Elastic Properties of Hard Sphere Crystals with Tripple (001) Nanolayer Inclusions within a Unit Supercell
Narojczyk Jakub W. 1*, Wojciechowski Krzysztof W. 1,2
1 Institute of Molecular Physics
Polish Academy of Sciences
ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland
∗E-mail: narojczyk@ifmpan.poznan.pl2 President Stanisław Wojciechowski University of Kalisz
Polytechnic Faculty
ul. Nowy Świat 4, 62-800 Kalisz, Poland
Received:
Received: 18 November 2024; accepted: 2 December 2024; published online: 20 December 2024
DOI: 10.12921/cmst.2024.0000020
Abstract:
In this work, the recent studies on hard particle systems containing nanolayer inclusions are extended. Earlier studies showed that systems with nanolayer inclusion can be used to coarse- or fine-tune the auxetic properties of cubic crystals. Here, the impact of spatial distribution of individual inclusion layers on elastic properties of hard sphere crystal of cubic symmetry has been investigated by numerical simulations. The Monte Carlo method with Parinello-Rahman approach in NpT ensemble has been used to evaluate the elastic constants and Poisson’s ratios for six different systems, each containing three nanolayers parallel to each other and orthogonal to [001]-direction. The obtained results are compared with reference systems studied earlier. The studies have been performed for selected thermodynamic conditions. It has been shown that elastic properties weakly depend on the distribution of the inclusions within the structure if the inclusions are not formed by neighbouring layers. Some distributions of the inclusion layers change the period of the structure, which indicates that this factor does not have a big impact on the elastic properties. It is worth stressing that in a particular case the Poisson’s ratio has been found to reach negative values in the [111][1¯10]-directions which are not auxetic in cubic crystals.
Key words:
auxetics, hard spheres, Monte Carlo simulations, nanoinclusions, nanolayers, negative Poisson’s ratio
References:
[1] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon Press, London, UK (1986).
[2] R.F. Almgren, An isotropic three-dimensional structure with Poisson’s ratio =-1, J. Elast. 15, 427–430 (1985).
[3] A.G. Kolpakov, Determination of the average characteristics of elastic frameworks, J. Appl. Math. Mech. 49, 739–745 (1985).
[4] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235, 1038–1040 (1987).
[5] 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).
[6] K.W. Wojciechowski, Two-dimensional isotropic model with a negative Poisson ratio, Phys. Lett. A 137, 60–64 (1989).
[7] K.W. Wojciechowski, A.C. Brańka, Negative Poisson ratio in a two-dimensional isotropic solid, Phys. Rev. A 40, 7222–7225 (1989).
[8] K.W. Wojciechowski, Non-chiral, molecular model of negative Poisson’s ratio in two dimensions, J. Phys. A: Math. Gen. 36, 11765–11778 (2003).
[9] K.E. Evans, Auxetic polymers: a new range of materials, Endeavour 15, 170–174 (1991).
[10] K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking!, Adv. Mater. 12, 617–628 (2000).
[11] A. Alderson, K.L. Alderson, Auxetic materials, Proc. Inst. Mech. Eng. Part G-J. Aerosp. Eng. 221, 565–575 (2007). [12] Y. Prawoto, Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio, Comput. Mater. Sci. 58, 140–153 (2012).
[13] K.E. Evans, M.A. Nkansah, I.J. Hutchinson, S.C. Rogers, Molecular network design, Nature 353, 124 (1991).
[14] A.C. Brańka, D.M. Heyes, K.W. Wojciechowski, Auxeticity of cubic materials under pressure, Phys. Status Solidi B 248, 96–104 (2011).
[15] J. Schwerdtfeger, F. Wein, G. Leugering, R.F. Singer, C. Körner, M. Stingl, F. Schury, Design of auxetic structures via mathematical optimization, Adv. Mater. 23, 2650–2654 (2011).
[16] O. Duncan, T. Shepherd, C. Moroney, Review of auxetic materials for sports applications: Expanding options in comfort and protection, Applied Sciences 8(6), 941 (2018).
[17] T.-C. Lim, Mechanics of Metamaterials with Negative Parameters, Springer, Singapore (2020).
[18] K. Alderson, S. Nazaré, A. Alderson, Large-scale extrusion of auxetic polypropylene fibre, Phys. Status Solidi B 253(7), 1279–1287 (2016).
[19] P. Verma, C.B. He, A.C. Griffin, Implications for auxetic response in liquid crystalline polymers: X-ray scattering and space-filling molecular modeling, Phys. Status Solidi B 257(10), 2000261 (2020).
[20] N. Novak, M. Vesenjak, G. Kennedy, N. Thadhani, Z. Ren, Response of chiral auxetic composite sandwich panel to fragment simulating projectile impact, Phys. Status Solidi B 257(10), 1900099 (2020).
[21] F. Portone, M. Amorini, M. Montanari, R. Pinalli, A. Pedrini, R. Verucchi, R. Brighenti, E. Dalcanale, Molecular auxetic polymer of intrinsic microporosity via conformational switching of a cavitand crosslinker, Adv. Funct. Mater. 33(51), 2307605 (2023).
[22] 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), 1700596 (2017).
[23] H.C. Cheng, F. Scarpa, T.H. Panzera, I. Farrow, H.-X. Peng, Shear stiffness and energy absorption of auxetic open cell foams as sandwich cores, Phys. Status Solidi B 256(1), 1800411 (2019).
[24] O. Duncan, F. Clegg, A. Essa, A.M.T. Bell, L. Foster, T. Allen, A. Alderson, Effects of heat exposure and volumetric compression on Poisson’s ratios, Young’s moduli, and polymeric composition during thermo-mechanical conversion of auxetic open cell polyurethane foam, Phys. Status Solidi B 256(1), 1800393 (2019).
[25] O. Duncan, A. Alderson, T. Allen, Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams, Smart Mater. Struct. 30(3), 035014 (2021).
[26] F. Usta, F. Scarpa, H.S. Turkmen, P. Johnson, A.W. Perriman, Y.Y. Chen, Multiphase lattice metamaterials with enhanced mechanical performance, Smart Mater. Struct. 30(2), 025014 (2021).
[27] P. Verma, K.B. Wagner, A.C. Griffin, M.L. Shofner, Reversibility of out-of-plane auxetic response in needle-punched nonwovens, Phys. Status Solidi B 259(12), 2200387 (2022).
[28] A. Zulifqar, H. Hu, Development of bi-stretch auxetic woven fabrics based on re-entrant hexagonal geometry, Phys. Status Solidi B 256(1), 1800172 (2019).
[29] N. Jiang, H. Hu, Auxetic yarn made with circular braiding technology, Phys. Status Solidi B 256(1), 1800168 (2019).
[30] A. Zulifqar, T. Hua, H. Hu, Single- and double-layered bistretch auxetic woven fabrics made of nonauxetic yarns based on foldable geometries, Phys. Status Solidi B 257(10), 1900156 (2020).
[31] D. Tahir, M. Zhang, H. Hu, Auxetic materials for personal protection: A review, Phys. Status Solidi B 259(12), 2200324 (2022).
[32] J. Smardzewski, R. Klos, B. Fabisiak, Design of small auxetic springs for furniture, Mater. Des. 51, 723–728 (2013).
[33] T. Kuskun, A. Kasal, G. Caglayan, E. Ceylan, M. Bulca, J. Smardzewski, Optimization of the cross-sectional geometry of auxetic dowels for furniture joints, Materials 16, 2838 (2023).
[34] J. Kaufman, K. Farsad, A. Chinubhai, C. Bonsignore, R. Al-Hakim, Auxetic stent increases venous inflow lumen in an ovine animal model, J. Vasc. Interv. Radiol. 34(3), S46–S47 (2023).
[35] L. Meeusen, S. Candidori, L.L. Micoli, G. Guidi, T. Stanković, S. Graziosi, Auxetic structures used in kinesiology tapes can improve form-fitting and personalization, Sci. Rep. 12, 13509 (2022).
[36] M. Kapnisi, C. Mansfield, C. Marijon, A.G. Guex, F. Perbellini, I. Bardi, E.J. Humphrey, J.L. Puetzer, D. Mawad, D.C. Koutsogeorgis, D.J. Stuckey, C.M. Terracciano, S.E. Harding, M.M. Stevens, Auxetic cardiac patches with tunable mechanical and conductive properties toward treating myocardial infarction, Adv. Funct. Mater. 28, 1800618 (2018).
[37] X. Ren, J. Shen, T. Phuong, T. Ngo, Y.M. Xie, Auxetic nail: Design and experimental study, Comp. Struct. 184, 288–298 (2018).
[38] 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).
[39] J.W. Narojczyk, M. Kowalik, K.W. Wojciechowski, Influence of nanochannels on Poisson’s ratio of degenerate crystal of hard dimers, Phys. Status Solidi B 253, 1324–1330 (2016).
[40] P.M. Piglowski, J.W. Narojczyk, K.W. Wojciechowski, K.V. Tretiakov, Auxeticity enhancement due to size polydispersity in fcc crystals of hard-core repulsive Yukawa particles, Soft Matter 13, 7916–7921 (2017).
[41] 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).
[42] K.V. Tretiakov, P.M. Piglowski, J.W. Narojczyk, M. Bilski, K.W. Wojciechowski, High partial auxeticity induced by nanochannels in [111]-direction in a simple model with Yukawa interactions, Materials 11, 2550 (2018).
[43] 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).
[44] J.W. Narojczyk, K.W. Wojciechowski, Poisson’s ratio of the f.c.c. hard sphere crystals with periodically stacked (001)-nanolayers of hard spheres of another diameter, Materials 12, 700 (2019).
[45] J.W. Narojczyk, K.W. Wojciechowski, J. Smardzewski, A.R. Imre, J.N. Grima, M. Bilski, Cancellation of auxetic properties in f.c.c. hard sphere crystals by hybrid layer-channel nanoinclusions filled by hard spheres of another diameter, Materials 14, 3008 (2021).
[46] J.W. Narojczyk, K.W. Wojciechowski, J. Smardzewski, K.V. Tretiakov, Auxeticity tuning by nanolayer inclusion ordering in hard sphere crystals, Materials 17, 4564 (2024).
[47] J.W. Narojczyk, J. Smardzewski, P. Kedziora, K.V. Tretiakov, K.W. Wojciechowski, Fine-tuning of elastic properties by modifying ordering of parallel nanolayer inclusions in hard sphere crystals, Phys. Status Solidi B 261, 2400529 (2024).
[48] R. Ali, M.R. Saleem, M. Roussey, J. Turunen, S. Honkanen, Fabrication of buried nanostructures by atomic layer deposition, Sci. Rep. 8, 15098 (2018).
[49] 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).
[50] D. Frenkel, Order through entropy, Nat. Mater. 14, 9–12 (2015).
[51] 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).
[52] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, Amsterdam, Netherlands, 2006.
[53] K.W. Wojciechowski, Poisson’s ratio of anisotropic systems, Comput. Methods Sci. Technol. 11(1), 73–79 (2005).
[54] M. Parrinello, A. Rahman, Polymorphic transitions in single crystals: A new molecular dynamics method, J. Appl. Phys. 52, 7182–7190 (1981).
[55] M. Parrinello, A. Rahman, Strain fluctuations and elastic constants, J. Chem. Phys. 76, 2662–2666 (1982).
[56] J.H. Weiner, Statistical Mechanics of Elasticity, Wiley, New York, USA, 1983.
[57] S.P. Tokmakova, Stereographic projections of Poisson’s ratio in auxetic crystals, Phys. Status Solidi B 242(3), 721–729 (2005).
[58] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Clarendon Press, Oxford, UK (1957).
In this work, the recent studies on hard particle systems containing nanolayer inclusions are extended. Earlier studies showed that systems with nanolayer inclusion can be used to coarse- or fine-tune the auxetic properties of cubic crystals. Here, the impact of spatial distribution of individual inclusion layers on elastic properties of hard sphere crystal of cubic symmetry has been investigated by numerical simulations. The Monte Carlo method with Parinello-Rahman approach in NpT ensemble has been used to evaluate the elastic constants and Poisson’s ratios for six different systems, each containing three nanolayers parallel to each other and orthogonal to [001]-direction. The obtained results are compared with reference systems studied earlier. The studies have been performed for selected thermodynamic conditions. It has been shown that elastic properties weakly depend on the distribution of the inclusions within the structure if the inclusions are not formed by neighbouring layers. Some distributions of the inclusion layers change the period of the structure, which indicates that this factor does not have a big impact on the elastic properties. It is worth stressing that in a particular case the Poisson’s ratio has been found to reach negative values in the [111][1¯10]-directions which are not auxetic in cubic crystals.
Key words:
auxetics, hard spheres, Monte Carlo simulations, nanoinclusions, nanolayers, negative Poisson’s ratio
References:
[1] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon Press, London, UK (1986).
[2] R.F. Almgren, An isotropic three-dimensional structure with Poisson’s ratio =-1, J. Elast. 15, 427–430 (1985).
[3] A.G. Kolpakov, Determination of the average characteristics of elastic frameworks, J. Appl. Math. Mech. 49, 739–745 (1985).
[4] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235, 1038–1040 (1987).
[5] 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).
[6] K.W. Wojciechowski, Two-dimensional isotropic model with a negative Poisson ratio, Phys. Lett. A 137, 60–64 (1989).
[7] K.W. Wojciechowski, A.C. Brańka, Negative Poisson ratio in a two-dimensional isotropic solid, Phys. Rev. A 40, 7222–7225 (1989).
[8] K.W. Wojciechowski, Non-chiral, molecular model of negative Poisson’s ratio in two dimensions, J. Phys. A: Math. Gen. 36, 11765–11778 (2003).
[9] K.E. Evans, Auxetic polymers: a new range of materials, Endeavour 15, 170–174 (1991).
[10] K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking!, Adv. Mater. 12, 617–628 (2000).
[11] A. Alderson, K.L. Alderson, Auxetic materials, Proc. Inst. Mech. Eng. Part G-J. Aerosp. Eng. 221, 565–575 (2007). [12] Y. Prawoto, Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio, Comput. Mater. Sci. 58, 140–153 (2012).
[13] K.E. Evans, M.A. Nkansah, I.J. Hutchinson, S.C. Rogers, Molecular network design, Nature 353, 124 (1991).
[14] A.C. Brańka, D.M. Heyes, K.W. Wojciechowski, Auxeticity of cubic materials under pressure, Phys. Status Solidi B 248, 96–104 (2011).
[15] J. Schwerdtfeger, F. Wein, G. Leugering, R.F. Singer, C. Körner, M. Stingl, F. Schury, Design of auxetic structures via mathematical optimization, Adv. Mater. 23, 2650–2654 (2011).
[16] O. Duncan, T. Shepherd, C. Moroney, Review of auxetic materials for sports applications: Expanding options in comfort and protection, Applied Sciences 8(6), 941 (2018).
[17] T.-C. Lim, Mechanics of Metamaterials with Negative Parameters, Springer, Singapore (2020).
[18] K. Alderson, S. Nazaré, A. Alderson, Large-scale extrusion of auxetic polypropylene fibre, Phys. Status Solidi B 253(7), 1279–1287 (2016).
[19] P. Verma, C.B. He, A.C. Griffin, Implications for auxetic response in liquid crystalline polymers: X-ray scattering and space-filling molecular modeling, Phys. Status Solidi B 257(10), 2000261 (2020).
[20] N. Novak, M. Vesenjak, G. Kennedy, N. Thadhani, Z. Ren, Response of chiral auxetic composite sandwich panel to fragment simulating projectile impact, Phys. Status Solidi B 257(10), 1900099 (2020).
[21] F. Portone, M. Amorini, M. Montanari, R. Pinalli, A. Pedrini, R. Verucchi, R. Brighenti, E. Dalcanale, Molecular auxetic polymer of intrinsic microporosity via conformational switching of a cavitand crosslinker, Adv. Funct. Mater. 33(51), 2307605 (2023).
[22] 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), 1700596 (2017).
[23] H.C. Cheng, F. Scarpa, T.H. Panzera, I. Farrow, H.-X. Peng, Shear stiffness and energy absorption of auxetic open cell foams as sandwich cores, Phys. Status Solidi B 256(1), 1800411 (2019).
[24] O. Duncan, F. Clegg, A. Essa, A.M.T. Bell, L. Foster, T. Allen, A. Alderson, Effects of heat exposure and volumetric compression on Poisson’s ratios, Young’s moduli, and polymeric composition during thermo-mechanical conversion of auxetic open cell polyurethane foam, Phys. Status Solidi B 256(1), 1800393 (2019).
[25] O. Duncan, A. Alderson, T. Allen, Fabrication, characterization and analytical modeling of gradient auxetic closed cell foams, Smart Mater. Struct. 30(3), 035014 (2021).
[26] F. Usta, F. Scarpa, H.S. Turkmen, P. Johnson, A.W. Perriman, Y.Y. Chen, Multiphase lattice metamaterials with enhanced mechanical performance, Smart Mater. Struct. 30(2), 025014 (2021).
[27] P. Verma, K.B. Wagner, A.C. Griffin, M.L. Shofner, Reversibility of out-of-plane auxetic response in needle-punched nonwovens, Phys. Status Solidi B 259(12), 2200387 (2022).
[28] A. Zulifqar, H. Hu, Development of bi-stretch auxetic woven fabrics based on re-entrant hexagonal geometry, Phys. Status Solidi B 256(1), 1800172 (2019).
[29] N. Jiang, H. Hu, Auxetic yarn made with circular braiding technology, Phys. Status Solidi B 256(1), 1800168 (2019).
[30] A. Zulifqar, T. Hua, H. Hu, Single- and double-layered bistretch auxetic woven fabrics made of nonauxetic yarns based on foldable geometries, Phys. Status Solidi B 257(10), 1900156 (2020).
[31] D. Tahir, M. Zhang, H. Hu, Auxetic materials for personal protection: A review, Phys. Status Solidi B 259(12), 2200324 (2022).
[32] J. Smardzewski, R. Klos, B. Fabisiak, Design of small auxetic springs for furniture, Mater. Des. 51, 723–728 (2013).
[33] T. Kuskun, A. Kasal, G. Caglayan, E. Ceylan, M. Bulca, J. Smardzewski, Optimization of the cross-sectional geometry of auxetic dowels for furniture joints, Materials 16, 2838 (2023).
[34] J. Kaufman, K. Farsad, A. Chinubhai, C. Bonsignore, R. Al-Hakim, Auxetic stent increases venous inflow lumen in an ovine animal model, J. Vasc. Interv. Radiol. 34(3), S46–S47 (2023).
[35] L. Meeusen, S. Candidori, L.L. Micoli, G. Guidi, T. Stanković, S. Graziosi, Auxetic structures used in kinesiology tapes can improve form-fitting and personalization, Sci. Rep. 12, 13509 (2022).
[36] M. Kapnisi, C. Mansfield, C. Marijon, A.G. Guex, F. Perbellini, I. Bardi, E.J. Humphrey, J.L. Puetzer, D. Mawad, D.C. Koutsogeorgis, D.J. Stuckey, C.M. Terracciano, S.E. Harding, M.M. Stevens, Auxetic cardiac patches with tunable mechanical and conductive properties toward treating myocardial infarction, Adv. Funct. Mater. 28, 1800618 (2018).
[37] X. Ren, J. Shen, T. Phuong, T. Ngo, Y.M. Xie, Auxetic nail: Design and experimental study, Comp. Struct. 184, 288–298 (2018).
[38] 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).
[39] J.W. Narojczyk, M. Kowalik, K.W. Wojciechowski, Influence of nanochannels on Poisson’s ratio of degenerate crystal of hard dimers, Phys. Status Solidi B 253, 1324–1330 (2016).
[40] P.M. Piglowski, J.W. Narojczyk, K.W. Wojciechowski, K.V. Tretiakov, Auxeticity enhancement due to size polydispersity in fcc crystals of hard-core repulsive Yukawa particles, Soft Matter 13, 7916–7921 (2017).
[41] 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).
[42] K.V. Tretiakov, P.M. Piglowski, J.W. Narojczyk, M. Bilski, K.W. Wojciechowski, High partial auxeticity induced by nanochannels in [111]-direction in a simple model with Yukawa interactions, Materials 11, 2550 (2018).
[43] 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).
[44] J.W. Narojczyk, K.W. Wojciechowski, Poisson’s ratio of the f.c.c. hard sphere crystals with periodically stacked (001)-nanolayers of hard spheres of another diameter, Materials 12, 700 (2019).
[45] J.W. Narojczyk, K.W. Wojciechowski, J. Smardzewski, A.R. Imre, J.N. Grima, M. Bilski, Cancellation of auxetic properties in f.c.c. hard sphere crystals by hybrid layer-channel nanoinclusions filled by hard spheres of another diameter, Materials 14, 3008 (2021).
[46] J.W. Narojczyk, K.W. Wojciechowski, J. Smardzewski, K.V. Tretiakov, Auxeticity tuning by nanolayer inclusion ordering in hard sphere crystals, Materials 17, 4564 (2024).
[47] J.W. Narojczyk, J. Smardzewski, P. Kedziora, K.V. Tretiakov, K.W. Wojciechowski, Fine-tuning of elastic properties by modifying ordering of parallel nanolayer inclusions in hard sphere crystals, Phys. Status Solidi B 261, 2400529 (2024).
[48] R. Ali, M.R. Saleem, M. Roussey, J. Turunen, S. Honkanen, Fabrication of buried nanostructures by atomic layer deposition, Sci. Rep. 8, 15098 (2018).
[49] 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).
[50] D. Frenkel, Order through entropy, Nat. Mater. 14, 9–12 (2015).
[51] 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).
[52] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, Amsterdam, Netherlands, 2006.
[53] K.W. Wojciechowski, Poisson’s ratio of anisotropic systems, Comput. Methods Sci. Technol. 11(1), 73–79 (2005).
[54] M. Parrinello, A. Rahman, Polymorphic transitions in single crystals: A new molecular dynamics method, J. Appl. Phys. 52, 7182–7190 (1981).
[55] M. Parrinello, A. Rahman, Strain fluctuations and elastic constants, J. Chem. Phys. 76, 2662–2666 (1982).
[56] J.H. Weiner, Statistical Mechanics of Elasticity, Wiley, New York, USA, 1983.
[57] S.P. Tokmakova, Stereographic projections of Poisson’s ratio in auxetic crystals, Phys. Status Solidi B 242(3), 721–729 (2005).
[58] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Clarendon Press, Oxford, UK (1957).
