Rise of the Poisson’s Ratio in f.c.c. Hard Sphere Crystals with the Narrowest Orthogonal Nanochannels Filled by Hard Spheres of Another Diameter
Narojczyk Jakub W. 1*, Tretiakov Konstantin V. 1,2†, Wojciechowski Krzysztof W. 1,2‡
1 Polish Academy of Sciences
Institute of Molecular Physics
Smoluchowskiego 17, 60-179 Poznań, Poland2 Calisia University – Kalisz
Nowy Świat 4, 62-800 Kalisz, Poland∗E-mail: narojczyk@ifmpan.poznan.pl
†E-mail: tretiakov@ifmpan.poznan.pl
‡E-mail: kww@ifmpan.poznan.pl
Received:
Received: 13 June 2022; revised: 20 June 2022; accepted: 21 June 2022; published online: 30 June 2022
DOI: 10.12921/cmst.2022.0000015
Abstract:
Auxetic materials, i.e. materials exhibiting negative Poisson’s ratio, stand to answer the demand for novel materials with unique and application-tailored properties. The vast range of potential applications motivates researchers to search for new materials with such properties, or to look for ways to modify the properties of existing materials. The study of systems with structural inclusions falls into the latter category. This work reports numerical investigations of elastic properties of hard sphere f.c.c. crystal. The investigations have been focused on Monte Carlo simulations of systems with arrays of inclusions filled by hard spheres of different diameter, resulting in binary systems, i.e. systems composed of two kinds of particles that differ only in size. Two different layouts of narrow nanoinclusions have been studied in the isobaricisothermal ensemble. It has been shown that even the narrowest inclusions can significantly alter elastic properties of hard particle crystal by eliminating auxetic properties while maintaining the effective cubic symmetry.
Key words:
auxetics, hard sphere crystals, mechanical metamaterials, Monte Carlo simulations, nanochannels, negative Poisson’s ratio
References:
[1] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, London, UK, Pergamon Press (1986).
[2] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235, 1038–1040 (1987).
[3] K.W. Wojciechowski, Constant thermodynamic tension Monte Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers, Mol. Phys. 61, 1247–1258 (1987).
[4] K.W. Wojciechowski, Two-dimensional isotropic model with a negative Poisson ratio, Phys. Lett. A, 137, 60–64 (1989).
[5] K.E. Evans, Auxetic polymers: a new range of materials, Endeavour 15, 170–174 (1991).
[6] L. Mizzi, D. Attard, A. Casha, J.N. Grima, R. Gatt, On the suitability of hexagonal honeycombs as stent geometries, Phys. Status Solidi B 251, 328–337 (2014).
[7] X. Ren, J. Shen, T. Phuong, T. Ngo, Y.M. Xie, Auxetic nail: Design and experimental study, Comp. Struct. 184, 288–298 (2018).
[8] Y.-C. Wang, H.-W. Lai, J.X. Ren, Enhanced auxetic and viscoelastic properties of filled reentrant honeycomb, Phys. Status Solidi B 257(10), 1900184 (2020).
[9] X.Y. Zhang, X. Ren, A simple methodology to generate metamaterials and structures with negative Poisson’s ratio, Phys. Status Solidi B 257(10), 2000439 (2020).
[10] A.C. Brańka, D.M. Heyes, K.W. Wojciechowski, Auxeticity of cubic materials under pressure, Phys. Status Solidi B 248, 96–104 (2011).
[11] R.S. Lakes, Advances in negative Poisson’s ratio materials, Adv. Mater. 5, 293–296 (1993).
[12] K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking!, Adv. Mater. 12, 617–628 (2000).
[13] R. Lakes, Negative-Poisson’s-ratio materials: Auxetic solids, Annual Review of Materials Research 47, 63–81 (2017).
[14] T.-C. Lim, Mechanics of Metamaterials with Negative Parameters, Springer, Singapore (2020).
[15] K.K. Dudek, R. Gatt, M.R. Dudek, J.N. Grima, Controllable hierarchical mechanical metamaterials guided by the hinge design, Materials 14(4), 758 (2021).
[16] O. Duncan, A. Alderson, T. Allen, Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams, Smart Mater. Struct. 30(3), 035014 (2021).
[17] T. Allen, T. Hewage, C. Newton-Mann, W. Wang, O. Duncan, A. Alderson, Fabrication of auxetic foam sheets for sports applications, Status Solidi B 254(12), 700596 (2017).
[18] R.H. Baughman, J.M. Shacklette, A.A. Zakhidov, S. Stafstrom, Negative Poisson’s ratios as a common feature of cubic metals, Nature 392, 362–365 (1998).
[19] D.Y. Fozdar, P. Soman, J.W. Lee, L.H. Han, S.C. Chen, Three-dimensional polymer constructs exhibiting a tunable negative Poisson’s ratio, Adv. Funct. Mater. 21, 2712–2720 (2011).
[20] K. Alderson, S. Nazaré, A. Alderson, Large-scale extrusion of auxetic polypropylene fibre, Phys. Status Solidi B 253(7), 1279–1287 (2016).
[21] K.L. Alderson, V.R. Simkins, V.L. Coenen, P.J. Davies, A.Alderson, K.E. Evans, How to make auxetic fibre reinforced composites, Phys. Status Solidi B 242(3), 509–518 (2005).
[22] D.T. Ho, H. Kim, S.-Y. Kwon, S.Y. Kim, Auxeticity of face-centered cubic metal (001) nanoplates, Phys. Status Solidi B 252(7), 1492–1501 (2015).
[23] R.V. Goldstein, V.A. Gorodtsov, D.S. Lisovenko, M.A. Volkov, Two-layered tubes from cubic crystals: Auxetic tubes, Phys. Status Solidi B 254(12), 1600815 (2017).
[24] V.A. Gorodtsov, M.A. Volkov, D.S. Lisovenko, Out-of-plane tension of thin two-layered plates of cubic crystals, Phys. Status Solidi B 258(12), 2100184, (2021).
[25] D.T. Ho, S. Park, S. Kwon, T. Han, S.Y. Kim, Negative Poisson’s ratio in cubic materials along principal directions, Phys. Status Solidi B 253(7), 1288–1294 (2016).
[26] D.S. Lisovenko, J.A. Baimova, L.K. Rysaeva, V.A. Gorodtsov, A.I. Rudskoy, S.V. Dmitriev, Equilibrium diamond-like carbon nanostructures with cubic anisotropy: Elastic properties, Phys. Status Solidi B 253(7), 1295–1302 (2016).
[27] J.N. Grima-Cornish, L. Vella-Z˙ arb, K.W. Wojciechowski, J.N. Grima, Shearing deformations of β-cristobalite-like boron arsenate, Symmetry 13, 977 (2021).
[28] A.A. Pozniak, K.W. Wojciechowski, J.N. Grima, L. Mizzi, Planar auxeticity from elliptic inclusions, Composites Part B 94, 379–388 (2016).
[29] T. Strek, J. Michalski, H. Jopek, Computational analysis of the mechanical impedance of the sandwich beam with auxetic metal foam core, Phys. Status Solidi B 256(1), 1800423 (2019).
[30] W.G. Hoover, C.G. Hoover, Searching for auxetics with DYNA3D and ParaDyn, Phys. Status Solidi B 242(3), 585– 594 (2005).
[31] K.V. Tretiakov, P.M. Piglowski, K. Hyzorek, K.W. Wojciechowski, Enhanced auxeticity in Yukawa systems due to introduction of nanochannels in [001]-direction, Smart Mater. Struct. 25, 054007 (2016).
[32] E. Pasternak, I. Shufrin, A.V. Dyskin, Thermal stresses in hybrid materials with auxetic inclusions, Comp. Struct. 138, 313–321 (2016).
[33] D.T. Ho, C.T. Nguyen, S.Y. Kwon, S.Y. Kim, Auxeticity in metals and periodic metallic porous structures induced by elastic instabilities, Phys. Status Solidi B 256, 1800122 (2018).
[34] J.W. Narojczyk, M. Bilski, J.N. Grima, P. Kedziora, D. Morozow, M. Rucki, K.W. Wojciechowski, Removing auxetic properties in f.c.c. hard sphere crystals by orthogonal nanochannels with hard spheres of another diameter, Materials 15, 1134 (2022).
[35] K.W. Wojciechowski, K.V. Tretiakov, M. Kowalik, Elastic properties of dense solid phases of hard cyclic pentamers and heptamers in two dimensions, Phys. Rev. E 67, 036121 (2003).
[36] D. Frenkel, Order through entropy, Nature Materials 14, 9–12 (2015).
[37] K.V. Tretiakov, K.W. Wojciechowski, Auxetic, partially auxetic, and nonauxetic behaviour in 2D crystals of hard cyclic tetramers, Phys. Status Solidi RRL 14, 2000198 (2020).
[38] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, Amsterdam, The Netherlands (2006).
[39] K.V. Tretiakov, K.W. Wojciechowski, Poisson’s ratio of the fcc hard sphere crystal at high densities, J. Chem. Phys. 123, 074509 (2005).
[40] J.W. Narojczyk, K.W. Wojciechowski, K.V. Tretiakov, J. Smardzewski, F. Scarpa, P.M. Piglowski, M. Kowalik, A.R. Imre, M. Bilski, Auxetic properties of a f.c.c. crystal of hard spheres with an array of [001]-nanochannels filled by hard spheres of another diameter, Phys. Status Solidi B 256, 1800611 (2019).
[41] K.W. Wojciechowski, A.C. Brańka, Negative Poisson ratio in a two-dimensional isotropic solid, Phys. Rev. A 40, 7222–7225 (1989).
[42] M. Parrinello, A. Rahman, Polymorphic transitions in single crystals: A new molecular dynamics method, J. Appl. Phys. 52, 7182–7190 (1981).
[43] M. Parrinello, A. Rahman, Strain fluctuations and elastic constants, J. Chem. Phys. 76, 2662–2666 (1982).
[44] S.P. Tokmakova, Stereographic projections of Poisson’s ratio in auxetic crystals, Phys. Status Solidi B 242(3), 721–729 (2005).
[45] J.F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices, Oxford, UK, Clarendon Press (1957).
[46] K.V. Tretiakov, P.M. Piglowski, J.W. Narojczyk, K.W. Wojciechowski, Selective enhancement of auxeticity through changing a diameter of nanochannels in Yukawa systems, Smart Mater. Struct. 27, 115021 (2018).
Auxetic materials, i.e. materials exhibiting negative Poisson’s ratio, stand to answer the demand for novel materials with unique and application-tailored properties. The vast range of potential applications motivates researchers to search for new materials with such properties, or to look for ways to modify the properties of existing materials. The study of systems with structural inclusions falls into the latter category. This work reports numerical investigations of elastic properties of hard sphere f.c.c. crystal. The investigations have been focused on Monte Carlo simulations of systems with arrays of inclusions filled by hard spheres of different diameter, resulting in binary systems, i.e. systems composed of two kinds of particles that differ only in size. Two different layouts of narrow nanoinclusions have been studied in the isobaricisothermal ensemble. It has been shown that even the narrowest inclusions can significantly alter elastic properties of hard particle crystal by eliminating auxetic properties while maintaining the effective cubic symmetry.
Key words:
auxetics, hard sphere crystals, mechanical metamaterials, Monte Carlo simulations, nanochannels, negative Poisson’s ratio
References:
[1] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, London, UK, Pergamon Press (1986).
[2] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235, 1038–1040 (1987).
[3] K.W. Wojciechowski, Constant thermodynamic tension Monte Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers, Mol. Phys. 61, 1247–1258 (1987).
[4] K.W. Wojciechowski, Two-dimensional isotropic model with a negative Poisson ratio, Phys. Lett. A, 137, 60–64 (1989).
[5] K.E. Evans, Auxetic polymers: a new range of materials, Endeavour 15, 170–174 (1991).
[6] L. Mizzi, D. Attard, A. Casha, J.N. Grima, R. Gatt, On the suitability of hexagonal honeycombs as stent geometries, Phys. Status Solidi B 251, 328–337 (2014).
[7] X. Ren, J. Shen, T. Phuong, T. Ngo, Y.M. Xie, Auxetic nail: Design and experimental study, Comp. Struct. 184, 288–298 (2018).
[8] Y.-C. Wang, H.-W. Lai, J.X. Ren, Enhanced auxetic and viscoelastic properties of filled reentrant honeycomb, Phys. Status Solidi B 257(10), 1900184 (2020).
[9] X.Y. Zhang, X. Ren, A simple methodology to generate metamaterials and structures with negative Poisson’s ratio, Phys. Status Solidi B 257(10), 2000439 (2020).
[10] A.C. Brańka, D.M. Heyes, K.W. Wojciechowski, Auxeticity of cubic materials under pressure, Phys. Status Solidi B 248, 96–104 (2011).
[11] R.S. Lakes, Advances in negative Poisson’s ratio materials, Adv. Mater. 5, 293–296 (1993).
[12] K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking!, Adv. Mater. 12, 617–628 (2000).
[13] R. Lakes, Negative-Poisson’s-ratio materials: Auxetic solids, Annual Review of Materials Research 47, 63–81 (2017).
[14] T.-C. Lim, Mechanics of Metamaterials with Negative Parameters, Springer, Singapore (2020).
[15] K.K. Dudek, R. Gatt, M.R. Dudek, J.N. Grima, Controllable hierarchical mechanical metamaterials guided by the hinge design, Materials 14(4), 758 (2021).
[16] O. Duncan, A. Alderson, T. Allen, Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams, Smart Mater. Struct. 30(3), 035014 (2021).
[17] T. Allen, T. Hewage, C. Newton-Mann, W. Wang, O. Duncan, A. Alderson, Fabrication of auxetic foam sheets for sports applications, Status Solidi B 254(12), 700596 (2017).
[18] R.H. Baughman, J.M. Shacklette, A.A. Zakhidov, S. Stafstrom, Negative Poisson’s ratios as a common feature of cubic metals, Nature 392, 362–365 (1998).
[19] D.Y. Fozdar, P. Soman, J.W. Lee, L.H. Han, S.C. Chen, Three-dimensional polymer constructs exhibiting a tunable negative Poisson’s ratio, Adv. Funct. Mater. 21, 2712–2720 (2011).
[20] K. Alderson, S. Nazaré, A. Alderson, Large-scale extrusion of auxetic polypropylene fibre, Phys. Status Solidi B 253(7), 1279–1287 (2016).
[21] K.L. Alderson, V.R. Simkins, V.L. Coenen, P.J. Davies, A.Alderson, K.E. Evans, How to make auxetic fibre reinforced composites, Phys. Status Solidi B 242(3), 509–518 (2005).
[22] D.T. Ho, H. Kim, S.-Y. Kwon, S.Y. Kim, Auxeticity of face-centered cubic metal (001) nanoplates, Phys. Status Solidi B 252(7), 1492–1501 (2015).
[23] R.V. Goldstein, V.A. Gorodtsov, D.S. Lisovenko, M.A. Volkov, Two-layered tubes from cubic crystals: Auxetic tubes, Phys. Status Solidi B 254(12), 1600815 (2017).
[24] V.A. Gorodtsov, M.A. Volkov, D.S. Lisovenko, Out-of-plane tension of thin two-layered plates of cubic crystals, Phys. Status Solidi B 258(12), 2100184, (2021).
[25] D.T. Ho, S. Park, S. Kwon, T. Han, S.Y. Kim, Negative Poisson’s ratio in cubic materials along principal directions, Phys. Status Solidi B 253(7), 1288–1294 (2016).
[26] D.S. Lisovenko, J.A. Baimova, L.K. Rysaeva, V.A. Gorodtsov, A.I. Rudskoy, S.V. Dmitriev, Equilibrium diamond-like carbon nanostructures with cubic anisotropy: Elastic properties, Phys. Status Solidi B 253(7), 1295–1302 (2016).
[27] J.N. Grima-Cornish, L. Vella-Z˙ arb, K.W. Wojciechowski, J.N. Grima, Shearing deformations of β-cristobalite-like boron arsenate, Symmetry 13, 977 (2021).
[28] A.A. Pozniak, K.W. Wojciechowski, J.N. Grima, L. Mizzi, Planar auxeticity from elliptic inclusions, Composites Part B 94, 379–388 (2016).
[29] T. Strek, J. Michalski, H. Jopek, Computational analysis of the mechanical impedance of the sandwich beam with auxetic metal foam core, Phys. Status Solidi B 256(1), 1800423 (2019).
[30] W.G. Hoover, C.G. Hoover, Searching for auxetics with DYNA3D and ParaDyn, Phys. Status Solidi B 242(3), 585– 594 (2005).
[31] K.V. Tretiakov, P.M. Piglowski, K. Hyzorek, K.W. Wojciechowski, Enhanced auxeticity in Yukawa systems due to introduction of nanochannels in [001]-direction, Smart Mater. Struct. 25, 054007 (2016).
[32] E. Pasternak, I. Shufrin, A.V. Dyskin, Thermal stresses in hybrid materials with auxetic inclusions, Comp. Struct. 138, 313–321 (2016).
[33] D.T. Ho, C.T. Nguyen, S.Y. Kwon, S.Y. Kim, Auxeticity in metals and periodic metallic porous structures induced by elastic instabilities, Phys. Status Solidi B 256, 1800122 (2018).
[34] J.W. Narojczyk, M. Bilski, J.N. Grima, P. Kedziora, D. Morozow, M. Rucki, K.W. Wojciechowski, Removing auxetic properties in f.c.c. hard sphere crystals by orthogonal nanochannels with hard spheres of another diameter, Materials 15, 1134 (2022).
[35] K.W. Wojciechowski, K.V. Tretiakov, M. Kowalik, Elastic properties of dense solid phases of hard cyclic pentamers and heptamers in two dimensions, Phys. Rev. E 67, 036121 (2003).
[36] D. Frenkel, Order through entropy, Nature Materials 14, 9–12 (2015).
[37] K.V. Tretiakov, K.W. Wojciechowski, Auxetic, partially auxetic, and nonauxetic behaviour in 2D crystals of hard cyclic tetramers, Phys. Status Solidi RRL 14, 2000198 (2020).
[38] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, Amsterdam, The Netherlands (2006).
[39] K.V. Tretiakov, K.W. Wojciechowski, Poisson’s ratio of the fcc hard sphere crystal at high densities, J. Chem. Phys. 123, 074509 (2005).
[40] J.W. Narojczyk, K.W. Wojciechowski, K.V. Tretiakov, J. Smardzewski, F. Scarpa, P.M. Piglowski, M. Kowalik, A.R. Imre, M. Bilski, Auxetic properties of a f.c.c. crystal of hard spheres with an array of [001]-nanochannels filled by hard spheres of another diameter, Phys. Status Solidi B 256, 1800611 (2019).
[41] K.W. Wojciechowski, A.C. Brańka, Negative Poisson ratio in a two-dimensional isotropic solid, Phys. Rev. A 40, 7222–7225 (1989).
[42] M. Parrinello, A. Rahman, Polymorphic transitions in single crystals: A new molecular dynamics method, J. Appl. Phys. 52, 7182–7190 (1981).
[43] M. Parrinello, A. Rahman, Strain fluctuations and elastic constants, J. Chem. Phys. 76, 2662–2666 (1982).
[44] S.P. Tokmakova, Stereographic projections of Poisson’s ratio in auxetic crystals, Phys. Status Solidi B 242(3), 721–729 (2005).
[45] J.F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices, Oxford, UK, Clarendon Press (1957).
[46] K.V. Tretiakov, P.M. Piglowski, J.W. Narojczyk, K.W. Wojciechowski, Selective enhancement of auxeticity through changing a diameter of nanochannels in Yukawa systems, Smart Mater. Struct. 27, 115021 (2018).
