Shape-Memory Effect and the Topology of Minimal Surfaces
Abstract
:1. Introduction
2. Minimal Surfaces
3. Deformation Paths of Martensitic Transformations
4. Density Functional Calculations of Martensitic Transformation Paths
5. Summary and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
SME | Shape-memory effect |
TPMS | Triply periodic minimal surface |
BCC | Body-centered cubic |
FCC | Face-centered cubic |
HCP | Hexagonal close-packed |
BCT | Body-centered tetragonal |
FCT | Face-centered–tetragonal |
KS | Kurdjumow–Sachs |
NW | Nishiyama–Wassermann |
BB | Bogers–Burgers |
OC | Olson–Cohen |
DFT | Density functional theory |
Appendix A
Phase | Surface | Parameters | ||
---|---|---|---|---|
B2 | Schwarz P | Equation (A3) [24,54] | 1 | |
B19′ | Family | Equation (A5) [24,54] | 1 | |
B19 | Family | Equation (A3) [24] | 1 | |
B33 | Family | Equation (A3) [24,54] | 1 | |
B32 | Schwarz D | Equation (A3) [15] | ||
R | Family | Equation (A5) [24,54] | 1 | |
109°–B19′ | Family | Equation (A5) [24,54] | 1 |
Appendix B
Phase | (T) | (T) | (T) | Parameters | |||
---|---|---|---|---|---|---|---|
B2 [35,56,57] | 1 | 1 | 1 | 1 | , | ||
B19′ [35,58] | 1.13 | 1.60 | 1.13 | 1.60 | 98° | 97.82° | , |
B19′ [35,59] | 1.13 | 1.60 | 1.13 | 1.60 | 97.78° | 97.70° | , |
B19′ [35] | 1.15 | 1.60 | 1.15 | 1.60 | 98° | 97.69° | , |
B19′ [56] | 1.16 | 1.60 | 1.16 | 1.60 | 98.97° | 97.92° | , |
B19 [35,60] | 1.56 | 1.07 | 1.56 | 1.07 | |||
B19 [35] | 1.62 | 1.11 | 1.62 | 1.11 | |||
B19 [56] | 1.75 | 1.08 | 1.75 | 1.08 | |||
B33 [39] | 1.24 | 1.67 | 1.24 | 1.67 | 107.4° | 107.4° | |
B33 [39] | 3.19 | 2.36 | 3.19 | 2.36 | |||
B32 [56] | 1 | 1 | 1 | 1 | , | ||
R1 [35,61] | 1.39 | 1 | 1.39 | 1 | 120° | 120° | |
R1 [35] | 1.36 | 1 | 1.36 | 1 | 120° | 120° | |
R2 [37] | 1.37 | 1 | 1.37 | 1 | 120° | 120° | |
R2 [35] | 1.39 | 1 | 1.39 | 1 | 120° | 120° | |
R3 [35] | 1.35 | 1 | 1.35 | 1 | 120° | 120° | |
109°–B19′ [37] | 1.13 | 1.61 | 1.13 | 1.61 | 108.88° | 109.9° | , |
References
- Olson, G.B.; Cohen, M. A perspective on martensitic nucleation. Ann. Rev. Mater. Sci. 1981, 11, 1–30. [Google Scholar] [CrossRef]
- Christian, J.W. The Theory of Transformations in Metals and Alloys, 3rd ed.; Pergamon: Amsterdam, The Netherlands, 2002. [Google Scholar]
- Battacharya, K. Microstructure of Martensite: Why It Forms, and How It Gives Rise to the Shape-Memory Effect; Oxford University Press: New York, NY, USA, 2003. [Google Scholar]
- Kelly, P.J.; Rose, L.R.F. The martensitic transformation in ceramics—Its role in transformation toughening. Prog. Mater. Sci. 2002, 47, 463–557. [Google Scholar] [CrossRef]
- Olson, G.B. New directions in martensite theory. Mater. Sci. Eng. A 1999, 273–275, 11–20. [Google Scholar] [CrossRef]
- Tadaki, K.; Otsuka, K.; Shimuzu, K. Shape memory alloys. Ann. Rev. Mater. Sci. 1988, 18, 25–45. [Google Scholar] [CrossRef]
- Ölander, A. An electrochemical investigation of solid cadmium-gold alloys. J. Am. Chem. Soc. 1932, 54, 3819–3833. [Google Scholar] [CrossRef]
- Buehler, W.J.; Gilfrich, J.V.; Wiley, R.C. Effect of low-temperature phase changes on the mechanical properties of alloys near composition TiNi. J. Appl. Phys. 1963, 34, 1475–1477. [Google Scholar] [CrossRef]
- Kauffman, G.B.; Mayo, I. The story of Nitinol: The serendipitous discovery of the memory metal and its applications. Chem. Educ. 1997, 2, 1–21. [Google Scholar] [CrossRef]
- Balasubramanian, M.; Srimath, R.; Vignesh, L.; Rajesh, S. Application of shape memory alloys in engineering—A review. J. Phys. Conf. Ser. 2021, 2054, 012078. [Google Scholar] [CrossRef]
- Andreasen, G.F.; Barrett, R.D. An evaluation of cobalt-substituted nitinol wire in orthodontics. Am. J. Orthod. 1973, 63, 462–470. [Google Scholar] [CrossRef]
- Simon, M.; Kaplow, R.; Salzman, E.; Freiman, D. A vena cava filter using thermal shape memory alloy. Radiology 1977, 125, 89–94. [Google Scholar] [CrossRef]
- Lipscomb, I.P.; Nokes, L.D.M. The Application of Shape Memory Alloys in Medicine; Wiley–Blackwell: Hoboken, NJ, USA, 1996. [Google Scholar]
- Yang, C.; Abanteriba, S.; Becker, A. A review of shape memory alloy based filtration devices. AIP Adv. 2020, 10, 060701. [Google Scholar] [CrossRef]
- Hyde, S.T.; Andersson, S. The martensite transition and differential geometry. Z. Kristallogr. 1986, 174, 225–236. [Google Scholar] [CrossRef]
- Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.; CRC Press: BocaRaton, FL, USA, 1997. [Google Scholar]
- O’Neill, B. Elementary Differential Geometry; Revised 2nd edn; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Lidin, S. Some results of the Bonnet transformation. J. Phys. Colloq. 1990, 51, C7-237–C7-242. [Google Scholar] [CrossRef]
- Koch, E.; Fischer, W. A crystallographic approach to 3-periodic minimal surfaces. In Statistical Thermodynamics and Differential Geometry of Microstructured Materials; Davis, H.T., Nitsche, J.C.C., Eds.; Springer: New York, NY, USA, 1993; pp. 14–48. [Google Scholar]
- Schwarz, H.A. Gesammelte Mathematische Abhandlungen; Springer: Berlin, Germany, 1890. [Google Scholar]
- Schoen, A.H. Infinite Periodic Minimal Surfaces without Self-Intersections; Tech. Rep. TN D-5541; NASA: Washington, DC, USA, 1970. [Google Scholar]
- Schoen, A.H. Reflections concerning triply-periodic minimal surfaces. Interface Focus 2012, 2, 658–668. [Google Scholar] [CrossRef] [PubMed]
- Karcher, H. Embedded minimal surfaces derived from Scherk’s examples. Manuscripta Math. 1989, 64, 291–357. [Google Scholar] [CrossRef]
- Chen, H.; Weber, M. An orthorhombic deformation family of Schwartz’ H surfaces. Trans. Am. Math. Soc. 2021, 374, 2057–2078. [Google Scholar] [CrossRef]
- Kocian, P.; Schenk, K.; Chapuis, G. Differential geometry: A natural tool for describing symmetry operations. Acta Crystallogr. 2009, A65, 329–341. [Google Scholar] [CrossRef]
- Nakahara, M. Geometry, Topology, and Physics, 2nd ed.; Institute of Physics: Bristol, UK, 2003. [Google Scholar]
- Bain, E.C.; Dunkirk, N.Y. The nature of martensite. Trans. AIME 1924, 70, 25–47. [Google Scholar]
- Kurdjumow, G.; Sachs, G. Über den mechanismus der stahlhärtung. Z. Physik 1930, 64, 325–343. [Google Scholar] [CrossRef]
- Nishiyama, Z. Martensitic Transformation; Academic: New York, NY, USA, 1978. [Google Scholar]
- Wassermann, G. Einfluß der α-γ-Umwandlung eines irreversiblen Nickelstahls auf Kristallorientierung und Zugfestigkeit. Archiv. Eisenhüttenwesen 1933, 6, 347–351. [Google Scholar] [CrossRef]
- Bogers, A.J.; Burgers, W.G. Partial dislocations on the {110} planes in the B.C.C. lattice and the transition of the F.C.C. into the B.C.C. lattice. Acta Metall. 1964, 12, 255–261. [Google Scholar] [CrossRef]
- Olson, G.; Cohen, M. A mechanism for the strain-induced nucleation of martensitic transformations. J. Less-Common Met. 1972, 107–118. [Google Scholar] [CrossRef]
- Sandoval, L.; Urbassek, H.M.; Entel, P. The Bain versus Nishiyama–Wassermann path in the martensitic transformation of Fe. New J. Phys. 2009, 11, 103027. [Google Scholar] [CrossRef]
- Zhang, L.H.; Cheng, M.J.; Shi, X.H.; Shuai, J.W.; Zhu, Z.Z. Bain and Nishiyama–Wassermann transition path separation in the martensitic transitions of Fe. RSC Adv. 2021, 11, 3043–3048. [Google Scholar] [CrossRef] [PubMed]
- Hatcher, N.; Kontsevoi, O.Y.; Freeman, A.J. Role of elastic and shear stabilities in the martensitic transformation path of NiTi. Phys. Rev. B 2009, 80, 144203. [Google Scholar] [CrossRef]
- Niu, J.G.; Geng, W.T. Anti-precursor effect of Fe on martensitic transformation in TiNi alloys. Acta Mater. 2016, 104, 18–24. [Google Scholar] [CrossRef]
- Hatcher, N.; Kontsevoi, O.Y.; Feemen, A.J. Martensitic transformation path of NiTi. Phys. Rev. B 2009, 79, 020202(R). [Google Scholar] [CrossRef]
- Morris, J.R.; Ye, Y.; Krcmar, M.; Fu, C.L. The Role of Phase Stability in Ductile, Ordered B2 Intermetallics. MRS Online Proc. Libr. 2006, 980, 610. [Google Scholar] [CrossRef]
- Krcmar, M.; Samolyuk, G.D.; Morris, J.R. Stacking faults and alternate crystal structures for the shape-memory alloy NiTi. Phys. Rev. Mater. 2020, 4, 103606. [Google Scholar] [CrossRef]
- Kibey, S.; Sehitoglu, H.; Johnson, D.D. Energy landscape for martensitic phase transformation in shape memory NiTi. Acta Mater. 2009, 57, 1624–1629. [Google Scholar] [CrossRef]
- Trinkle, D.R.; Hatch, D.M.; Stokes, H.T.; Hennig, R.G.; Albers, R.C. Systematic pathway generation and sorting in martensitic transformations: Titanium α to ω. Phys, Rev. B 2005, 72, 014105. [Google Scholar] [CrossRef]
- Hennig, R.G.; Trinkle, D.R.; Bouchet, J.; Srinivasan, S.G.; Albers, R.C.; Wilkins, J.W. Impurities block the α to ω martensitic transformation in titanium. Nat. Mater. 2005, 4, 129–133. [Google Scholar] [CrossRef]
- Ghosh, P.S.; Arya, A.; Tewari, R.; Dey, G.K. Alpha to omega martensitic phase transformation pathways in pure Zr. J. Alloys Compd. 2014, 586, 693–698. [Google Scholar] [CrossRef]
- Suzuki, T.; Ledbetter, H.M. Barrier energy for the b c.c.-f.c.c. martensitic transition in sodium. Phil. Mag. 1983, 48, 83–94. [Google Scholar] [CrossRef]
- Okatov, S.V.; Kuznetsov, A.R.; Gornostyrev, Y.N.; Urtsev, V.N.; Katsnelson, M.I. Effect of magnetic state on the γ-α transition in iron: First-principles calculations of the Bain transformation path. Phys. Rev. B 2009, 79, 094111. [Google Scholar] [CrossRef]
- Wang, T.L.; Du, J.L.; Wei, S.Z.; Liu, F. Ab-initio investigation for the microscopic thermodynamics and kinetics of martensitic transformation. Prog. Nat. Sci. Mater. Int. 2021, 31, 121–128. [Google Scholar] [CrossRef]
- Wang, L.G.; Šob, M. Structural stability of higher-energy phases and its relation to the atomic configurations of extended defects: The example of Cu. Phys. Rev. B 1999, 60, 844–850. [Google Scholar] [CrossRef]
- Jeong, S. Structural properties of bulk copper: Pseudopotential plane-wave-basis study. Phys. Rev. B 1996, 53, 13973–13976. [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]
- Mishin, Y.; Farkas, D.; Mehl, M.J.; Papaconstantopoulos, D.A. Interatomic potentials for monoatomic metals from experimental data and ab initio calculations. Phys. Rev. B 1999, 59, 3393–3407. [Google Scholar] [CrossRef]
- Bhattacharya, K.; Conti, S.; Zanzotto, G.; Zimmer, J. Crystal symmetry and the reversibility of martensitic transformations. Nature 2004, 428, 55–59. [Google Scholar] [CrossRef] [PubMed]
- Weierstrass, K. Mathematische Werke; Mayer & Müller: Berlin, Germany, 1903. [Google Scholar]
- Enneper, A. Analytisch-Geometrische Untersuchung. Z. Math Phys. 1864, 9, 96–125. [Google Scholar]
- Weyhaupt, A.G. New Families of Embedded Triply Periodic Minimal Surfaces of Genus Three in Euclidean Space. Ph.D. Thesis, Indiana University, Bloomington, IN, USA, 2006. [Google Scholar]
- Stein, E.M.; Shakarchi, R. Complex Analysis; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Liu, G.T.; Liu, H.Y.; Feng, X.L.; Redfern, S.A.T. High-pressure phase transitions of nitinol NiTi to a semiconductor with an unusual topological structure. Phys. Rev. B 2018, 97, 140104. [Google Scholar] [CrossRef]
- Sittner, P.; Lukás, P.; Neov, D.; Novák, V.; Toebbens, D.M. The martensite transition and differential geometry. J. Phys. IV (Fr.) 2003, 112, 709–711. [Google Scholar] [CrossRef]
- Prokoshkin, S.D.; Korotitskiy, A.V.; Brailovski, V.; Turenne, S.; Khmelevskaya, I.Y.; Trubitsyna, I.B. On the lattice parameters of phases in binary Ti–Ni shape memory alloys. Acta Mater. 2004, 52, 4479–4492. [Google Scholar] [CrossRef]
- Kudoh, Y.; Tokonami, M.; Miyazaki, S.; Otsuka, K. Crystal structure of the martensite in Ti-49.2 at. %Ni alloy analyzed by the single crystal X-ray diffraction method. Acta Metall. 1985, 33, 2049–2056. [Google Scholar] [CrossRef]
- Nam, T.H.; Saburi, T.; Nakata, Y.; Shimizu, K.I. Shape Memory Characteristics and Lattice Deformation in Ti–Ni–Cu Alloys. Mater. Trans. JIM 1990, 31, 1050–1056. [Google Scholar] [CrossRef]
- Hara, T.; Ohba, T.; Okinushi, E.; Otsuka, K. Structural study of R-phase in Ti-50.23 at. %Ni and Ti-47.75 at. %Ni-1.50 at. %Fe Alloys. Mater. Trans. JIM 1997, 38, 11–17. [Google Scholar] [CrossRef]
Material | Initial | Path | Final | |
---|---|---|---|---|
NiTi [35] | B2 | basal shear | B19′ | ∼37 |
B2 | bilayer shear | B19′ | ∼22 | |
B2 | 109°–B19′ | B19′ | 0 | |
NiTi [36] | B2 | [111] elongation | R | 0 |
NiTi [35,37,38,39] | B2 | alternate bilayer shear | B33 | 0 |
NiTi [40] | B2 | basal shear | B19 | ∼13 |
Ti [41] | HCP() | , | Hexagonal () | ∼9 |
Ti [42] | HCP() | , | Hexagonal () | ∼9.5 |
Zr [43] | HCP() | , | Hexagonal () | 22 |
Na [44] | BCC | Bain | FCC | ∼0.79 |
Fe [34,45] | -Fe | Bain (FCT) | -Fe | ∼20 |
–Fe | Bain | –Fe | ∼45 | |
Fe [34] | –Fe | NW (FCT) | –Fe | ∼37 |
Fe [46] | –Fe | BB | –Fe | ∼25 |
Cu [47] | BCT(BCC) | Bain | FCC | 1 |
Cu [48] | BCT(BCC) | Bain | FCC | 1.3 |
Cu [49] | FCC | BB | BCC | ∼700 |
Al [50] | BCT(BCC) | Bain | FCC | ∼30 |
Al [50] | FCC | BB | BCC | ∼100 |
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. |
© 2024 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
Yin, M.; Vvedensky, D.D. Shape-Memory Effect and the Topology of Minimal Surfaces. Symmetry 2024, 16, 1187. https://doi.org/10.3390/sym16091187
Yin M, Vvedensky DD. Shape-Memory Effect and the Topology of Minimal Surfaces. Symmetry. 2024; 16(9):1187. https://doi.org/10.3390/sym16091187
Chicago/Turabian StyleYin, Mengdi, and Dimitri D. Vvedensky. 2024. "Shape-Memory Effect and the Topology of Minimal Surfaces" Symmetry 16, no. 9: 1187. https://doi.org/10.3390/sym16091187