- Email: [email protected]
Critical stress map for ZrO2 tetragonal to monoclinic phase transformation in ZrO2-toughened glass-ceramics
Materialia 9 (2020) 100548
Contents lists available at ScienceDirect
Materialia journal homepage: www.elsevier.com/locate/mtla
Full Length Article
Critical stress map for ZrO2 tetragonal to monoclinic phase transformation in ZrO2 -toughened glass-ceramics Binghui Deng a,∗, Jian Luo b, Jason T. Harris a, Charlene M. Smith b a b
Manufacturing Technology & Engineering, Corning Incorporated, Corning, NY 14831, USA Science & Technology, Corning Research and Development Corporation, Corning, NY 14831, USA
a r t i c l e
i n f o
Keywords: ZrO2 phase transformation Fracture toughness Critical stress Molecular dynamics Glass-ceramics
a b s t r a c t Inspired by recent observation of compressive stress induced tetragonal to monoclinic (t-m) phase transformation in yttria-stabilized zirconia, we conduct a systematic study to investigate the critical stress map for ZrO2 transformation under various loading conditions. The results show that (1) uniaxial tensile and compressive loadings can trigger the transformation, but not hydrostatic tensile and compressive loadings; (2) shear and equal biaxial tensile loading are most favored to trigger the transformation. Additionally, simulation on a model ZrO2 -containing SiO2 -based glass-ceramic shows that the stress fields around the running crack front can help triggering ZrO2 phase transformation, and consequently change the stress in ZrO2 from tensile to compressive and deflect the crack propagation. The insights obtained from these simulations can help us better understand ZrO2 transformation toughening mechanism, and further leverage it to enhance the mechanical performance of materials in many real applications.
1. Introduction Glass-ceramics are inorganic, non-metallic materials that contain a residual glassy phase and one or more embedded crystalline phases produced by controlled crystallization via different processing methods [1–3]. The ability for them to be easily manufactured using various glass-forming processes (e.g., blowing, rolling, floating, and pressing), together with the superior mechanical properties (e.g., high flexural strength and high fracture toughness) compared with their parent glass have been attracting more and more attention and investment, both in academia and industry [4]. Different toughening mechanisms, such as crack deflection, crack bowing, interface debonding, microcrack generation, crack bridging, phase transformation toughening, etc., have been widely observed and studied in different glass-ceramic systems over the past few decades [4–9]. Among these mechanisms is the advantageous behavior of ZrO2 -containing glass-ceramics with stress-activated tetragonal (t) to monoclinic (m) phase transformation. The transformation is usually accompanied with 3%–4% volume expansion that is believed to generate a compressive stress zone around an active crack which consequently compensates for a tensile stress, impedes crack propagation, and increases fracture toughness [10–14]. It has been widely believed that tensile stress and shear stress are the primary driving forces to trigger ZrO2 t-m transformation in various applications [15,16]. Interestingly, Allahkarami and Hanan [17] recently observed high compressive stress induced ZrO2 t-m transformation in ∗
zirconia core dental crowns, which echoes the study by Piascik et al. [18] in which they observed compressive stress (because of substrate bias) induced ZrO2 t-m transformation during sputter deposition of yttria-stabilized zirconia. Similar behavior was also observed in ZrO2 micropillars compression experiments conducted by Camposilvan and Anglada [19]. The ability to efficiently take advantage of ZrO2 t-m transformation to improve material performance in many different industrial applications requires a full understanding of stress conditions, i.e., shear, hydrostatic, uniaxial, likely to trigger the transformation. Clearly, it is extremely challenging to draw such a critical map for of ZrO2 t-m transformation taking into considerations of available complicated stress states by only relying on experiments. Molecular dynamics simulations, however, could potentially answer these questions through careful simulation design and high-throughput systematic study. To this end, we aim to investigate a critical stress map for an ideal ZrO2 single crystal t-m transformation under various loading conditions. The complications arising from grain boundaries in polycrystal, interfaces and other various defects in real systems are not at the primary scope of this study. Nevertheless, reasonable inferences might be made based on the observations for the single crystal. In addition, simulations on real-time ZrO2 t-m transformation in a model glass-ceramics sample are also conducted to study how the complicated stress around a running crack tip helps to activate the ZrO2 transformation, and consequently how the crack propagation path is affected.
Corresponding author. E-mail address: [email protected] (B. Deng).
https://doi.org/10.1016/j.mtla.2019.100548 Received 21 October 2019; Accepted 27 November 2019 Available online 28 November 2019 2589-1529/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 1. (a) The ZrO2 single crystal volume evolution over the course of uniaxial tensile loading. The discontinuity indicates first-order phase transformation. (b) The corresponding stress–strain curve over the course of loading. (c) Sample snapshot (top view) before ZrO2 phase transformation. (d) Sample snapshot after ZrO2 phase transformation.
2. Simulation method Podone force field is used to capture the short-range interactions. For the long-range columbic interactions, the damped shifted force (DSF) method with a cutoff of 8 Å and damping parameter of 0.25 Å−1 are used to speed up calculations following our previous conventions [20]. The force field is proven to be able to capture the ZrO2 phase transformation behavior. Pure single tetragonal ZrO2 crystal is made following the lattice parameter of (3.61 Å, 3.61 Å, 5.21 Å) and basis reported in [21]. The sample size is 8.67 × 8.67 × 8.34 nm3 with a total of 55,296 atoms. The sample is brought to mechanical tests under different loading conditions at 300 K with an engineering strain rate of 0.2 ns−1 . Periodic boundary conditions (PBC) are applied in all the sample dimensions over the course of testing. In addition, we also prepare a model SiO2 -ZrO2 glass-ceramic sample to study the real-time ZrO2 phase transformation at the presence of a running crack in glass-ceramics following our previous approach [22]. The glass phase is pure fused SiO2 , and the crystal phase is tetragonal
ZrO2 . The glass is initially made by first randomly inserting approximate 300,000 atoms into a slab simulation box (54.5 × 54.5 × 2.7 nm3 ) and equilibrating them at 3000 K for 2 ns in the Canonical (NVT) ensemble, then continuously cooling down the melt to 300 K over a period of 8 ns in the Isothermal-isobaric (NPT) ensemble under atmosphere with an average cooling rate of 337.5 K.ns−1 . The crystal is cut out of a sheet of ZrO2 with a radius of 10.0 nm, and then inserted into the center of glass sample as shown in Fig. 6. To heal the fresh interface between crystalline nanoparticles and the glass, the samples are heated up to 2600 K and relaxed for 2 ns in the NPT ensemble under atmosphere pressure, then cooled down to 300 K following the same procedure that was used to create the glass sample. The elevated temperature is deliberately chosen such that the surface atoms of crystalline nanoparticles are melted and well mixed with glass atoms to form a strong interface but do not diffuse excessively to obfuscate the phase boundary. All the simulations including sample preparations are conducted using LAMMPS [23] with a timestep of 2 fs. Temperature and pressure are well controlled via Nose–Hoover [24,25] thermostat and
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 2. (a) The ZrO2 single crystal volume evolution over the course of uniaxial compressive loading. The discontinuity indicates first-order phase transformation. (b) The corresponding stress–strain curve over the course of loading.
Fig. 3. (a) The ZrO2 single crystal volume evolution over the course of shear loading. The discontinuity indicates a first-order phase transformation. (b) The corresponding stress–strain curve over the course of loading.
Table 1 Young’s modulus (E), bulk modulus (K), shear modulus (G), and Poisson’s ratio (𝜈) of tetragonal ZrO2 single crystal measured with both tension and compression tests in the x, y, and z directions.
ZrO2 <1 0 0 > ZrO2 < 0 1 0> ZrO2 < 0 0 1>
E (GPa)
K (GPa)
G (GPa)
𝜈
369.8 369.8 602.1
267.1 267.1 276.2
145.7 145.7 264.8
0.27 0.27 0.14
barostat, respectively. Elastic moduli of each phase are calculated and averaged with both tension and compression tests in the x, y, and z directions within 1% engineering strain range. The results are detailed in Table 1. Cleary, the tetragonal ZrO2 exhibits the same elastic properties in the [100] and [010] direction due to the structure symmetry,
while has higher elastic moduli and lower Poisson’s ratio in the [001] direction. 3. Results and discussions Fig. 1(a) shows the volume evolution of tetragonal ZrO2 over the course of uniaxial tensile loading in the [100] direction. The sudden jump at the strain of around 0.06 indicates the occurrence of a firstorder phase transformation. In addition, Fig. 1(b) shows the corresponding stress–strain curve over the course of loading. The stress suddenly drops from tensile, at critical tensile stress threshold of around 14.5 GPa, to compressive because of the phase transformation. Fig. 1(c) and (d) shows snapshots of a ZrO2 single crystal in the X–Y (001) plane before and after phase transformation, respectively. Clearly, the tetragonal ZrO2 transforms to a monoclinic crystal structure triggered by uniaxial loading. Note that we also conduct uniaxial loading in the [001] direction, however, no transformation is observed.
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 4. The ZrO2 single crystal volume evolution over the course of (a) hydrostatic tension loading and (b) hydrostatic compression loading. Continuous curve indicates no presence of ZrO2 phase transformation.
Fig. 5. The critical stress map for ZrO2 single crystal t-m phase transformation. Each point represents an individual numerical test with a constant loading ratio 𝑟 = 𝜎𝑥 ∕𝜎𝑦 . The ratio ranges from −1 to 1, covering loading conditions from pure shear (r = −1), uniaxial (r = 0), to equal biaxial (r = 1). Note that no stress component in the [001] direction is included in the stress map due to its inability to trigger the transformation.
Similarly, Fig. 2(a) shows the volume evolution of tetragonal ZrO2 over the course of uniaxial compressive loading. The discontinuity at the strain of around 0.1 indicates the occurrence of a first-order phase transformation. Fig. 2(b) shows the corresponding stress–strain curve over the course of loading. Clearly, the stress suddenly drops as the result of phase transformation at the critical stress of around 37.5 GPa. Although the compressive loading does trigger the transformation, it takes larger strain and higher stress. In other words, compressive stress is less favorable to enable the transformation. Fig. 3(a) shows the volume evolution of tetragonal single crystal ZrO2 over the course of shear loading. The shear loading is implemented by simultaneously applying tensile loading in the [100] direction, while applying the same magnitude of compressive loading in the [010] direction. Again, the discontinuity at the strain of around 0.06 indicates the occurrence of a first-order phase transformation. Fig. 3(b) shows the corresponding stress–strain curve. Clearly, the stress suddenly jumps from tensile regime to compressive regime as the result of the phase transformation at the critical tensile stress of around 9.2 GPa.
Fig. 4(a) and (b) shows the volume evolution of ZrO2 single crystal over the course of tensile and compressive loadings, respectively. Interestingly, both curves show no signs of discontinuity, indicating the absence of ZrO2 phase transformation under these two loading conditions. It further suggests that some sort of shear stress component is required to enable the ZrO2 t-m transformation. To completely understand stress conditions able to trigger a transformation, we move forward to load the sample under various stress states. Fig. 5 shows the critical stress map for ZrO2 single crystal t-m phase transformation. Each point represents an individual test with a constant loading ratio 𝑟 = 𝜎𝑥 ∕𝜎𝑦 . The ratio ranges from −1 to 1, covering loading conditions from pure shear (r = −1), uniaxial (r = 0), to equal biaxial (r = 1). The distance between each point to the origin represents the effective stress magnitude to trigger the phase transformation. Clearly, we can see that stress states in the shear and equal biaxial tension are most favored to trigger the phase transformation. In many real applications where ZrO2 is used as a fracture toughness enhancement phase, it is critical to understand how the complicated stress around a running crack tip activates the ZrO2 transformation, and consequently how the crack propagation path is affected. Fig. 6(a) shows the snapshots of a model SiO2 -based glass-ceramic sample with a tetragonal ZrO2 particle embedded into SiO2 glass matrix under mechanical loading. The sample is loaded in the X direction with a pre-existing crack of 5 nm at the sample bottom. Periodic boundary conditions are applied in the X and Z direction, while free surface in the Y direction. A very thin layer of atoms on each free surface is gripped to mimic the complex loading conditions in many real circumstances. It seems that the ZrO2 nanocrystal undergoes phase transformation under the applied tensile load, then the propagating crack is deflected due to the inhomogeneous stress filed around the ZrO2 nanocrystal after the transformation. The ZrO2 t-m phase transformation is further clearly evidenced by Fig. 6(c) in which it shows the ZrO2 crystal structure before and after the phase transformation. Interestingly, it seems that the transformation nucleates at multiple sites at the interface, then propagates inward, and forms impinging boundaries which is shown clearly at Fig. 6 (c-2) and (c-3). In addition, we also probe the stress map evolution of the whole sample over the course of loading. Fig. 6 (d-1) and (d-2) shows the sample stress map right before and after the ZrO2 t-m phase transformation. Clearly, tensile stress is intensively accumulated inside the ZrO2 crystal with increasing tensile loading, which becomes compressive after triggering of the phase transformation, which echoes many previous studies on the toughening mechanism of ZrO2 -containing glass-ceramics.
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 6. (a) Snapshots of SiO2 glass-ceramic sample with a tetragonal ZrO2 particle embedded into SiO2 glass matrix over the course of loading. The sample size is 54.5 × 54.5 × 2.7 nm3 , and the radius of ZrO2 particle is around 10.0 nm. The sample is loaded in the X direction orthogonal to a pre-existing crack of 5 nm. (b) The accompanying stress–strain curve. (c) Snapshot of ZrO2 transforming from tetragonal structure (c-1) to monoclinic crystal structure (c-2) with the closeup shown in (c-3). (d) The sample stress map right before (d-1) and after (d-2) the phase transformation.
4. Conclusion
References
Motived by recent observation of considerable presence of compressive stress induced tetragonal to monoclinic phase transformation in yttria-stabilized zirconia [17,18], we conduct a systematic study to investigate the critical stress map for ZrO2 phase transformation under various loading conditions using molecular dynamics simulations. The results on a single ZrO2 crystal shows that (1) uniaxial tensile and compressive loadings are able to trigger the phase transformation; (2) shear and equal biaxial tensile loading are most favorable loading conditions to trigger the phase transformation; (3) hydrostatic tensile and compressive loading are not able to trigger the phase transformation; Note that the aforementioned loading conditions are applicable in either [100] or [001] direction due to the symmetrical nature of tetragonal crystal structure, while any loading in the [001] direction is not able to trigger the phase transformation. In addition, we also conduct simulations on a model SiO2 -based glass-ceramic sample with a tetragonal ZrO2 particle embedded into SiO2 glass matrix under mechanical loading. The results show that the stress fields at the running crack front can help triggering the ZrO2 phase transformation, and consequently transform the stress state in ZrO2 from tensile to compressive, which retards and then deflects crack propagation. Although all observations are taken on an ideal ZrO2 single crystal, insights obtained from the simulations can help us better understand its transformation likelihood in real applications, and further inspire us to better leverage the ZrO2 transformation toughening mechanism to improve many material properties.
[1] G.H. Beall, Design and properties of glass-ceramics, Annu. Rev. Mater. Sci. 22 (1992) 91–119. [2] J. Deubener, M. Allix, M.J. Davis, A. Duran, T. Höche, T. Honma, T. Komatsu, S. Krüger, I. Mitra, R. Müller, S. Nakane, M.J. Pascual, J.W.P. Schmelzer, E.D. Zanotto, S. Zhou, Updated definition of glass-ceramics, J. Non Cryst. Solids 501 (2018) 3–10. [3] R.D. Rawlings, J.P. Wu, A.R. Boccaccini, Glass-ceramics: their production from wastes—a review, J. Mater. Sci. 41 (3) (2006) 733–761. [4] Q. Fu, G.H. Beall, C.M. Smith, Nature-inspired design of strong, tough glass-ceramics, MRS Bull. 42 (03) (2017) 220–225. [5] F.C. Serbena, I. Mathias, C.E. Foerster, E.D. Zanotto, Crystallization toughening of a model glass-ceramic, Acta Mater. 86 (2015) 216–228. [6] D. Li, X.C. Li, M. Meng, R. Wei, L. He, S.F. Zhang, Strengthening of a lithium disilicate glass-ceramic by rapid cooling, Ceram. Int. 44 (10) (2018) 11650–11657. [7] D. Li, J.W. Guo, X.S. Wang, S.F. Zhang, L. He, Effects of crystal size on the mechanical properties of a lithium disilicate glass-ceramic, Mater. Sci. Eng. A 669 (2016) 332–339. [8] L. Hallmann, P. Ulmer, M. Kern, Effect of microstructure on the mechanical properties of lithium disilicate glass-ceramics, J. Mech. Behav. Biomed. Mater. 82 (2018) 355–370. [9] B. Deng, Y. Shi, The embrittlement and toughening of metallic glasses from nano-crystallization, J. Appl. Phys. 125 (14) (2019) 145101. [10] P.M.K. Richard, H.J. Hannink, B.C. Muddle, Transformation toughening in zirconia‐containing ceramics, J. Am. Ceram. Soc. 83 (3) (2000) 461–487. [11] T. Zhao, J. Zhu, J. Luo, Study of crack propagation behavior in single crystalline tetragonal zirconia with the phase field method, Eng. Fract. Mech. 159 (2016) 155–173. [12] M.N.M. Tomozawa, ZrO2‐Transformation‐Toughened glass‐ceramics prepared by the sol‐gel process from metal alkoxides, J. Am. Ceram. Soc. 69 (2) (1986) 99–102. [13] R.D. Sarno, M. Tomozawa, Toughening mechanisms for a zirconia-lithium aluminosilicate glass-ceramic, J. Mater. Sci. 30 (17) (1995) 4380–4388. [14] M. Mamivand, M. Asle Zaeem, H. El Kadiri, L.-Q. Chen, Phase field modeling of the tetragonal-to-monoclinic phase transformation in zirconia, Acta Mater. 61 (14) (2013) 5223–5235. [15] P.M. Kelly, L.R. Francis Rose, The martensitic transformation in ceramics - its role in transformation toughening, Prog. Mater. Sci. 47 (5) (2002) 463–557. [16] M. Mamivand, M. Asle Zaeem, H. El Kadiri, Phase field modeling of stress-induced tetragonal-to-monoclinic transformation in zirconia and its effect on transformation toughening, Acta Mater. 64 (2014) 208–219. [17] M. Allahkarami, J.C. Hanan, Mapping the tetragonal to monoclinic phase transformation in zirconia core dental crowns, Dent. Mater. 27 (12) (2011) 1279–1284. [18] J.R. Piascik, Q. Zhang, C.A. Bower, J.Y. Thompson, B.R. Stoner, Evidence of stress-induced tetragonal-to-monoclinic phase transformation during sputter deposition of yttria-stabilized zirconia, J. Mater. Res. 22 (4) (2007) 1105–1111. [19] E. Camposilvan, M. Anglada, Size and plasticity effects in zirconia micropillars compression, Acta Mater. 103 (2016) 882–892.
Declaration of Competing Interest The authors declare that there are no known competing interests regarding to this work. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful for the technical support from the Corning Scientific Computing group.
B. Deng, J. Luo and J.T. Harris et al. [20] B. Deng, J. Luo, J.T. Harris, C.M. Smith, M.E. McKenzie, Molecular dynamics simulations on fracture toughness of Al2O3-SiO2 glass-ceramics, Scr. Mater. 162 (2019) 277–280. [21] Y.I. Naoki Igawa, Crystal structure of metastable tetragonal zirconia up to 1473K, J. Am. Ceram. Soc. 84 (5) (2001) 1169–1171. [22] B. Deng, J. Luo, J.T. Harris, C.M. Smith, M.E. McKenzie, Toughening of Li2O‐2SiO2 glass‐ceramics induced by intriguing deformation behavior of lithium disilicate nanocrystal, J. Am. Ceram. Soc. 00 (2019) 1–8.
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Contents lists available at ScienceDirect
Materialia journal homepage: www.elsevier.com/locate/mtla
Full Length Article
Critical stress map for ZrO2 tetragonal to monoclinic phase transformation in ZrO2 -toughened glass-ceramics Binghui Deng a,∗, Jian Luo b, Jason T. Harris a, Charlene M. Smith b a b
Manufacturing Technology & Engineering, Corning Incorporated, Corning, NY 14831, USA Science & Technology, Corning Research and Development Corporation, Corning, NY 14831, USA
a r t i c l e
i n f o
Keywords: ZrO2 phase transformation Fracture toughness Critical stress Molecular dynamics Glass-ceramics
a b s t r a c t Inspired by recent observation of compressive stress induced tetragonal to monoclinic (t-m) phase transformation in yttria-stabilized zirconia, we conduct a systematic study to investigate the critical stress map for ZrO2 transformation under various loading conditions. The results show that (1) uniaxial tensile and compressive loadings can trigger the transformation, but not hydrostatic tensile and compressive loadings; (2) shear and equal biaxial tensile loading are most favored to trigger the transformation. Additionally, simulation on a model ZrO2 -containing SiO2 -based glass-ceramic shows that the stress fields around the running crack front can help triggering ZrO2 phase transformation, and consequently change the stress in ZrO2 from tensile to compressive and deflect the crack propagation. The insights obtained from these simulations can help us better understand ZrO2 transformation toughening mechanism, and further leverage it to enhance the mechanical performance of materials in many real applications.
1. Introduction Glass-ceramics are inorganic, non-metallic materials that contain a residual glassy phase and one or more embedded crystalline phases produced by controlled crystallization via different processing methods [1–3]. The ability for them to be easily manufactured using various glass-forming processes (e.g., blowing, rolling, floating, and pressing), together with the superior mechanical properties (e.g., high flexural strength and high fracture toughness) compared with their parent glass have been attracting more and more attention and investment, both in academia and industry [4]. Different toughening mechanisms, such as crack deflection, crack bowing, interface debonding, microcrack generation, crack bridging, phase transformation toughening, etc., have been widely observed and studied in different glass-ceramic systems over the past few decades [4–9]. Among these mechanisms is the advantageous behavior of ZrO2 -containing glass-ceramics with stress-activated tetragonal (t) to monoclinic (m) phase transformation. The transformation is usually accompanied with 3%–4% volume expansion that is believed to generate a compressive stress zone around an active crack which consequently compensates for a tensile stress, impedes crack propagation, and increases fracture toughness [10–14]. It has been widely believed that tensile stress and shear stress are the primary driving forces to trigger ZrO2 t-m transformation in various applications [15,16]. Interestingly, Allahkarami and Hanan [17] recently observed high compressive stress induced ZrO2 t-m transformation in ∗
zirconia core dental crowns, which echoes the study by Piascik et al. [18] in which they observed compressive stress (because of substrate bias) induced ZrO2 t-m transformation during sputter deposition of yttria-stabilized zirconia. Similar behavior was also observed in ZrO2 micropillars compression experiments conducted by Camposilvan and Anglada [19]. The ability to efficiently take advantage of ZrO2 t-m transformation to improve material performance in many different industrial applications requires a full understanding of stress conditions, i.e., shear, hydrostatic, uniaxial, likely to trigger the transformation. Clearly, it is extremely challenging to draw such a critical map for of ZrO2 t-m transformation taking into considerations of available complicated stress states by only relying on experiments. Molecular dynamics simulations, however, could potentially answer these questions through careful simulation design and high-throughput systematic study. To this end, we aim to investigate a critical stress map for an ideal ZrO2 single crystal t-m transformation under various loading conditions. The complications arising from grain boundaries in polycrystal, interfaces and other various defects in real systems are not at the primary scope of this study. Nevertheless, reasonable inferences might be made based on the observations for the single crystal. In addition, simulations on real-time ZrO2 t-m transformation in a model glass-ceramics sample are also conducted to study how the complicated stress around a running crack tip helps to activate the ZrO2 transformation, and consequently how the crack propagation path is affected.
Corresponding author. E-mail address: [email protected] (B. Deng).
https://doi.org/10.1016/j.mtla.2019.100548 Received 21 October 2019; Accepted 27 November 2019 Available online 28 November 2019 2589-1529/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 1. (a) The ZrO2 single crystal volume evolution over the course of uniaxial tensile loading. The discontinuity indicates first-order phase transformation. (b) The corresponding stress–strain curve over the course of loading. (c) Sample snapshot (top view) before ZrO2 phase transformation. (d) Sample snapshot after ZrO2 phase transformation.
2. Simulation method Podone force field is used to capture the short-range interactions. For the long-range columbic interactions, the damped shifted force (DSF) method with a cutoff of 8 Å and damping parameter of 0.25 Å−1 are used to speed up calculations following our previous conventions [20]. The force field is proven to be able to capture the ZrO2 phase transformation behavior. Pure single tetragonal ZrO2 crystal is made following the lattice parameter of (3.61 Å, 3.61 Å, 5.21 Å) and basis reported in [21]. The sample size is 8.67 × 8.67 × 8.34 nm3 with a total of 55,296 atoms. The sample is brought to mechanical tests under different loading conditions at 300 K with an engineering strain rate of 0.2 ns−1 . Periodic boundary conditions (PBC) are applied in all the sample dimensions over the course of testing. In addition, we also prepare a model SiO2 -ZrO2 glass-ceramic sample to study the real-time ZrO2 phase transformation at the presence of a running crack in glass-ceramics following our previous approach [22]. The glass phase is pure fused SiO2 , and the crystal phase is tetragonal
ZrO2 . The glass is initially made by first randomly inserting approximate 300,000 atoms into a slab simulation box (54.5 × 54.5 × 2.7 nm3 ) and equilibrating them at 3000 K for 2 ns in the Canonical (NVT) ensemble, then continuously cooling down the melt to 300 K over a period of 8 ns in the Isothermal-isobaric (NPT) ensemble under atmosphere with an average cooling rate of 337.5 K.ns−1 . The crystal is cut out of a sheet of ZrO2 with a radius of 10.0 nm, and then inserted into the center of glass sample as shown in Fig. 6. To heal the fresh interface between crystalline nanoparticles and the glass, the samples are heated up to 2600 K and relaxed for 2 ns in the NPT ensemble under atmosphere pressure, then cooled down to 300 K following the same procedure that was used to create the glass sample. The elevated temperature is deliberately chosen such that the surface atoms of crystalline nanoparticles are melted and well mixed with glass atoms to form a strong interface but do not diffuse excessively to obfuscate the phase boundary. All the simulations including sample preparations are conducted using LAMMPS [23] with a timestep of 2 fs. Temperature and pressure are well controlled via Nose–Hoover [24,25] thermostat and
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 2. (a) The ZrO2 single crystal volume evolution over the course of uniaxial compressive loading. The discontinuity indicates first-order phase transformation. (b) The corresponding stress–strain curve over the course of loading.
Fig. 3. (a) The ZrO2 single crystal volume evolution over the course of shear loading. The discontinuity indicates a first-order phase transformation. (b) The corresponding stress–strain curve over the course of loading.
Table 1 Young’s modulus (E), bulk modulus (K), shear modulus (G), and Poisson’s ratio (𝜈) of tetragonal ZrO2 single crystal measured with both tension and compression tests in the x, y, and z directions.
ZrO2 <1 0 0 > ZrO2 < 0 1 0> ZrO2 < 0 0 1>
E (GPa)
K (GPa)
G (GPa)
𝜈
369.8 369.8 602.1
267.1 267.1 276.2
145.7 145.7 264.8
0.27 0.27 0.14
barostat, respectively. Elastic moduli of each phase are calculated and averaged with both tension and compression tests in the x, y, and z directions within 1% engineering strain range. The results are detailed in Table 1. Cleary, the tetragonal ZrO2 exhibits the same elastic properties in the [100] and [010] direction due to the structure symmetry,
while has higher elastic moduli and lower Poisson’s ratio in the [001] direction. 3. Results and discussions Fig. 1(a) shows the volume evolution of tetragonal ZrO2 over the course of uniaxial tensile loading in the [100] direction. The sudden jump at the strain of around 0.06 indicates the occurrence of a firstorder phase transformation. In addition, Fig. 1(b) shows the corresponding stress–strain curve over the course of loading. The stress suddenly drops from tensile, at critical tensile stress threshold of around 14.5 GPa, to compressive because of the phase transformation. Fig. 1(c) and (d) shows snapshots of a ZrO2 single crystal in the X–Y (001) plane before and after phase transformation, respectively. Clearly, the tetragonal ZrO2 transforms to a monoclinic crystal structure triggered by uniaxial loading. Note that we also conduct uniaxial loading in the [001] direction, however, no transformation is observed.
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 4. The ZrO2 single crystal volume evolution over the course of (a) hydrostatic tension loading and (b) hydrostatic compression loading. Continuous curve indicates no presence of ZrO2 phase transformation.
Fig. 5. The critical stress map for ZrO2 single crystal t-m phase transformation. Each point represents an individual numerical test with a constant loading ratio 𝑟 = 𝜎𝑥 ∕𝜎𝑦 . The ratio ranges from −1 to 1, covering loading conditions from pure shear (r = −1), uniaxial (r = 0), to equal biaxial (r = 1). Note that no stress component in the [001] direction is included in the stress map due to its inability to trigger the transformation.
Similarly, Fig. 2(a) shows the volume evolution of tetragonal ZrO2 over the course of uniaxial compressive loading. The discontinuity at the strain of around 0.1 indicates the occurrence of a first-order phase transformation. Fig. 2(b) shows the corresponding stress–strain curve over the course of loading. Clearly, the stress suddenly drops as the result of phase transformation at the critical stress of around 37.5 GPa. Although the compressive loading does trigger the transformation, it takes larger strain and higher stress. In other words, compressive stress is less favorable to enable the transformation. Fig. 3(a) shows the volume evolution of tetragonal single crystal ZrO2 over the course of shear loading. The shear loading is implemented by simultaneously applying tensile loading in the [100] direction, while applying the same magnitude of compressive loading in the [010] direction. Again, the discontinuity at the strain of around 0.06 indicates the occurrence of a first-order phase transformation. Fig. 3(b) shows the corresponding stress–strain curve. Clearly, the stress suddenly jumps from tensile regime to compressive regime as the result of the phase transformation at the critical tensile stress of around 9.2 GPa.
Fig. 4(a) and (b) shows the volume evolution of ZrO2 single crystal over the course of tensile and compressive loadings, respectively. Interestingly, both curves show no signs of discontinuity, indicating the absence of ZrO2 phase transformation under these two loading conditions. It further suggests that some sort of shear stress component is required to enable the ZrO2 t-m transformation. To completely understand stress conditions able to trigger a transformation, we move forward to load the sample under various stress states. Fig. 5 shows the critical stress map for ZrO2 single crystal t-m phase transformation. Each point represents an individual test with a constant loading ratio 𝑟 = 𝜎𝑥 ∕𝜎𝑦 . The ratio ranges from −1 to 1, covering loading conditions from pure shear (r = −1), uniaxial (r = 0), to equal biaxial (r = 1). The distance between each point to the origin represents the effective stress magnitude to trigger the phase transformation. Clearly, we can see that stress states in the shear and equal biaxial tension are most favored to trigger the phase transformation. In many real applications where ZrO2 is used as a fracture toughness enhancement phase, it is critical to understand how the complicated stress around a running crack tip activates the ZrO2 transformation, and consequently how the crack propagation path is affected. Fig. 6(a) shows the snapshots of a model SiO2 -based glass-ceramic sample with a tetragonal ZrO2 particle embedded into SiO2 glass matrix under mechanical loading. The sample is loaded in the X direction with a pre-existing crack of 5 nm at the sample bottom. Periodic boundary conditions are applied in the X and Z direction, while free surface in the Y direction. A very thin layer of atoms on each free surface is gripped to mimic the complex loading conditions in many real circumstances. It seems that the ZrO2 nanocrystal undergoes phase transformation under the applied tensile load, then the propagating crack is deflected due to the inhomogeneous stress filed around the ZrO2 nanocrystal after the transformation. The ZrO2 t-m phase transformation is further clearly evidenced by Fig. 6(c) in which it shows the ZrO2 crystal structure before and after the phase transformation. Interestingly, it seems that the transformation nucleates at multiple sites at the interface, then propagates inward, and forms impinging boundaries which is shown clearly at Fig. 6 (c-2) and (c-3). In addition, we also probe the stress map evolution of the whole sample over the course of loading. Fig. 6 (d-1) and (d-2) shows the sample stress map right before and after the ZrO2 t-m phase transformation. Clearly, tensile stress is intensively accumulated inside the ZrO2 crystal with increasing tensile loading, which becomes compressive after triggering of the phase transformation, which echoes many previous studies on the toughening mechanism of ZrO2 -containing glass-ceramics.
B. Deng, J. Luo and J.T. Harris et al.
Materialia 9 (2020) 100548
Fig. 6. (a) Snapshots of SiO2 glass-ceramic sample with a tetragonal ZrO2 particle embedded into SiO2 glass matrix over the course of loading. The sample size is 54.5 × 54.5 × 2.7 nm3 , and the radius of ZrO2 particle is around 10.0 nm. The sample is loaded in the X direction orthogonal to a pre-existing crack of 5 nm. (b) The accompanying stress–strain curve. (c) Snapshot of ZrO2 transforming from tetragonal structure (c-1) to monoclinic crystal structure (c-2) with the closeup shown in (c-3). (d) The sample stress map right before (d-1) and after (d-2) the phase transformation.
4. Conclusion
References
Motived by recent observation of considerable presence of compressive stress induced tetragonal to monoclinic phase transformation in yttria-stabilized zirconia [17,18], we conduct a systematic study to investigate the critical stress map for ZrO2 phase transformation under various loading conditions using molecular dynamics simulations. The results on a single ZrO2 crystal shows that (1) uniaxial tensile and compressive loadings are able to trigger the phase transformation; (2) shear and equal biaxial tensile loading are most favorable loading conditions to trigger the phase transformation; (3) hydrostatic tensile and compressive loading are not able to trigger the phase transformation; Note that the aforementioned loading conditions are applicable in either [100] or [001] direction due to the symmetrical nature of tetragonal crystal structure, while any loading in the [001] direction is not able to trigger the phase transformation. In addition, we also conduct simulations on a model SiO2 -based glass-ceramic sample with a tetragonal ZrO2 particle embedded into SiO2 glass matrix under mechanical loading. The results show that the stress fields at the running crack front can help triggering the ZrO2 phase transformation, and consequently transform the stress state in ZrO2 from tensile to compressive, which retards and then deflects crack propagation. Although all observations are taken on an ideal ZrO2 single crystal, insights obtained from the simulations can help us better understand its transformation likelihood in real applications, and further inspire us to better leverage the ZrO2 transformation toughening mechanism to improve many material properties.
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Declaration of Competing Interest The authors declare that there are no known competing interests regarding to this work. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful for the technical support from the Corning Scientific Computing group.
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