.N. Transmission Electron Microscopy of Multilayer Thin Films. Annu. Rev. Mater. Res.
**2008**, 38, 559. [Google Scholar] [CrossRef] [Green Version] - Piraux, L.; George, J.M.; Despres, J.F.; Leroy, C.; Ferain, E.; Legras, R.; Ounadjela, K.; Fert, A. Giant magnetoresistance in magnetic multilayered nanowires. Appl. Phys. Lett.
**1994**, 65, 2484. [Google Scholar] [CrossRef] - Mikhaylova, M.; Kim, D.K.; Bobrysheva, N.; Osmolowsky, M.; Semenov, V.; Tsakalakos, T.; Muhammed, M. Superparamagnetism of magnetite nanoparticles: Dependence on surface modification. Langmuir
**2004**, 20, 2472. [Google Scholar] [CrossRef] [PubMed] - Cuenot, S.; Frétigny, C.; Demoustier-Champagne, S.; Nysten, B. Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B
**2004**, 69, 165410. [Google Scholar] [CrossRef] [Green Version] - Zhou, L.G.; Huang, H. Are surfaces elastically softer or stiffer? Appl. Phys. Lett.
**2004**, 84, 1940–1942. [Google Scholar] [CrossRef] - Dingreville, R.; Qu, J.; Cherkaoui, M. Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids
**2005**, 53, 1827–1854. [Google Scholar] [CrossRef] - Petrova, H.; Perez-Juste, J.; Zhang, Z.Y.; Zhang, J.; Kosel, T.; Hartland, G.V. Crystal structure dependence of the elastic constants of gold nanorods. J. Mater. Chem.
**2006**, 16, 3957–3963. [Google Scholar] [CrossRef] - Olsson, P.A.T.; Melin, S.; Persson, C. Atomistic simulations of tensile and bending properties of single-crystal bcc iron nanobeams. Phys. Rev. B
**2007**, 76, 224112. [Google Scholar] [CrossRef] - Richter, G.; Hillerich, K.; Gianola, D.S.; Mönig, R.; Kraft, O.; Volkert, C.A. Ultrahigh Strength Single Crystalline Nanowhiskers Grown by Physical Vapor Deposition. Nano Lett.
**2009**, 9, 3048–3052. [Google Scholar] [CrossRef] [Green Version] - Olsson, P.A.T. Transverse resonant properties of strained gold nanowires. J. Appl. Phys.
**2010**, 108, 034318. [Google Scholar] [CrossRef] - Olsson, P.A.T.; Park, H.S.; Lidström, P.C. The Influence of shearing and rotary inertia on the resonant properties of gold nanowires. J. Appl. Phys.
**2010**, 108, 104312. [Google Scholar] [CrossRef] [Green Version] - Olsson, P.A.T.; Park, H.S. Atomistic study of the buckling of gold nanowires. Acta Mater.
**2011**, 59, 3883–3894. [Google Scholar] [CrossRef] - Olsson, P.A.T.; Park, H.S. On the importance of surface elastic contributions to the flexural rigidity of nanowires. J. Mech. Phys. Solids
**2012**, 60, 2064–2083. [Google Scholar] [CrossRef] - Schiøtz, J.; Di Tolla, F.; Jacobsen, K. Softening of nanocrystalline metals at very small grain sizes. Nature
**1998**, 391, 561–563. [Google Scholar] [CrossRef] - Schiøtz, J.; Jacobsen, K.W. A Maximum in the Strength of Nanocrystalline Copper. Science
**2003**, 301, 1357. [Google Scholar] [CrossRef] [Green Version] - Gupta, A.; Gruber, J.; Rajaram, S.S.; Thompson, G.B.; McDowell, D.L.; Tucker, G.J. On the mechanistic origins of maximum strength in nanocrystalline metals. npj Comput. Mater.
**2020**, 6, 153. [Google Scholar] [CrossRef] - Liu, Y.; Zhang, S.; Han, Z.; Zhao, Y. Grain-size-dependent thermal conductivity of nanocrystalline materials. J. Nanopart. Res.
**2016**, 18, 296. [Google Scholar] [CrossRef] - Daniel, R.; Holec, D.; Bartosik, M.; Keckes, J.; Mitterer, C. Size effect of thermal expansion and thermal/intrinsic stresses in nanostructured thin films: Experiment and model. Acta Mater.
**2011**, 59, 6631. [Google Scholar] [CrossRef] - Klam, H.J.; Hahn, H.; Gleiter, H. The thermal expansion of grain boundaries. Acta Metall.
**1987**, 35, 2101–2104. [Google Scholar] [CrossRef] - Lu, T.J.; Fleck, N.A. The thermal shock resistance of solids. Acta Mater.
**1998**, 46, 4755. [Google Scholar] [CrossRef] [Green Version] - Janssen, G.C.A.M. Stress and strain in polycrystalline thin films. Thin Solid Films
**2007**, 515, 6654. [Google Scholar] [CrossRef] - Kuru, Y.; Wohlschlögel, M.; Welzel, U.; Mittemeijer, E.J. Crystallite size dependence of the coefficient of thermal expansion of metals. Appl. Phys. Lett.
**2007**, 90, 243113. [Google Scholar] [CrossRef] - Kuru, Y.; Wohlschlögel, M.; Welzel, U.; Mittemeijer, E.J. Coefficients of thermal expansion of thin metal films investigated by non-ambient X-ray diffraction stress analysis. Surf. Coat. Technol.
**2008**, 202, 2306–2309. [Google Scholar] [CrossRef] - Bogatyrenko, S.; Kryshtal, A. Thermal expansion coefficients of Ag, Cu and diamond nanoparticles: In situ TEM diffraction and EELS measurements. Mater. Charact.
**2021**, 178, 111296. [Google Scholar] [CrossRef] - Birringer, R.; Gleiter, H. Nanocrystalline Materials. In Encyclopedia of Materials Science and Engineering: Supplementary, Vol. 1 (Advances in Materials Science and and Engineering); Cahn, R.W., Ed.; Pergamon Press: Oxford, UK, 1988; pp. 339–349. [Google Scholar]
- Qin, X.Y. Thermal expansion behavior of nanocrystalline silver at high temperatures. Acta Phys. Sin.
**1995**, 44, 244–250. [Google Scholar] - Lu, K.; Sui, M.L. Thermal expansion behaviors in nanocrystalline materials with a wide grain size range. Acta Mater.
**1995**, 43, 3325–3332. [Google Scholar] [CrossRef] - Zhao, Y.H.; Lu, K. Grain-size dependence of thermal properties of nanocrystalline elemental selenium studied by x-ray diffraction. Phys. Rev. B
**1997**, 56, 14330–14337. [Google Scholar] [CrossRef] [Green Version] - Yang, C.C.; Xiao, M.X.; Li, W.; Jiang, Q. Size effects on Debye temperature, Einstein temperature, and volume thermal expansion coefficient of nanocrystals. Solid State Commun.
**2006**, 139, 148. [Google Scholar] [CrossRef] - Yang, L.; Ge, T.; Guo, G.Q.; Huang, C.L.; Meng, X.F.; Wei, S.H.; Chen, D.; Chen, L.Y. Atomic and cluster level dense packing contributes to the high glass-forming ability in metallic glasses. Intermetallics
**2013**, 34, 106. [Google Scholar] [CrossRef] - Eastman, J.A.; Fitzsimmons, M.R.; Thompson, L.J. The thermal properties of nanocrystalline Pd from 16 to 300 K. Philos. Mag. B
**1992**, 66, 667–696. [Google Scholar] [CrossRef] - Eastman, J.A.; Fitzsimmons, M.R.; Thompson, L.J.; Lawson, A.C.; Robinson, R.A. Diffraction studies of the thermal properties of nanocrystalline Pd and Cr. Nanostruct. Mater.
**1992**, 1, 465–470. [Google Scholar] [CrossRef] [Green Version] - Turi, T.; Erb, U. Thermal expansion and heat capacity of porosity-free nanocrystalline materials. Mater. Sci. Eng. A
**1995**, 204, 34–38. [Google Scholar] [CrossRef] - Panigrahi, B.B.; Dabhade, V.V.; Godkhindi, M.M. Thermal expansion behaviour of nanocrystalline titanium powder compacts. Mater. Lett.
**2005**, 59, 2539–2541. [Google Scholar] [CrossRef] - Fang, W.; Lo, C.-Y. On the thermal expansion coefficients of thin films. Sens. Actuator A Phys.
**2000**, 84, 310–314. [Google Scholar] [CrossRef] - Chang, I.-L.; Chang, F.-R. The atomistic study on the thermal expansion behaviors of nanowires. Comput. Mater. Sci.
**2012**, 54, 266. [Google Scholar] [CrossRef] - Zhou, X.-Y.; Huang, B.-L.; Zhang, T.-Y. Size- and temperature-dependent Young’s modulus and size-dependent thermal expansion coefficient of thin films. Phys. Chem. Chem. Phys.
**2016**, 18, 21508. [Google Scholar] [CrossRef] - Plimpton, S.J. Fast parallel algorithms for short-range molecular dynamics. J. Chem. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Thompson, A.P.; Aktulga, H.M.; Berger, R.; Bolintineanu, D.S.; Brown, W.M.; Crozier, P.S.; in ’t Veld, P.J.; Kohlmeyer, A.; Moore, S.G.; Nguyen, T.D.; et al. LAMMPS—A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comput. Phys. Comm.
**2002**, 271, 108171. [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] - Hallberg, H.; Olsson, P.A.T. Investigation of microstructure evolution during self-annealing in thin Cu films by combining mesoscale level set and ab initio modeling. J. Mech. Phys. Solids
**2016**, 90, 160–178. [Google Scholar] [CrossRef] [Green Version] - Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys.
**1984**, 81, 511–519. [Google Scholar] [CrossRef] [Green Version] - Nosé, S. A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys.
**1984**, 52, 255–268. [Google Scholar] [CrossRef] - Hoover, W.G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A
**1985**, 31, 1695. [Google Scholar] [CrossRef] [Green Version] - Evans, D.J.; Holian, B.L. The Nosé–Hoover thermostat. J. Chem. Phys.
**1985**, 83, 4069–4074. [Google Scholar] [CrossRef] - Foley, D.; Coleman, S.P.; Tucker, G.; Tschopp, M.A. Voronoi Based Nanocrystalline Generation Algorithm for Atomistic Simulations; ARL Technical Note. ARL-TN-0806; US Army Research Laboratory: Adelphi, MD, USA, 2016. [Google Scholar]
- Fortunato, M.E.; Mattson, J.; Taylor, D.E.; Larentzos, J.P.; Brennan, J.K. Pre- and Post-Processing Tools to Create and Characterize Particle-Based Composite Model Structures; ARL Technical Report, ARL-TR-8213; US Army Research Laboratory: Adelphi, MD, USA, 2017. [Google Scholar]
- Ackland, G.J.; Jones, A.P. Applications of local crystal structure measures in experiment and simulation. Phys. Rev. B
**2006**, 73, 054104. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO—The Open Visualization Tool. Model. Simul. Mat. Sci. Eng.
**2009**, 18, 015012. [Google Scholar] [CrossRef] - Ozaki, T.; Kino, H. Efficient projector expansion for the ab initio LCAO method. Phys. Rev. B
**2005**, 72, 045121. [Google Scholar] [CrossRef] - Ozaki, T. Variationally optimized atomic orbitals for large-scale electronic structures. Phys. Rev. B
**2003**, 67, 155108. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865. [Google Scholar] [CrossRef] [Green Version] - Ozaki, T.; Kino, H. Numerical atomic basis orbitals from H to Kr. Phys. Rev. B
**2004**, 69, 195113. [Google Scholar] [CrossRef] - Music, D.; Geyer, R.W.; Schneider, J.M. Recent progress and new directions in density functional theory based design of hard coatings. Surf. Coat. Technol.
**2016**, 286, 178–190. [Google Scholar] [CrossRef] - Söderlind, P.; Nordström, L.; Yongming, L.; Johansson, B. Relativistic effects on the thermal expansion of the actinide elements. Phys. Rev. B
**1990**, 42, 4544. [Google Scholar] [CrossRef] [PubMed] - Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B
**1993**, 47, 558–561. [Google Scholar] [CrossRef] - Kresse, G.; Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal amorphous-semiconductor transition in germanium. Phys. Rev. B
**1994**, 49, 14251–14269. [Google Scholar] [CrossRef] [PubMed] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] [PubMed] - Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci.
**1996**, 6, 15–50. [Google Scholar] [CrossRef] - Dronskowski, R.; Blöchl, P.E. Crystal Orbital Hamilton Populations (COHP). Energy-Resolved Visualization of Chemical Bonding in Solids based on Density-Functional Calculations. J. Phys. Chem.
**1993**, 97, 8617–8624. [Google Scholar] [CrossRef] - Deringer, V.L.; Tchougreeff, A.L.; Dronskowski, R. Crystal Orbital Hamilton Population (COHP) Analysis as Projected from Plane-Wave Basis Sets. J. Phys. Chem. A
**2011**, 115, 5461–5466. [Google Scholar] [CrossRef] - Maintz, S.; Deringer, V.L.; Tchougreeff, A.L.; Dronskowski, R. Analytic Projection from Plane-Wave and PAW Wavefunctions and Application to Chemical-Bonding Analysis in Solids. J. Comput. Chem.
**2013**, 34, 2557–2567. [Google Scholar] [CrossRef] - Nelson, R.; Ertural, C.; George, J.; Deringer, V.L.; Hautier, G.; Dronskowski, R. LOBSTER: Local orbital projections, atomic charges, and chemical-bonding analysis from projector-augmented-wave-based density-functional theory. J. Comput. Chem.
**2020**, 41, 1931–1940. [Google Scholar] [CrossRef] [PubMed] - Manz, T.A. Introducing DDEC6 atomic population analysis: Part 3. Comprehensive method to compute bond orders. RSC Adv.
**2017**, 7, 45552–45581. [Google Scholar] [CrossRef] [Green Version] - Blöchl, P.E.; Jepsen, O.; Andersen, O.K. Improved tetrahedron method for Brillouin-zone integrations. Phys. Rev. B
**1994**, 49, 16223–16233. [Google Scholar] [CrossRef] [PubMed] - Blöchl, P.E. Projector augmented-wave method. Phys. Rev. B
**1994**, 50, 17953–17979. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B
**1999**, 59, 1758–1775. [Google Scholar] [CrossRef] - Pearson, W.B. A Handbook of Lattice Spacings and Structures of Metals and Alloys, 1st ed.; Pergamon: Oxford, UK, 1958; pp. 123–130. [Google Scholar]
- Carvill, J. Mechanical Engineer’s Data Handbook, 1st ed.; Butterworth-Heinemann: Oxford, UK, 1993; pp. 218–266. [Google Scholar]
- Wang, L.; Teng, J.; Liu, P.; Hirata, A.; Ma, E.; Zhang, Z.; Chen, M.; Han, X. Grain rotation mediated by grain boundary dislocations in nanocrystalline platinum. Nat. Commun.
**2014**, 5, 4402. [Google Scholar] [CrossRef] [Green Version] - Music, D.; Schmidt, P.; Czigány, Z.; Greczynski, G.; Geyer, R.W.; Hans, M. Electrical resistivity modulation of thermoelectric iron based nanocomposites. Vacuum
**2018**, 157, 384–390. [Google Scholar] [CrossRef] - O’Connell, K.; Regalbuto, J.R. High Sensitivity Silicon Slit Detectors for 1 nm Powder XRD Size Detection Limit. Catal. Lett.
**2015**, 145, 777–783. [Google Scholar] [CrossRef] - Munro, J.M.; Latimer, K.; Horton, M.K.; Dwaraknath, S.; Persson, K.A. An improved symmetry-based approach to reciprocal space path selection in band structure calculations. npj Comput. Mater.
**2020**, 6, 112. [Google Scholar] [CrossRef] - Mavračić, J.; Mocanu, F.C.; Deringer, V.L.; Csányi, G.; Elliott, S.R. Similarity Between Amorphous and Crystalline Phases: The Case of TiO
_{2}. J. Phys. Chem. Lett.**2018**, 9, 2985–2990. [Google Scholar] [CrossRef] - Jaiswal, R.L.; Pandey, B.K.; Mishra, D.; Fatma, H. Thermo-physical Behavior of Nanomaterials with the Change in Size and Shape. Int. J. Thermodyn.
**2021**, 24, 1–7. [Google Scholar] [CrossRef] - Goyal, M.; Gupta, B.R.K. Shape, size and temperature dependency of thermal expansion, lattice parameter and bulk modulus in nanomaterials. Pramana J. Phys.
**2018**, 90, 80. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) An amorphous configuration and (

**b**) a nanocrystal comprising 13 grains with the averaged diameter of 12.9 nm. The Cu atoms are colored using the Ackland-Jones characterization approach [49] such that green, red, blue, and grey particles have FCC, HCP, BCC, and unknown coordination, respectively. The images were generated using the OVITO software [50].

**Figure 2.**(

**a**) Thermally-induced strain for a fully amorphous (AM) configuration and single-crystal (SC) Cu as computed by means of CMD and DFT. Here the reference state corresponds to that at room temperature. (

**b**) Temperature dependent CTE for AM and SC. (

**c**) Radial distribution function and (

**d**) structure factor for AM Cu.

**Figure 3.**(

**a**) Thermally-induced strain for an individual grain as computed by means of CMD for a grain with a $d=25.7$ nm, along with the experimental data for a microstructure with $d=25$ nm [23]. (

**b**) Thermal expansion at room temperature for nanocrystalline Cu as function of the averaged grain diameter. The vertical error bars correspond to one standard deviation. The vertical dotted line indicates the amorphous/crystalline limit below which the material is considered fully amorphous and the horizontal dashed lines correspond to the linear thermal expansion of single-crystalline Cu as computed by means of CMD and DFT. The experimental data are from [23]. (

**c**) CTE map for nanocrystalline Cu as function of the averaged grain diameter for different temperatures, estimated by herein generated DFT data fitted to Equation (2).

**Figure 4.**(

**a**) Total DOS and (

**b**–

**e**) projected DOS for single-crystalline and amorphous Cu. The Fermi level is chosen as reference.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Olsson, P.A.T.; Awala, I.; Holmberg-Kasa, J.; Krause, A.M.; Tidefelt, M.; Vigstrand, O.; Music, D.
Grain Size-Dependent Thermal Expansion of Nanocrystalline Metals. *Materials* **2023**, *16*, 5032.
https://doi.org/10.3390/ma16145032

**AMA Style**

Olsson PAT, Awala I, Holmberg-Kasa J, Krause AM, Tidefelt M, Vigstrand O, Music D.
Grain Size-Dependent Thermal Expansion of Nanocrystalline Metals. *Materials*. 2023; 16(14):5032.
https://doi.org/10.3390/ma16145032

**Chicago/Turabian Style**

Olsson, Pär A. T., Ibrahim Awala, Jacob Holmberg-Kasa, Andreas M. Krause, Mattias Tidefelt, Oscar Vigstrand, and Denis Music.
2023. "Grain Size-Dependent Thermal Expansion of Nanocrystalline Metals" *Materials* 16, no. 14: 5032.
https://doi.org/10.3390/ma16145032