1 Nature Materials 2011 Vol: 10(11):823-837. DOI: 10.1038/nmat3134

Poisson's ratio and modern materials

In comparing a material's resistance to distort under mechanical load rather than to alter in volume, Poisson's ratio offers the fundamental metric by which to compare the performance of any material when strained elastically. The numerical limits are set by ½ and −1, between which all stable isotropic materials are found. With new experiments, computational methods and routes to materials synthesis, we assess what Poisson's ratio means in the contemporary understanding of the mechanical characteristics of modern materials. Central to these recent advances, we emphasize the significance of relationships outside the elastic limit between Poisson's ratio and densification, connectivity, ductility and the toughness of solids; and their association with the dynamic properties of the liquids from which they were condensed and into which they melt.

Mentions
Figures
Figure 1: Poisson's ratio: physical significance, materials characteristics and the Milton map.a, Numerical window of Poisson's ratio ν, from −1 to ½, plotted as a function of the ratio of the bulk and shear moduli B/G for a wide range of isotropic classes of materials (see text for details). b, Milton map11 of bulk modulus versus shear modulus, showing the regimes of ν, and the differences in material characteristics. The stability boundaries are pertinent to phase transformations11, 12. c, Jiang and Dai plot32 of bulk modulus versus shear modulus for metallic glasses and polycrystalline metals. Straight lines refer to thresholds for brittle–ductile transition in metals. The ratio B/G = 5/3 (ν = 0.25) relates to the lower limit of Poisson's ratio for most metals. RE, rare earth. Ni- indicates nickel-based, and so on. Figure adapted with permission from: b, ref. 11, © 1992 Elsevier; ref. 12, © 1993 Wiley; c, ref. 32, © 2010 Taylor & Francis. Figure 2: Varying Poisson's ratio structurally.a, Foams: conventional polyurethane foam with ν ≈ 0.3 (A). Negative Poisson's ratio foam with folded-in (re-entrant) cells fabricated from polyurethane foam (B) and from copper foam (C), after Lakes9. Figure adapted with permission from ref. 9, © 1987 AAAS. b, Auxetic geometries: hierarchical laminate (left), after Milton11. Rotating hinged triangle and square structures (right) after Grima et al.25, 64. Left panel adapted with permission from ref. 11, © 1992 Elsevier; right panel reproduced with permission from ref. 64, © 2005 Wiley. c, Atomic motifs: crystal structure of α-cristobalite projected along the a axis (upper), after Haeri et al.7. Under compression the bridging oxygen pairs on opposite sides of the six-fold ring — a and b, c and d, e and f — move conversely through tetrahedral rotations as shown. Changing geometry of silver atoms in the van der Waals solid Ag3[Co(CN)6] (lower), showing the formation of a re-entrant honeycomb under 0.23 GPa pressure75. Co(CN)6 anions occupy adjacent interstitial locations and the arrows indicate the action underlying the displacive transition. Figure reproduced with permission from: Top panel, ref. 7, © 1992 AAAS; bottom panels, ref. 75, © 2008 NAS. d, Cellular structures: honeycomb structure found in cork66 normal to the direction of growth (upper) where ν ≤ 0 and parallel to this (middle) where ν > 0. Semi-re-entrant honeycomb structure proposed to model zero Poisson's ratio materials (lower)67. Figure reproduced with permission from: Top panel, ref. 66, © 2005 Maney; bottom panel, ref. 67, © 2010 Wiley. e, Time-dependent Poisson's ratio for poly(methyl methacrylate), PMMA, showing the gradual rise in ν(t) with relaxation under uniaxial shear from the instantaneous elastic value of 0.33 to the viscoelastic and eventual incompressible bulk value of 0.5. Figure reproduced with permission from ref. 20, © 1997 Wiley. Figure 3: Poisson's ratio and atomic packing.Poisson's ratio ν as a function of the atomic packing density Cg = ρΣifiVi/ΣiMi, where ρ is the specific mass, N is Avogadro's number, rA and rB are the ionic radii, fi is the molar fraction and Mi is the molar mass. For the ith constituent with chemical formula AxBy, Vi = (4/3)πN(xrA3 +yrB3). The distinct symbols show that there are monotonic and nearly linear increases of ν with Cg for each separate chemical system42. Figure adapted with permission from ref. 42, © 2007 Wiley. Figure 4: Connectivity in glasses and liquids.Poisson's ratio ν in glasses and liquids reflects their structure and its dependence on temperature. a, In glasses ν varies with the average coordination number <n> in chalcogenide glasses, or the number of bridging oxygens per glass-forming cation nBO in oxide glasses (ref. 42). <n> is defined as <n> = Σifini, where fi and ni are, respectively, the atomic fraction and the coordination number of the ith constituent, and nBO = 4 − ΣiMizi/ ΣjFj, where Mi and zi are respectively the atomic fraction (after deduction of the number of charge compensators) and the valence of the ith modifying cation, and Fj is the fraction of the jth glass-forming cation. b, The temperature dependence of ν for different glass-forming systems. The vertical bar marks the glass transition temperature Tg, above which sharp rises are observed, particularly for fragile supercooled liquids42. Figure reproduced with permission from ref. 42, © 2007 Wiley. Figure 5: Poisson's ratio and phase transformations.a, Bulk modulus B, shear modulus G and Poisson's ratio ν of a polymer gel versus temperature associated with a volume phase transition close to a critical point measured optically82. b, Bulk modulus, shear modulus and Poisson's ratio associated with the α–β first-order transition in quartz versus temperature, measured with resonant ultrasound spectroscopy (RUS)84. c, Melting of metals at ultra-high temperatures and pressures. Poisson's ratio of cerium97, aluminium88, iron36 and molybdenum34, 35 versus pressure during shock-melting experiments, with the incipient melting temperatures Tm indicated. The outer and inner pressure limits of the Earth's core are indicated, as is the brittle–ductile threshold from c and c. d, Poisson's ratio of silica glass during compression and decompression measured with Brillouin scattering39. Figure reproduced with permission from: a, ref. 82, © 1990 ACS; b, ref. 84, © 2008 IOP; d, ref. 39, © 1994 APS. Figure 6: Indentation of glasses: densification or shear flow.The effects of ν are seen in the stages of deformation under indentation. The surface profiles are shown for the indenter at maximum load (solid lines) and after unloading (dotted lines). Arrows indicate matter displacement. σ is the mean contact pressure. Reproduced with permission from ref. 46, © 2010 AIP. Figure 7: Poisson's ratio, non-ergodicity and fracture toughness.a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31. Thresholds for polycrystalline metals are indicated by vertical lines30. d, Fracture toughness log Efracture versus m for bulk metallic glasses, differentiating ductile from brittle character, with a sharp threshold close to m ≈ 30. Dashed curves in c and d are included to guide the eye. Figure reproduced with permission from: a, ref. 79, © 2007 Taylor & Francis; b, ref. 106, © 2010 APS; c, ref. 50, © 2005 Taylor & Francis. Figure 8: Boson peak and melt fragility.a, Reduction in low-frequency collective terahertz band during the collapse of zeolite Y with densification94 (left) ending in the formation of a glass. Zeolitic subunits α and β cages and double six-fold rings (D6R) features are retained in the LDA phase, even when 90% of the zeolite has amorphized. Reproduced with permission from ref. 94, © 2005 AAAS. Micrographs of zeolite and final glass (right) reveal onset of viscous flow as part of the LDL–HDL transition. Reproduced from ref. 93, © 2003 NPG. b, Reduction in the size of the boson peak ABP = D(ω)/ω calculated for a 2D glass-forming system under increasing pressure P, where D(ω) is the VDOS for acoustic modes103. c, Temperature dependence of the viscosity η versus Tg/T of the HDL and LDL supercooled phases together with the HDA and LDA glasses associated with the collapse of zeolite A. This shows how the increase in melt fragility m follows the LDA–HDA increase in densification93. The classical strong liquid SiO2, whose fragility falls between those of the two liquid phases, is included for comparison. d, Temperature dependence (Tg/T) of the structural relaxation time τ calculated for the 2D glass in b, showing the increase in melt fragility with increasing pressure P (ref. 103). Note that the fragility is given by m = [∂logη(T)/(∂Tg/T)]T=Tg and that η and τ are related by η = G∞τα (Box 2). Figure reproduced or adapted from: b,d, ref. 103, © 2008 NPG; c, ref. 93, © 2003 NPG.
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References
  1. Link , .
    • . . . Taking this wide perspective, we show how Poisson's ratio1 has provided inspiration for creating new solids and liquids, and challenges in understanding existing ones . . .
  2. Poisson, S. D. Traité de Mécanique 2, 476 (1811) .
    • . . . In the 200th year since the publication of Poisson's Traité de Mécanique2 (Box 1), this is a good time to take stock of the utility of Poisson's ratio. . . .
  3. Poisson, S. D. Ann. Chim. Phys. 36, 384-385 (1827) .
  4. Cauchy, A. L. Sur les équations qui expriment les conditions d'équilibre, ou les lois du mouvement intérieur d'un corps solide élastique ou non élastique Exercices de Mathématiques 3, (1828) .
    • . . . Once it was recognized that elastic moduli are independent4, 5, it could be seen that the two most appropriate for formulating ν are the isothermal bulk modulus, B = −VdP/dV = 1/κ, where κ is the isothermal compressibility, and the shear modulus G = σt/(2et) (ref. 6), as these are representative of the change in size and shape respectively . . .
  5. Voigt, W. Allgemeine Formeln für die Bestimmung der Elasticitätsconstanten von Krystallen durch die Beobachtung der Biegung und Drillung von Prismen Ann. Phys. 16, 273-310-398-415 (1882) .
    • . . . Once it was recognized that elastic moduli are independent4, 5, it could be seen that the two most appropriate for formulating ν are the isothermal bulk modulus, B = −VdP/dV = 1/κ, where κ is the isothermal compressibility, and the shear modulus G = σt/(2et) (ref. 6), as these are representative of the change in size and shape respectively . . .
  6. Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity , (1944) .
    • . . . Once it was recognized that elastic moduli are independent4, 5, it could be seen that the two most appropriate for formulating ν are the isothermal bulk modulus, B = −VdP/dV = 1/κ, where κ is the isothermal compressibility, and the shear modulus G = σt/(2et) (ref. 6), as these are representative of the change in size and shape respectively . . .
    • . . . The physical significance of ν is revealed by various interrelations between theoretical elastic properties6 . . .
    • . . . Likewise, metals are stiff6, 30, 31, B/G ranging from 1.7 to 5.6 and ν from 0.25 to 0.42 (ref. 32; Fig. 1c) . . .
    • . . . Through the elastic moduli, Poisson's ratio can also be expressed in terms of the transverse (shear) and longitudinal (compressive) speeds of sound, Vt and Vl respectively6 (Box 2) . . .
    • . . . In inorganic materials, anisotropic auxetic behaviour was suggested for iron pyrites6 with ν = 1/7 in some directions . . .
  7. Haeri, A. Y.; Weidner, D. J.; Parise, J. B. Elasticity of α-cristobalite: a silicon dioxide with a negative Poisson's ratio Science 257, 650-652 (1992) .
    • . . . Left panel adapted with permission from ref. 11, © 1992 Elsevier; right panel reproduced with permission from ref. 64, © 2005 Wiley. c, Atomic motifs: crystal structure of α-cristobalite projected along the a axis (upper), after Haeri et al.7 . . .
    • . . . Figure reproduced with permission from: Top panel, ref. 7, © 1992 AAAS; bottom panels, . . .
    • . . . For gases, ν = 0, and network structures can exhibit ν < 0 (ref. 7) . . .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . When B/G << 1 and ν −1 in Fig. 1b (horizontal axis), materials are extremely compressible, examples being re-entrant foams and molecular structures7, 11, 14, 16, 23, 24 . . .
    • . . . Rigid mechanical models, such as rotating hinged squares and triangles, replicate auxetic behaviour and serve as models for molecular structures and designs (Fig. 2b)7, 8, 25, 63, 64 . . .
    • . . . In α-cristobalite, Poisson's ratios range from +0.08 to −0.5, depending on direction (Fig. 2c)7, 71, with an aggregate expected to be negative25 . . .
  8. Evans, K. E.; Nkansah, M. A.; Hutchinson, I. J.; Rogers, S. C. Molecular network design Nature 353, 124 (1991) .
    • . . . Materials with negative Poisson's ratio are called 'auxetic'8 . . .
    • . . . Rigid mechanical models, such as rotating hinged squares and triangles, replicate auxetic behaviour and serve as models for molecular structures and designs (Fig. 2b)7, 8, 25, 63, 64 . . .
    • . . . Other anticipated enhancements include improved shear stiffness, self-adaptive vibrational damping and shock absorption, with applications in body armour, increased-sensitivity piezoelectric composites, fibre composites with greater pull-out resistance, a natural tendency to form dome-shaped surfaces (synclastic curvature), foams with improved filter performance and drug release, more comfortable textiles with reduced clothing pressure and so on8, 25, 64. . . .
    • . . . For anisotropic single crystals such as KH2PO4, the shear elastic tensor element C66 softens to zero during phase transformation8 . . .
  9. Lakes, R. S. Foam structures with a negative Poisson's ratio Science 235, 1038-1040 (1987) .
    • . . . Negative Poisson's ratio foam with folded-in (re-entrant) cells fabricated from polyurethane foam (B) and from copper foam (C), after Lakes9 . . .
    • . . . Figure adapted with permission from ref. 9, © 1987 AAAS. b, Auxetic geometries: hierarchical laminate (left), after Milton11 . . .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . Although for years the possibility of negative ν was excluded61, re-entrant foams (Fig. 2a) achieved values as negative as −0.8 (refs 9,10,11,12,13) . . .
  10. Caddock, B. D.; Evans, K. E. Microporous materials with negative Poisson's ratios. I: Microstructure and mechanical properties J. Phys. D. 22, 1877-1882 (1989) .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . Although for years the possibility of negative ν was excluded61, re-entrant foams (Fig. 2a) achieved values as negative as −0.8 (refs 9,10,11,12,13) . . .
  11. Milton, G. Composite materials with Poisson's ratios close to −1 J. Mech. Phys. Solids 40, 1105-1137 (1992) .
    • . . . a, Numerical window of Poisson's ratio ν, from −1 to ½, plotted as a function of the ratio of the bulk and shear moduli B/G for a wide range of isotropic classes of materials (see text for details). b, Milton map11 of bulk modulus versus shear modulus, showing the regimes of ν, and the differences in material characteristics . . .
    • . . . The stability boundaries are pertinent to phase transformations11, 12. c, Jiang and Dai plot32 of bulk modulus versus shear modulus for metallic glasses and polycrystalline metals . . .
    • . . . Figure adapted with permission from: b, ref. 11, © 1992 Elsevier; ref. 12, © 1993 Wiley; c, ref. 32, © 2010 Taylor & Francis. . . .
    • . . . Figure adapted with permission from ref. 9, © 1987 AAAS. b, Auxetic geometries: hierarchical laminate (left), after Milton11 . . .
    • . . . Left panel adapted with permission from ref. 11, © 1992 Elsevier; right panel reproduced with permission from ref. 64, © 2005 Wiley. c, Atomic motifs: crystal structure of α-cristobalite projected along the a axis (upper), after Haeri et al.7 . . .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . The huge diversity of elastic properties of modern and natural materials can also be viewed in plots of B versus G (ref. 11), as shown in Fig. 1b . . .
    • . . . These are illustrated in the Milton map of bulk isothermal modulus B versus shear modulus G (Fig. 1b)11, 22 . . .
    • . . . When B/G << 1 and ν −1 in Fig. 1b (horizontal axis), materials are extremely compressible, examples being re-entrant foams and molecular structures7, 11, 14, 16, 23, 24 . . .
    • . . . Although for years the possibility of negative ν was excluded61, re-entrant foams (Fig. 2a) achieved values as negative as −0.8 (refs 9,10,11,12,13) . . .
    • . . . It is not necessary to have empty space in a microstructure to achieve the effect: hierarchical two-phase laminates with a chevron structure and multiple length scales can approach the isotropic lower limit ν = −1 (ref. 11).These microstructures are played out in the α-cristobalite network (Fig. 2c) . . .
    • . . . Softening of the shear or the bulk modulus of perovskite minerals11 has been reported, and martensitic transformations are also characterized by a change in shear elastic moduli associated with changes in crystal structure . . .
    • . . . The elegant work of Jiang and Dai32 (Fig. 1c) explores relations between elastic moduli in crystalline and glassy metals in terms of a Milton map11 . . .
  12. Lakes, R. S. Advances in negative Poisson's ratio materials Adv. Mater. 5, 293-296 (1993) .
    • . . . The stability boundaries are pertinent to phase transformations11, 12. c, Jiang and Dai plot32 of bulk modulus versus shear modulus for metallic glasses and polycrystalline metals . . .
    • . . . Figure adapted with permission from: b, ref. 11, © 1992 Elsevier; ref. 12, © 1993 Wiley; c, ref. 32, © 2010 Taylor & Francis. . . .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . Although for years the possibility of negative ν was excluded61, re-entrant foams (Fig. 2a) achieved values as negative as −0.8 (refs 9,10,11,12,13) . . .
  13. Alderson, K. L.; Evans, K. E. The fabrication of microporous polyethylene having negative Poisson's ratio Polymer 33, 4435-4438 (1992) .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . Although for years the possibility of negative ν was excluded61, re-entrant foams (Fig. 2a) achieved values as negative as −0.8 (refs 9,10,11,12,13) . . .
  14. Baughman, R. H.; Shacklette, J-M.; Zakhidev, A. A.; Stafström, S. Negative Poisson's ratio as a common feature of cubic metals Nature 392, 362-365 (1998) .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . When B/G << 1 and ν −1 in Fig. 1b (horizontal axis), materials are extremely compressible, examples being re-entrant foams and molecular structures7, 11, 14, 16, 23, 24 . . .
    • . . . Many cubic metals when stretched in the [110] direction become auxetic14 . . .
  15. Sanchez-Valle, C. Negative Poisson's ratios in siliceous zeolite MFI-silicalite J. Chem. Phys. 128, 184503 (2008) .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . During the collapse of microporous crystals, ν rises from directionally auxetic values15, 25 to isotropic values of 0.2 typical of many glasses38 . . .
    • . . . Likewise, many zeolites show off-axis swings in ν on rotation15, 25 . . .
  16. Hall, L. J. Sign change of Poisson's ratio for carbon nanotube sheets Science 320, 504-507 (2008) .
    • . . . Re-entrant foams were the first reported9 (Fig. 2a) but subsequently it was shown that auxeticity is a common feature of a variety of honeycomb structures and networks (Fig. 2b–d), where ν can take both positive and negative values, depending on orientation7, 9, 10, 11, 12, 13, 14, 15, 16, with aggregate values that can be negative . . .
    • . . . When B/G << 1 and ν −1 in Fig. 1b (horizontal axis), materials are extremely compressible, examples being re-entrant foams and molecular structures7, 11, 14, 16, 23, 24 . . .
    • . . . Carbon nanotube sheets16, for example, exhibit in-plane auxeticity . . .
  17. Smith, C. W.; Wootton, R. J.; Evans, K. E. Interpretation of experimental data for Poisson's ratio of highly nonlinear materials Exp. Mech. 39, 356-362 (1999) .
    • . . . Nonlinear regime The concept of Poisson's ratio can be extended into the nonlinear regime17, 18, 19, to describe elastomers such as rubbers as well as glass fibres, when subjected to gigapascal tensile stresses . . .
    • . . . For instance, ν was found to decrease from 0 to −14 for an anisotropic expanded PTFE in a true strain range of 0.03 (ref. 17) . . .
  18. Tschoegl, N. W.; Knauss, W. J.; Emri, I. Poisson's ratio in linear viscoelasticity, a critical review Mech. Time-Depend. Mater. 6, 3-51 (2002) .
    • . . . Nonlinear regime The concept of Poisson's ratio can be extended into the nonlinear regime17, 18, 19, to describe elastomers such as rubbers as well as glass fibres, when subjected to gigapascal tensile stresses . . .
    • . . . In particular, ν*(f) is a complex function of frequency f or a function ν(t) of time t which can be obtained from creep and stress relaxation functions18, 19 . . .
    • . . . In sharp contrast, polymers are compliant and yet they share similar values: that is, B/G ≈ 8/3 and ν ≈ 0.33 (refs 18,19,20), the difference relating to the magnitude of the elastic moduli, decades smaller than for inorganic materials (see Fig. 5). . . .
    • . . . For the glass-to-rubber transition in polymers the shear modulus may change by more than a factor of a thousand, and the bulk modulus by about a factor of two18 . . .
  19. Lakes, R. S.; Wineman, A. On Poisson's ratio in linearly viscoelastic solids J. Elast. 85, 46-63 (2006) .
    • . . . Nonlinear regime The concept of Poisson's ratio can be extended into the nonlinear regime17, 18, 19, to describe elastomers such as rubbers as well as glass fibres, when subjected to gigapascal tensile stresses . . .
    • . . . In sharp contrast, polymers are compliant and yet they share similar values: that is, B/G ≈ 8/3 and ν ≈ 0.33 (refs 18,19,20), the difference relating to the magnitude of the elastic moduli, decades smaller than for inorganic materials (see Fig. 5). . . .
  20. Lu, H.; Zhang, X.; Krauss, W. G. Uniaxial, shear, and Poisson relaxation and their conversion to bulk relaxation: studies on poly(methyl methacrylate) Polym. Eng. Sci. 37, 1053-1064 (1997) .
    • . . . Figure reproduced with permission from ref. 20, © 1997 Wiley. . . .
    • . . . Despite these complexities an increase of ν(t) with time is often reported20, Poisson's ratio tending to ½ for most polymer materials . . .
    • . . . In sharp contrast, polymers are compliant and yet they share similar values: that is, B/G ≈ 8/3 and ν ≈ 0.33 (refs 18,19,20), the difference relating to the magnitude of the elastic moduli, decades smaller than for inorganic materials (see Fig. 5). . . .
    • . . . As illustrated in Fig. 2e, ν is about 0.3 in the glassy regime rising to nearly 0.5 in the rubbery regime20 . . .
  21. Link , .
    • . . . Conversely, ν*(f) decreases with increasing frequency because the elastic regime is favoured at high rates21. . . .
  22. Wang, Y. C.; Lakes, R. S. Composites with inclusions of negative bulk modulus: extreme damping and negative Poisson's ratio J. Comp. Mater. 39, 1645-1657 (2005) .
    • . . . These are illustrated in the Milton map of bulk isothermal modulus B versus shear modulus G (Fig. 1b)11, 22 . . .
  23. Smith, C. W.; Grima, J. N.; Evans, K. E. A novel mechanism for generating auxetic behaviour in reticulated foams: missing rib foam model Acta. Mater. 48, 4349-4356 (2000) .
    • . . . When B/G << 1 and ν −1 in Fig. 1b (horizontal axis), materials are extremely compressible, examples being re-entrant foams and molecular structures7, 11, 14, 16, 23, 24 . . .
  24. Baughman, R. H. Negative Poisson's ratios for extreme states of matter Science 288, 2018-2022 (2000) .
    • . . . When B/G << 1 and ν −1 in Fig. 1b (horizontal axis), materials are extremely compressible, examples being re-entrant foams and molecular structures7, 11, 14, 16, 23, 24 . . .
    • . . . Microstructures such as composites with rotating discs and units, magnetic films, hypothetical granular structures or plasmas in neutron stars can also exhibit such effects24, 64, 65. . . .
  25. Grima, J. N.; Jackson, R.; Alderson, A.; Evans, K. E. Do zeolites have negative Poisson's ratios? Adv. Mater. B 12, 1912-1918 (2000) .
    • . . . Rotating hinged triangle and square structures (right) after Grima et al.25, 64 . . .
    • . . . Materials possessing stiff arms or struts in directions normal to the loading axis, such as honeycomb structures loaded along the c axis, will resist transverse contraction and exhibit ν ≈ 0 as cork does (Fig. 2d) or ν < 0 as some zeolites might (Fig. 2b)25 . . .
    • . . . During the collapse of microporous crystals, ν rises from directionally auxetic values15, 25 to isotropic values of 0.2 typical of many glasses38 . . .
    • . . . Rigid mechanical models, such as rotating hinged squares and triangles, replicate auxetic behaviour and serve as models for molecular structures and designs (Fig. 2b)7, 8, 25, 63, 64 . . .
    • . . . Other anticipated enhancements include improved shear stiffness, self-adaptive vibrational damping and shock absorption, with applications in body armour, increased-sensitivity piezoelectric composites, fibre composites with greater pull-out resistance, a natural tendency to form dome-shaped surfaces (synclastic curvature), foams with improved filter performance and drug release, more comfortable textiles with reduced clothing pressure and so on8, 25, 64. . . .
    • . . . In α-cristobalite, Poisson's ratios range from +0.08 to −0.5, depending on direction (Fig. 2c)7, 71, with an aggregate expected to be negative25 . . .
    • . . . Likewise, many zeolites show off-axis swings in ν on rotation15, 25 . . .
    • . . . Such anisotropic auxetic behaviour might well influence adsorption chemistry, if crystals are stressed along specific directions25, 73. . . .
    • . . . Notwithstanding the considerable directional swings exhibited by single crystals25, 74, Poisson's ratio for these low-density crystalline materials typically averages out around 0.2 . . .
  26. Poirier, J-P. Introduction to the Physics of the Earth's Interior , (2000) .
    • . . . For ceramics, glasses and semiconductors, B/G ≈ 5/3 and ν 1/4 (refs 26,27,28,29) . . .
    • . . . These speeds and their ratio are pertinent to seismic waves studied in geophysics26 . . .
    • . . . Shock-wave melting of metals Despite its importance across the physical sciences, melting remains incompletely understood in terms of temperature–pressure melting curves and the underlying equations of state26 . . .
    • . . . Melting is traditionally understood to occur as the average root-mean-square displacement of atoms in the solid state approaches ~10% of the average interatomic separation—the so-called Lindemann criterion26 . . .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
  27. Cohen, M. L. Calculation of bulk moduli of diamond and zinc–blende solids Phys. Rev. B 32, 7988-7991 (1985) .
    • . . . For ceramics, glasses and semiconductors, B/G ≈ 5/3 and ν 1/4 (refs 26,27,28,29) . . .
    • . . . In the case of crystalline structures, the valence electron density plays a key role, and ab-initio calculation is becoming an increasingly popular approach following the early work of Cohen27 . . .
  28. Fukumoto, A. First-principles pseudopotential calculations of the elastic properties of diamond, Si, and Ge Phys. Rev. B 42, 7462-7469 (1990) .
    • . . . For ceramics, glasses and semiconductors, B/G ≈ 5/3 and ν 1/4 (refs 26,27,28,29) . . .
    • . . . Subsequently, the cases of diamond semiconductors (νSi = 0.22 and νGe = 0.28) were reported28, as were the fullerites29, molybdenum40 and most recently dental amalgams41 . . .
  29. Perottoni, C. A.; Da Jornada, J. A. H. First-principles calculation of the structure and elastic properties of a 3D-polymerized fullerite Phys. Rev. B 65, 224208 (2002) .
    • . . . For ceramics, glasses and semiconductors, B/G ≈ 5/3 and ν 1/4 (refs 26,27,28,29) . . .
    • . . . Subsequently, the cases of diamond semiconductors (νSi = 0.22 and νGe = 0.28) were reported28, as were the fullerites29, molybdenum40 and most recently dental amalgams41 . . .
  30. Cottrell, A. H. Advances in Physical Metallurgy , (1990) .
    • . . . Thresholds for polycrystalline metals are indicated by vertical lines30. d, Fracture toughness log Efracture versus m for bulk metallic glasses, differentiating ductile from brittle character, with a sharp threshold close to m ≈ 30 . . .
    • . . . Likewise, metals are stiff6, 30, 31, B/G ranging from 1.7 to 5.6 and ν from 0.25 to 0.42 (ref. 32; Fig. 1c) . . .
    • . . . Indeed, Poisson's ratio provides a sharp criterion for differentiating brittleness from ductility in crystalline30, 31 and in amorphous50, 51 metals (see Fig. 7c) . . .
    • . . . Unlike glasses which are generally homogeneous and isotropic but lack atomic long-range order79, 107, 109, polycrystalline materials have unit cell symmetry but atomic periodicity is broken internally by dislocations and impurities, and externally by grain boundaries30, 31 . . .
    • . . . Although there is no simple link between interatomic potentials and mechanical toughness in polycrystalline materials, Poisson's ratio ν has proved valuable for many years as a criterion for the brittle–ductile transition exhibited by metals30, 31, 32, 115, just as it is now helping to distinguish brittle glasses from ductile glasses (Fig. 7c) which, from Fig. 7a, are associated with strong and fragile melts respectively79, 100, 101. . . .
    • . . . The strengths of pure metals at low temperatures are known to be mainly governed by the strengths of grain boundaries30 . . .
    • . . . The brittle–ductile threshold occurs close to B/G ≈ 2.4 (ν = 0.32) and is generally higher than the thresholds reported for polycrystalline metals30, 31(Fig. 7c) . . .
    • . . . Plastic flow in metallic glasses occurs very locally in shear bands51, 114, compared with polycrystalline metals where flow is dislocation-mediated, and delocalized by associated work-hardening30. . . .
  31. Kelly, A.; Tyson, W. R.; Cottrell, A. H. Ductile and brittle crystals Phil. Mag. 15, 567-586 (1967) .
    • . . . Likewise, metals are stiff6, 30, 31, B/G ranging from 1.7 to 5.6 and ν from 0.25 to 0.42 (ref. 32; Fig. 1c) . . .
    • . . . Conversely, if the collective dynamics of connected polyhedra are significant, ν will be reduced with increased G encouraging embrittlement beyond the elastic limit31, 51 . . .
    • . . . Indeed, Poisson's ratio provides a sharp criterion for differentiating brittleness from ductility in crystalline30, 31 and in amorphous50, 51 metals (see Fig. 7c) . . .
    • . . . Unlike glasses which are generally homogeneous and isotropic but lack atomic long-range order79, 107, 109, polycrystalline materials have unit cell symmetry but atomic periodicity is broken internally by dislocations and impurities, and externally by grain boundaries30, 31 . . .
    • . . . These usually exceed the crystalline cleavage strength, which is governed by the dynamics of dislocations generated at the crack tip31 . . .
    • . . . The brittle–ductile threshold occurs close to B/G ≈ 2.4 (ν = 0.32) and is generally higher than the thresholds reported for polycrystalline metals30, 31(Fig. 7c) . . .
  32. Jiang, M. Q.; Dai, L. H. Short-range-order effects on the intrinsic plasticity of metallic glasses Phil. Mag. Lett. 90, 269-277 (2010) .
    • . . . The stability boundaries are pertinent to phase transformations11, 12. c, Jiang and Dai plot32 of bulk modulus versus shear modulus for metallic glasses and polycrystalline metals . . .
    • . . . Figure adapted with permission from: b, ref. 11, © 1992 Elsevier; ref. 12, © 1993 Wiley; c, ref. 32, © 2010 Taylor & Francis. . . .
    • . . . Likewise, metals are stiff6, 30, 31, B/G ranging from 1.7 to 5.6 and ν from 0.25 to 0.42 (ref. 32; Fig. 1c) . . .
    • . . . Although there is no simple link between interatomic potentials and mechanical toughness in polycrystalline materials, Poisson's ratio ν has proved valuable for many years as a criterion for the brittle–ductile transition exhibited by metals30, 31, 32, 115, just as it is now helping to distinguish brittle glasses from ductile glasses (Fig. 7c) which, from Fig. 7a, are associated with strong and fragile melts respectively79, 100, 101. . . .
    • . . . The elegant work of Jiang and Dai32 (Fig. 1c) explores relations between elastic moduli in crystalline and glassy metals in terms of a Milton map11 . . .
  33. McQueen, R. G.; Hopson, J. W.; Fritz, J. N. Optical technique for determining rarefaction wave velocities at very high pressures Rev. Sci. Instrum. 53, 245-250 (1982) .
    • . . . For example, when metals melt, ν increases from ~0.3 to 0.5 (refs 33,34,35,36,37) . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
  34. Santamaría-Pérez, D. X-ray diffraction measurements of Mo melting at 119 GPa and the high pressure phase diagram J. Chem. Phys. 130, 124509 (2009) .
    • . . . Poisson's ratio of cerium97, aluminium88, iron36 and molybdenum34, 35 versus pressure during shock-melting experiments, with the incipient melting temperatures Tm indicated . . .
    • . . . For example, when metals melt, ν increases from ~0.3 to 0.5 (refs 33,34,35,36,37) . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
    • . . . Originally interpreted as a bcc to hcp transition35, recent X-ray measurements of the melting curve for Mo, made in a diamond-anvil cell, suggest that values of Poisson's ratio before Tm, which are unusually high for bcc metals (Fig. 1c), point to a new intermediate non-crystalline phase34—perhaps a low-density polyamorph . . .
  35. Hixson, R. S.; Boness, D. A.; Shaner, J. W. Acoustic velocities and phase transitions in molybdenum under strong shock compression Phys. Rev. Lett. 62, 637-640 (1989) .
    • . . . Poisson's ratio of cerium97, aluminium88, iron36 and molybdenum34, 35 versus pressure during shock-melting experiments, with the incipient melting temperatures Tm indicated . . .
    • . . . For example, when metals melt, ν increases from ~0.3 to 0.5 (refs 33,34,35,36,37) . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
    • . . . Shock melting is accompanied by a sharp increase in Poisson's ratio to the liquid value of 0.5, pinpointed by the intersection of the shock adiabatic, or Hugoniot, with the melting curve obtained from the equations of state35 . . .
    • . . . Originally interpreted as a bcc to hcp transition35, recent X-ray measurements of the melting curve for Mo, made in a diamond-anvil cell, suggest that values of Poisson's ratio before Tm, which are unusually high for bcc metals (Fig. 1c), point to a new intermediate non-crystalline phase34—perhaps a low-density polyamorph . . .
  36. Nguyen, J. H.; Holmes, N. C. Melting of iron at the physical conditions of the Earth's core Nature 427, 339-342 (2004) .
    • . . . Poisson's ratio of cerium97, aluminium88, iron36 and molybdenum34, 35 versus pressure during shock-melting experiments, with the incipient melting temperatures Tm indicated . . .
    • . . . For example, when metals melt, ν increases from ~0.3 to 0.5 (refs 33,34,35,36,37) . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
    • . . . The inner and outer pressure limits of the Earth's core surround the melting of Fe, underlining the relevance of these experiments for determining the temperature of the core by parameterizing the equations of state36 . . .
  37. Jensen, B. J.; Cherne, F. J.; Cooley, J. C. Shock melting of cerium Phys. Rev. B 81, 214109 (2010) .
    • . . . For example, when metals melt, ν increases from ~0.3 to 0.5 (refs 33,34,35,36,37) . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
  38. Greaves, G. N. Zeolite collapse and polyamorphism J. Phys. Cond. Mat. 19, 415102 (2007) .
    • . . . During the collapse of microporous crystals, ν rises from directionally auxetic values15, 25 to isotropic values of 0.2 typical of many glasses38 . . .
    • . . . Polyamorphism is not in dispute, but whether or not transformations between polyamorphic phases are of first order is controversial38, 58, 59, 79, 80, 89, 90 . . .
    • . . . An alternative approach has been discovered that involves the amorphization of microporous crystalline materials, such as zeolites38, 93, 94, 95, tungstates96 or metal–organic frameworks97 . . .
    • . . . In situ small-angle X-ray scattering (SAXS)93 and inelastic X-ray scattering experiments38, 98 reveal that Poisson's ratio drops considerably during amorphization before the final HDA glass is formed, by which time ν is close to 0.25, similar to other aluminosilicate glasses42 . . .
    • . . . Poisson's ratios for these ordered glasses are expected to be significantly smaller than their melt-quenched HDA counterparts38. . . .
    • . . . The overall distribution narrows towards the glasses associated with the strongest liquids38, 79 . . .
    • . . . Although silica is usually considered the strongest of these, experiments on zeolite amorphisation reveal that for new low-density liquids (LDL) m ≤ 10 (ref. 93; Fig. 8c), with the associated LDA glasses being topologically ordered38, 79, 99 . . .
  39. Zha, C-S.; Hemley, R. J.; Mao, H-K.; Duffy, T. S.; Meade, C. Acoustic velocities and refractive index of SiO2 glass to 57.5 GPa by Brillouin scattering Phys. Rev. B 50, 13105-13112 (1994) .
    • . . . The outer and inner pressure limits of the Earth's core are indicated, as is the brittle–ductile threshold from Fig. 1c and Fig. 7c. d, Poisson's ratio of silica glass during compression and decompression measured with Brillouin scattering39 . . .
    • . . . Figure reproduced with permission from: a, ref. 82, © 1990 ACS; b, ref. 84, © 2008 IOP; d, ref. 39, © 1994 APS. . . .
    • . . . With densification Poisson's ratio for glasses continues to rise, for silica increasing from 0.19 to 0.33 (ref. 39; see Fig. 5d). . . .
    • . . . Nevertheless, a direct comparison of Poisson's ratio for specimens of identical composition and initial structure but submitted to treatments affecting the packing density (isostatic pressing, agitation, annealing to promote structural relaxation) provides clear evidence for a sensitive increase of ν in most cases39, 44, 45 . . .
    • . . . After treatments under high pressure (typically above 10 GPa), ν increases to 0.25 (ref. 46), reaching 0.33 above 30 GPa (ref. 39; see Fig. 5d), the effect of densification depending on the initial ν value47 . . .
    • . . . For both HDA ice58 and densified silica39, careful in situ sound velocity experiments reveal major changes in elastic moduli at these phase transformations . . .
    • . . . Decompression returns a densified tetrahedral phase39. . . .
  40. Zeng, Z-Y.; Hu, C-E.; Cai L-C.; Chen, X-R.; Jing, F-Q. Lattice dynamics and thermodynamics of molybdenum from first-principles calculations J. Phys. Chem. B 114, 298-310 (2010) .
    • . . . Subsequently, the cases of diamond semiconductors (νSi = 0.22 and νGe = 0.28) were reported28, as were the fullerites29, molybdenum40 and most recently dental amalgams41 . . .
  41. Davies, R. A. Geometric, electronic and elastic properties of dental silver amalgam γ-(Ag3Sn), γ1-(Ag2Hg3), γ2-(Sn8Hg) phases, comparison of experiment and theory Intermetallics 18, 756-760 (2010) .
    • . . . Subsequently, the cases of diamond semiconductors (νSi = 0.22 and νGe = 0.28) were reported28, as were the fullerites29, molybdenum40 and most recently dental amalgams41 . . .
  42. Rouxel, T. Elastic properties and short-to-medium range order in glasses J. Am. Ceram. Soc. 90, 3019-3039 (2007) .
    • . . . The distinct symbols show that there are monotonic and nearly linear increases of ν with Cg for each separate chemical system42 . . .
    • . . . Figure adapted with permission from ref. 42, © 2007 Wiley. . . .
    • . . . Poisson's ratio ν in glasses and liquids reflects their structure and its dependence on temperature. a, In glasses ν varies with the average coordination number <n> in chalcogenide glasses, or the number of bridging oxygens per glass-forming cation nBO in oxide glasses (ref. 42). <n> is defined as <n> = Σifini, where fi and ni are, respectively, the atomic fraction and the coordination number of the ith constituent, and nBO = 4 − ΣiMizi/ ΣjFj, where Mi and zi are respectively the atomic fraction (after deduction of the number of charge compensators) and the valence of the ith modifying cation, and Fj is the fraction of the jth glass-forming cation. b, The temperature dependence of ν for different glass-forming systems . . .
    • . . . The vertical bar marks the glass transition temperature Tg, above which sharp rises are observed, particularly for fragile supercooled liquids42 . . .
    • . . . Eventually, within limited compositional ranges, linear trends can be observed42 . . .
    • . . . Highly crosslinked networks, such as silica glass, lead to small Poisson's ratios (0.19), whereas weakly correlated networks, such as chain-based chalcogenide glasses or cluster-based metallic glasses, show values between 0.3 and 0.4 (Fig. 4a)42 . . .
    • . . . Hence, as already noticed by Bridge and Higazy52 in a study limited to some oxide glasses, and by Sreeram et al.53 for chalcogenide glasses and later generalized by Rouxel42 for a wide range of glasses including covalent and metallic ones, ν depends almost linearly on connectivity and increases as the dimensionality of the structural units decreases. . . .
    • . . . A steep increase in Poisson's ratio reveals rapid network depolymerization, as is the case for organic chain polymers such as glycerol or polystryrene and also for a-B2O3 (ref. 42) which transform into fragile liquids (Box 2) . . .
    • . . . In situ small-angle X-ray scattering (SAXS)93 and inelastic X-ray scattering experiments38, 98 reveal that Poisson's ratio drops considerably during amorphization before the final HDA glass is formed, by which time ν is close to 0.25, similar to other aluminosilicate glasses42 . . .
  43. Makishima, A.; Mackenzie, J. D. Calculation of bulk modulus, shear modulus and Poisson's ratio of glass J. Non-Cryst. Sol. 17, 147-157 (1975) .
    • . . . Much earlier, Makishima had proposed that ν = ½ − 1/7.2Cg for silicate glasses43 but without knowledge of accurate atomic glass network parameters, such as interatomic distance and coordination number. . . .
  44. Antao, S. M. Network rigidity in GeSe2 glass at high pressure Phys. Rev. Lett. 100, 115501 (2008) .
    • . . . Nevertheless, a direct comparison of Poisson's ratio for specimens of identical composition and initial structure but submitted to treatments affecting the packing density (isostatic pressing, agitation, annealing to promote structural relaxation) provides clear evidence for a sensitive increase of ν in most cases39, 44, 45 . . .
  45. Nicholas, J.; Sinogeikin, S.; Kieffer, J.; Bass J. A high pressure Brillouin scattering study of vitreous boron oxide up to 57 GPa J. Non-Cryst. Sol. 349, 30-34 (2004) .
    • . . . Nevertheless, a direct comparison of Poisson's ratio for specimens of identical composition and initial structure but submitted to treatments affecting the packing density (isostatic pressing, agitation, annealing to promote structural relaxation) provides clear evidence for a sensitive increase of ν in most cases39, 44, 45 . . .
  46. Rouxel, T.; Ji, H.; Guin, J. P.; Augereau, F.; Rufflé, B. Indentation deformation mechanism in glass: densification versus shear flow J. Appl. Phys. 107, 094903 (2010) .
    • . . . Reproduced with permission from ref. 46, © 2010 AIP. . . .
    • . . . After treatments under high pressure (typically above 10 GPa), ν increases to 0.25 (ref. 46), reaching 0.33 above 30 GPa (ref. 39; see Fig. 5d), the effect of densification depending on the initial ν value47 . . .
    • . . . The increase of ν with Cg is observed either along elastic loading paths or after permanent densification46 . . .
    • . . . A direct manifestation of the correlations between ν and Cg (Fig. 3) and densification47 concerns the way matter deforms under high contact pressure, such as during indentation or scratch loading46 . . .
    • . . . For high-Cg materials (precious metals, metallic glasses, clay), deformation is nearly isochoric or involves some dilation, and proceeds by localized shear (Fig. 6)46, 49, 50, 51 . . .
  47. Rouxel, T.; Ji, H.; Hammouda, T.; Moreac, A. Poisson's ratio and the densification of glass under high pressure Phys. Rev. Lett. 100, 225501 (2008) .
    • . . . After treatments under high pressure (typically above 10 GPa), ν increases to 0.25 (ref. 46), reaching 0.33 above 30 GPa (ref. 39; see Fig. 5d), the effect of densification depending on the initial ν value47 . . .
    • . . . A direct manifestation of the correlations between ν and Cg (Fig. 3) and densification47 concerns the way matter deforms under high contact pressure, such as during indentation or scratch loading46 . . .
  48. Das, B. M. Advanced Soil Mechanics , (2002) .
    • . . . The same situation is observed with sand: for loose sand ν is typically around 0.2 whereas dense sand reaches 0.45 and saturated cohesive soils are almost incompressible (ν = 0.5)48 . . .
  49. Ji, H.; Robin, E.; Rouxel, T. Physics and mechanics of the deformation of plasticine: macroscopic indentation behaviour for temperature between 103–293 K J. Mech. Mat. 41, 199-209 (2009) .
    • . . . For high-Cg materials (precious metals, metallic glasses, clay), deformation is nearly isochoric or involves some dilation, and proceeds by localized shear (Fig. 6)46, 49, 50, 51 . . .
  50. Lewandowski, J. J.; Wang, W. H.; Greer, A. L. Intrinsic plasticity or brittleness of metallic glasses Phil. Mag. Lett 85, 77-87 (2005) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . Figure reproduced with permission from: a, ref. 79, © 2007 Taylor & Francis; b, ref. 106, © 2010 APS; c, ref. 50, © 2005 Taylor & Francis. . . .
    • . . . For high-Cg materials (precious metals, metallic glasses, clay), deformation is nearly isochoric or involves some dilation, and proceeds by localized shear (Fig. 6)46, 49, 50, 51 . . .
    • . . . Indeed, Poisson's ratio provides a sharp criterion for differentiating brittleness from ductility in crystalline30, 31 and in amorphous50, 51 metals (see Fig. 7c) . . .
    • . . . In metallic glasses, though—particularly bulk metallic glasses (BMG) that are cast by slow cooling like conventional glasses118—sufficient data on elastic moduli and toughness50 are available, as well as on melt fragility102, to test these ideas quantitatively . . .
    • . . . From the most brittle metallic glasses to the toughest, fracture energies Efracture increase by as much as four decades50 . . .
  51. Lewandowski, J. J.; Greer, A. L. Temperature rise at shear bands in metallic glasses Nature Mater. 5, 15-18 (2006) .
    • . . . For high-Cg materials (precious metals, metallic glasses, clay), deformation is nearly isochoric or involves some dilation, and proceeds by localized shear (Fig. 6)46, 49, 50, 51 . . .
    • . . . Conversely, if the collective dynamics of connected polyhedra are significant, ν will be reduced with increased G encouraging embrittlement beyond the elastic limit31, 51 . . .
    • . . . Plastic flow in metallic glasses occurs very locally in shear bands51, 114, compared with polycrystalline metals where flow is dislocation-mediated, and delocalized by associated work-hardening30. . . .
    • . . . Recalling that fracture energy and melt fragility are correlated (Fig. 7d), its seems likely that the ductile and brittle properties of metals, whether they are glasses or crystals, are intimately related to the viscous time-dependent properties of their supercooled antecedents, either constrained in metallic glass shear bands51 or, more speculatively, in polycrystalline grain boundaries117. . . .
  52. Bridge, B.; Higazy, A. A. A model of the compositional dependence of the elastic moduli of multicomponent oxide glasses Phys. Chem. Glasses 27, 1-14 (1986) .
    • . . . Hence, as already noticed by Bridge and Higazy52 in a study limited to some oxide glasses, and by Sreeram et al.53 for chalcogenide glasses and later generalized by Rouxel42 for a wide range of glasses including covalent and metallic ones, ν depends almost linearly on connectivity and increases as the dimensionality of the structural units decreases. . . .
  53. Sreeram, A. N.; Varshneya, A. K.; Swiler, D. R. Molar volume and elastic properties of multicomponent chalcogenide glasses J. Non-Cryst. Sol. 128, 294-309 (1991) .
    • . . . Hence, as already noticed by Bridge and Higazy52 in a study limited to some oxide glasses, and by Sreeram et al.53 for chalcogenide glasses and later generalized by Rouxel42 for a wide range of glasses including covalent and metallic ones, ν depends almost linearly on connectivity and increases as the dimensionality of the structural units decreases. . . .
  54. Moysan, C.; Riedel, R.; Harshe, R.; Rouxel, T.; Augereau, F. Mechanical characterization of a polysiloxane-derived SiOC glass J. Europ. Ceram. Soc. 27, 397-403 (2007) .
    • . . . Extreme cases are illustrated on the one hand by silicon oxycarbide glasses, where the formation of CSi4 tetrahedra based on fourfold covalent carbon atoms further enhances the network cross-linking over that of a-SiO2, with ν reaching 0.11 for the polymer-derived SiO1.6C0.8 composition54 . . .
  55. Miracle, D. B. A structural model for metallic glasses Nature Mater. 3, 697-701 (2004) .
    • . . . On the other hand, for precious-metal-based metallic glasses, which are considered to consist of quasi-equivalent cluster-type units (0D) eventually packed with icosahedral-like medium range order55, 56, ν approaches 0.4 . . .
  56. Sheng, H. W.; Luo, W. K.; Alamgir, F. M.; Bai, J. M.; Ma, E. Atomic packing density and short-to-medium range order in metallic glasses Nature 439, 419-425 (2006) .
    • . . . On the other hand, for precious-metal-based metallic glasses, which are considered to consist of quasi-equivalent cluster-type units (0D) eventually packed with icosahedral-like medium range order55, 56, ν approaches 0.4 . . .
  57. Hessinger, J.; White, B. E.; Pohl, R. O. Elastic properties of amorphous and crystalline ice films Planet. Space Sci. 44, 937-944 (1996) .
    • . . . Judging from a Poisson's ratio of 0.3, the glassy ice network might be based on chain-like hydrogen-bonding of water molecules or contain cluster-like structural units, for example icosahedral clusters as suggested by Hessinger et al.57 . . .
  58. Loerting, T.; Giovambattista, N. Amorphous ices: experiments and numerical simulations J. Phys. Cond. Mat. 18, R919-R977 (2006) .
    • . . . However, diffraction studies of low-density amorphous (LDA) ice reveal a hydrogen-bonded tetrahedral network mainly comprising sixfold rings58, rather like water . . .
    • . . . This differs from LDA ice by the presence of interstitial molecules58 that interrupt the network topology. . . .
    • . . . At different temperatures and pressures, crystalline materials can undergo phase transitions78 and, attracting considerable debate, so too can glasses and liquids58, 79, 80, 81 . . .
    • . . . Polyamorphism is not in dispute, but whether or not transformations between polyamorphic phases are of first order is controversial38, 58, 59, 79, 80, 89, 90 . . .
    • . . . In addition to LDA and HDA amorphous ice58, 59, polyamorphism has been reported in many tetrahedral glasses89 . . .
  59. Mishima, O.; Calvert, L. D.; Whalley, E. 'Melting ice' I at 77 K and 10 kbar: a new method of making amorphous materials Nature 310, 393-395 (1984) .
    • . . . With the application of pressure, crystalline hexagonal ice transforms into high-density amorphous (HDA) ice59 . . .
    • . . . Polyamorphism is not in dispute, but whether or not transformations between polyamorphic phases are of first order is controversial38, 58, 59, 79, 80, 89, 90 . . .
  60. Gibson, L. J.; Ashby, M. F. Cellular Solids , (1997) .
    • . . . Indeed the analysis of foam structures60 predicts that ν increases as φ decreases . . .
  61. Beer, F. P.; Johnston, E. R. Mechanics of Materials , (1992) .
    • . . . Although for years the possibility of negative ν was excluded61, re-entrant foams (Fig. 2a) achieved values as negative as −0.8 (refs 9,10,11,12,13) . . .
    • . . . In mixed valence transitions in YbInCu4 (ref. 61), the cubic crystal structure does not change; but the bulk modulus softens and Poisson's ratio drops over a narrow range of temperature near 67 K . . .
  62. Lakes, R. S. Negative Poisson's ratio materials Science 238, 551 (1987) .
    • . . . A coarse cell structure is not required to predict a positive or negative Poison's ratio; classical elastic properties have no length scale62 . . .
  63. Wojciechowski, K. W. Two-dimensional isotropic system with a negative Poisson ratio Phys. Lett. A 137, 60-64 (1989) .
    • . . . Rigid mechanical models, such as rotating hinged squares and triangles, replicate auxetic behaviour and serve as models for molecular structures and designs (Fig. 2b)7, 8, 25, 63, 64 . . .
  64. Grima, J. N.; Alderson, A.; Evans, K. E. Auxetic behaviour from rotating rigid units Phys. Stat. Solidi B 242, 561-75 (2005) .
    • . . . Rotating hinged triangle and square structures (right) after Grima et al.25, 64 . . .
    • . . . Left panel adapted with permission from ref. 11, © 1992 Elsevier; right panel reproduced with permission from ref. 64, © 2005 Wiley. c, Atomic motifs: crystal structure of α-cristobalite projected along the a axis (upper), after Haeri et al.7 . . .
    • . . . Rigid mechanical models, such as rotating hinged squares and triangles, replicate auxetic behaviour and serve as models for molecular structures and designs (Fig. 2b)7, 8, 25, 63, 64 . . .
    • . . . Other anticipated enhancements include improved shear stiffness, self-adaptive vibrational damping and shock absorption, with applications in body armour, increased-sensitivity piezoelectric composites, fibre composites with greater pull-out resistance, a natural tendency to form dome-shaped surfaces (synclastic curvature), foams with improved filter performance and drug release, more comfortable textiles with reduced clothing pressure and so on8, 25, 64. . . .
  65. Rothenburg, L.; Berlin, A. A.; Bathurst, R. J. Microstructure of isotropic materials with negative Poisson's ratio Nature 354, 470-472 (1991) .
    • . . . Microstructures such as composites with rotating discs and units, magnetic films, hypothetical granular structures or plasmas in neutron stars can also exhibit such effects24, 64, 65. . . .
  66. Silva, S. P. Cork: properties, capabilities and applications Int. Mater. Rev. 50, 345-365 (2005) .
    • . . . Figure reproduced with permission from: Top panel, ref. 7, © 1992 AAAS; bottom panels, ref. 75, © 2008 NAS. d, Cellular structures: honeycomb structure found in cork66 normal to the direction of growth (upper) where ν ≤ 0 and parallel to this (middle) where ν > 0 . . .
    • . . . Figure reproduced with permission from: Top panel, ref. 66, © 2005 Maney; bottom panel, ref. 67, © 2010 Wiley. e, Time-dependent Poisson's ratio for poly(methyl methacrylate), PMMA, showing the gradual rise in ν(t) with relaxation under uniaxial shear from the instantaneous elastic value of 0.33 to the viscoelastic and eventual incompressible bulk value of 0.5 . . .
    • . . . If tensile stress is applied normal to the radial growth, ν ≤ 0 in the direction of growth and ν > 0 orthogonal to this66 . . .
  67. Grima, J. N. Hexagonal honeycombs with zero Poisson's ratios and enhanced stiffness Adv. Eng. Mater. 12, 855-862 (2010) .
    • . . . Semi-re-entrant honeycomb structure proposed to model zero Poisson's ratio materials (lower)67 . . .
    • . . . Figure reproduced with permission from: Top panel, ref. 66, © 2005 Maney; bottom panel, ref. 67, © 2010 Wiley. e, Time-dependent Poisson's ratio for poly(methyl methacrylate), PMMA, showing the gradual rise in ν(t) with relaxation under uniaxial shear from the instantaneous elastic value of 0.33 to the viscoelastic and eventual incompressible bulk value of 0.5 . . .
    • . . . By alternating entrant and re-entrant honeycomb layers, Grima et al. have shown how structures with ν ≈ 0 can be generated in particular directions, leading to cylindrical shaped curvatures67. . . .
  68. Barré de Saint-Venant Resumé des Leçons sur l'application de la mécanique à l'établissement des constructions et des machines , (1848) .
    • . . . Directional auxetic properties were originally envisaged by Saint-Venant68 for anisotropic materials and later by Lempriere69 for composites . . .
  69. Lempriere, B. M. Poisson's ratio in orthotropic materials AIAA J. 6, 2226-2227 (1968) .
    • . . . Directional auxetic properties were originally envisaged by Saint-Venant68 for anisotropic materials and later by Lempriere69 for composites . . .
  70. Gunton, D. J.; Saunders, G. A. The Young's modulus and Poisson's ratio of arsenic, antimony, and bismuth J. Mater. Sci. 7, 1061-1068 (1972) .
    • . . . Arsenic, antimony and bismuth70 are highly anisotropic in single-crystal form and their calculated Poisson's ratios are negative in some directions . . .
  71. Kimizuka, H.; Kaburaki, H.; Kogure, Y. Mechanism for negative Poisson ratios over the α-β transition of cristobalite, SiO2: a molecular-dynamics study Phys. Rev. Lett. 84, 5548-5551 (2000) .
    • . . . In α-cristobalite, Poisson's ratios range from +0.08 to −0.5, depending on direction (Fig. 2c)7, 71, with an aggregate expected to be negative25 . . .
  72. Williams, J. J.; Smith, C. W.; Evans, K. E. Off-axis elastic properties and the effect of extraframework species on structural flexibility of the NAT-type zeolites: simulations of structure and elastic properties Chem. Mater. 19, 2423-2434 (2007) .
    • . . . Influenced by the presence of extra-framework water and also templating molecules, recent calculations predict oscillations in ν of 0.5 to −0.5 for 45° rotations72 . . .
  73. Lee, Y.; Vogt, T.; Hriljac, J. A.; Parise, J. B.; Artioli, G. J. Am. Chem. Soc. 124, 5466-5475 (2002) .
    • . . . Such anisotropic auxetic behaviour might well influence adsorption chemistry, if crystals are stressed along specific directions25, 73. . . .
  74. Lethbridge, Z. A. D.; Walton R. I.; Marmier, A. S. H.; Smith, C.; Evans, K. E. Elastic anisotropy and extreme Poisson's ratios in single crystals Acta Mater. 58, 6444-6451 (2010) .
    • . . . Broadly speaking, extreme Poisson's ratios in single crystals are found to be strongly correlated with elastic anisotropy74 . . .
    • . . . Notwithstanding the considerable directional swings exhibited by single crystals25, 74, Poisson's ratio for these low-density crystalline materials typically averages out around 0.2 . . .
  75. Goodwin, A. L.; Keen, D. A.; Tucker, G. Large negative linear compressibility of Ag3[Co(CN)6] Proc. Natl Acad. Sci. USA 105, 18708-18713 (2008) .
    • . . . Changing geometry of silver atoms in the van der Waals solid Ag3[Co(CN)6] (lower), showing the formation of a re-entrant honeycomb under 0.23 GPa pressure75 . . .
    • . . . Figure reproduced with permission from: Top panel, ref. 7, © 1992 AAAS; bottom panels, ref. 75, © 2008 NAS. d, Cellular structures: honeycomb structure found in cork66 normal to the direction of growth (upper) where ν ≤ 0 and parallel to this (middle) where ν > 0 . . .
    • . . . For example, in Ag3[Co(CN)6] the metals form an alternate layered structure containing auxetic motifs (Fig. 2c) which expands along the c axis but contracts in the basal plane under isotropic compression75: that is, the compressibility κ has different signs in different crystallographic directions . . .
  76. Goodwin, A. L. Colossal positive and negative thermal expansion in the framework material Ag3[Co(CN)6] Science 319, 794 (2008) .
    • . . . This behaviour is also expected to be associated with anisotropic Grüneisen parameters, whereby negative linear compressibility in specific directions should be accompanied by negative thermal expansion, which is also observed76 and is huge compared with related compounds like H3[Co(CN)6], ZrW2O8 or Cd(CN)2 (ref. 77). . . .
  77. Mary, T. A.; Evans, J. S. O.; Vogt, T.; Sleight, A. W. Negative thermal expansion from 0.3 K to 1050 K in ZrW2O8 Science 272, 90-92 (1996) .
    • . . . This behaviour is also expected to be associated with anisotropic Grüneisen parameters, whereby negative linear compressibility in specific directions should be accompanied by negative thermal expansion, which is also observed76 and is huge compared with related compounds like H3[Co(CN)6], ZrW2O8 or Cd(CN)2 (ref. 77). . . .
  78. Bridgman, P. W. The Physics of High Pressure , (1949) .
    • . . . At different temperatures and pressures, crystalline materials can undergo phase transitions78 and, attracting considerable debate, so too can glasses and liquids58, 79, 80, 81 . . .
  79. Greaves, G. N.; Sen, S. Inorganic glasses, glass-forming liquids and amorphising solids Adv. Phys. 56, 1-166 (2007) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . Figure reproduced with permission from: a, ref. 79, © 2007 Taylor & Francis; b, ref. 106, © 2010 APS; c, ref. 50, © 2005 Taylor & Francis. . . .
    • . . . At different temperatures and pressures, crystalline materials can undergo phase transitions78 and, attracting considerable debate, so too can glasses and liquids58, 79, 80, 81 . . .
    • . . . Polyamorphic phases in tetrahedral glasses Polyamorphism occurs when amorphous phases in the same liquid or supercooled state share the same composition, but differ in density and entropy79, 89 . . .
    • . . . Polyamorphism is not in dispute, but whether or not transformations between polyamorphic phases are of first order is controversial38, 58, 59, 79, 80, 89, 90 . . .
    • . . . Since these discoveries, a wealth of new densified HDA phases obtained from crystalline precursors have been reported79, 92 . . .
    • . . . As ISAXS is in turn proportional to the isothermal compressibility κT (ref. 79) the ISAXS peak points to a substantial minimum in B, and therefore in ν, consistent with the first-order phase transitions illustrated in Fig. 5a and b . . .
    • . . . Atomistic models of the low-entropy LDA or 'perfect glasses'79 formed from other microporous crystalline materials have been reported95, 96, 97 . . .
    • . . . In glasses this feature in the VDOS is generically referred to as the boson peak (because it generally scales with the Bose–Einstein function), and its origin has been fiercely argued over for the past two decades79 . . .
    • . . . This densification behaviour of the low-frequency VDOS is reflected more generally across the whole of the glassy state79, 91, 100, the frequency ωBP increasing and the size ABP decreasing with increasing ρ . . .
    • . . . There is a lot of experimental evidence that the boson peak in the glassy state has transverse character, with a suspicion that its frequency ωBP may be linked to the Ioffe–Regel limit, where the mean free path of phonons approaches their wavelength and beyond which they no longer propagate79, 101, 102, 103 . . .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
    • . . . In particular, liquid fragility is defined by the steepness of shear viscosity η as a function of reciprocal temperature as Tg is approached79, 107, 108, 109, differentiating 'strong' liquids such as silica from 'fragile' liquids like molecular melts . . .
    • . . . The ratio B/G relates directly to Poisson's ratio ν and α to the non-ergodicity factor f0 (ref. 79; Box 2), which describes the extent of the departure from thermodynamic equilibrium . . .
    • . . . Although objections to this relationship were originally voiced104, 110, many of these have been overcome as more glasses have been added79, 101, 102, 106, 111, 112 and the central proposition that “the fragility of a liquid (might be) embedded in the properties of its glass”105 has generally been strengthened . . .
    • . . . Although it is true that the point scatter in the m versus B/G plots is considerable, it is also clear that groups of glasses can be differentiated79, 111, forming an approximately radial arrangement from modified oxide glasses through single glass formers to the dense metallic glasses . . .
    • . . . The overall distribution narrows towards the glasses associated with the strongest liquids38, 79 . . .
    • . . . Specifically, f0 equates with the autocorrelation function of the liquid density fluctuations over the longest timescales79, 107, 109 . . .
    • . . . Unlike glasses which are generally homogeneous and isotropic but lack atomic long-range order79, 107, 109, polycrystalline materials have unit cell symmetry but atomic periodicity is broken internally by dislocations and impurities, and externally by grain boundaries30, 31 . . .
    • . . . Although there is no simple link between interatomic potentials and mechanical toughness in polycrystalline materials, Poisson's ratio ν has proved valuable for many years as a criterion for the brittle–ductile transition exhibited by metals30, 31, 32, 115, just as it is now helping to distinguish brittle glasses from ductile glasses (Fig. 7c) which, from Fig. 7a, are associated with strong and fragile melts respectively79, 100, 101. . . .
  80. Poole, P. H.; Grande, T.; Angell, C. A.; McMillan, P. F. Science 275, 322 (1997) .
    • . . . At different temperatures and pressures, crystalline materials can undergo phase transitions78 and, attracting considerable debate, so too can glasses and liquids58, 79, 80, 81 . . .
    • . . . Polyamorphism is not in dispute, but whether or not transformations between polyamorphic phases are of first order is controversial38, 58, 59, 79, 80, 89, 90 . . .
  81. Greaves, G. N. Detection of first order liquid–liquid phase transitions in yttrium oxide–aluminium oxide melts Science 322, 566-570 (2008) .
    • . . . At different temperatures and pressures, crystalline materials can undergo phase transitions78 and, attracting considerable debate, so too can glasses and liquids58, 79, 80, 81 . . .
  82. Hirotsu, S. Elastic anomaly near the critical point of volume phase transition in polymer gels Macromolecules 23, 903-905 (1990) .
    • . . . a, Bulk modulus B, shear modulus G and Poisson's ratio ν of a polymer gel versus temperature associated with a volume phase transition close to a critical point measured optically82. b, Bulk modulus, shear modulus and Poisson's ratio associated with the α–β first-order transition in quartz versus temperature, measured with resonant ultrasound spectroscopy (RUS)84. c, Melting of metals at ultra-high temperatures and pressures . . .
    • . . . Figure reproduced with permission from: a, ref. 82, © 1990 ACS; b, ref. 84, © 2008 IOP; d, ref. 39, © 1994 APS. . . .
    • . . . Figure 5a illustrates this for a volume transition in a polymer gel close to a critical point82 and Fig. 5b for the α–β transition in crystalline quartz83, 84 . . .
  83. Lakshtanov, D. L.; Sinogeikin, S. V.; Bass, J. D. High-temperature phase transitions and elasticity of silica polymorphs Phys. Chem. Miner. 34, 11-22 (2007) .
    • . . . Figure 5a illustrates this for a volume transition in a polymer gel close to a critical point82 and Fig. 5b for the α–β transition in crystalline quartz83, 84 . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
  84. McKnight, R. E. A. Grain size dependence of elastic anomalies accompanying the alpha-beta phase transition in polycrystalline quartz J. Phys. Cond. Mat. 20, 075229 (2008) .
    • . . . a, Bulk modulus B, shear modulus G and Poisson's ratio ν of a polymer gel versus temperature associated with a volume phase transition close to a critical point measured optically82. b, Bulk modulus, shear modulus and Poisson's ratio associated with the α–β first-order transition in quartz versus temperature, measured with resonant ultrasound spectroscopy (RUS)84. c, Melting of metals at ultra-high temperatures and pressures . . .
    • . . . Figure reproduced with permission from: a, ref. 82, © 1990 ACS; b, ref. 84, © 2008 IOP; d, ref. 39, © 1994 APS. . . .
    • . . . Figure 5a illustrates this for a volume transition in a polymer gel close to a critical point82 and Fig. 5b for the α–β transition in crystalline quartz83, 84 . . .
    • . . . Similar behaviour is reported for different gel concentrations85 and for different quartz grain sizes84, both of which affect the phase transition temperatures . . .
  85. Li, C.; Hu, Z.; Li, Y. Poisson's ratio in polymer gels near the phase-transition point Phys. Rev. E 48, 603-606 (1993) .
    • . . . Similar behaviour is reported for different gel concentrations85 and for different quartz grain sizes84, both of which affect the phase transition temperatures . . .
  86. Dong, L.; Stone, D. S.; Lakes, R. S. Softening of bulk modulus and negative Poisson's ratio in barium titanate ceramic near the Curie point Phil. Mag. Lett. 90, 23-33 (2010) .
    • . . . Softening of the bulk modulus also occurs for the ferroelastic cubic–tetragonal transition in BaTiO3, in the vicinity of the Curie point, with an auxetic minimum in ν (ref. 86). . . .
  87. Alefeld, G.; Volkl, J.; Schaumann, G. Elastic diffusion relaxation Phys. Status Solidi 37, 337-351 (1970) .
    • . . . For transformations governed by stress-induced diffusion, the compressibility is predicted to diverge87, the bulk modulus softening to zero corresponding to a substantial lowering of ν . . .
  88. Boehler, R.; Ross, M. Melting curve of aluminum in a diamond cell to 0.8 Mbar: implications for iron Earth Planet. Sci. Lett. 153, 223 (1997) .
    • . . . Poisson's ratio of cerium97, aluminium88, iron36 and molybdenum34, 35 versus pressure during shock-melting experiments, with the incipient melting temperatures Tm indicated . . .
    • . . . Nevertheless, significant experimental advances have been made in high-pressure/high-temperature physics where Poisson's ratio offers a unifying approach33, 34, 35, 36, 37, 83, 88 . . .
  89. McMillan, P. F. Polyamorphism and liquid–liquid phase transitions: challenges for experiment and theory J. Phys. Cond. Mat. 19, 415101 (2007) .
    • . . . Polyamorphic phases in tetrahedral glasses Polyamorphism occurs when amorphous phases in the same liquid or supercooled state share the same composition, but differ in density and entropy79, 89 . . .
    • . . . The HDA phase generally has higher entropy than the LDA phase89, in which case the slope of the phase boundary dT/dP = dV/dS < 0 . . .
    • . . . In addition to LDA and HDA amorphous ice58, 59, polyamorphism has been reported in many tetrahedral glasses89 . . .
  90. Greaves, G. N. Composition and polyamorphism in supercooled yttria–alumina melts J. Non-Cryst. Solids 357, 435-441 (2011) .
    • . . . Polyamorphism is not in dispute, but whether or not transformations between polyamorphic phases are of first order is controversial38, 58, 59, 79, 80, 89, 90 . . .
  91. Inamura, Y.; Katyama, Y.; Ursumi, W.; Funakoshi, K. I. Transformations in the intermediate-range structure of SiO2 glass under high pressure and temperature Phys. Rev. Lett. 93, 015501 (2004) .
    • . . . For silica under pressure Si–O polyhedra change from tetrahedral coordination eventually to octahedral coordination91 reminiscent of the high-density SiO2 crystalline phase stishovite . . .
    • . . . This densification behaviour of the low-frequency VDOS is reflected more generally across the whole of the glassy state79, 91, 100, the frequency ωBP increasing and the size ABP decreasing with increasing ρ . . .
    • . . . Densified silica provides a good example91 . . .
  92. Richet, P.; Gillet, P. Pressure-induced amorphisation of minerals: a review Eur. J. Mineral 9, 589-600 (1997) .
    • . . . Since these discoveries, a wealth of new densified HDA phases obtained from crystalline precursors have been reported79, 92 . . .
  93. Greaves, G.N. Rheology of collapsing zeolites amorphised by temperature and pressure Nature Mater. 2, 622-629 (2003) .
    • . . . Reproduced from ref. 93, © 2003 NPG. b, Reduction in the size of the boson peak ABP = D(ω)/ω calculated for a 2D glass-forming system under increasing pressure P, where D(ω) is the VDOS for acoustic modes103. c, Temperature dependence of the viscosity η versus Tg/T of the HDL and LDL supercooled phases together with the HDA and LDA glasses associated with the collapse of zeolite A . . .
    • . . . This shows how the increase in melt fragility m follows the LDA–HDA increase in densification93 . . .
    • . . . An alternative approach has been discovered that involves the amorphization of microporous crystalline materials, such as zeolites38, 93, 94, 95, tungstates96 or metal–organic frameworks97 . . .
    • . . . Amorphization occurs at pressures of a few gigapascals93, 94, 95, 96 at ambient temperature and close to Tg at ambient pressure93, 97 . . .
    • . . . In situ small-angle X-ray scattering (SAXS)93 and inelastic X-ray scattering experiments38, 98 reveal that Poisson's ratio drops considerably during amorphization before the final HDA glass is formed, by which time ν is close to 0.25, similar to other aluminosilicate glasses42 . . .
    • . . . Interestingly, this is an exothermic process93 during which the SAXS intensity ISAXS rises and falls by orders of magnitude93 . . .
    • . . . The increase in fragility can be seen in Fig. 8c for the LDA–HDA transitions underpinning the collapse of a low-density zeolite93; the viscosity of the corresponding fragile HDL liquid meets the glass transition Tg from above far more steeply than the viscosity of the LDA phase from below Tg . . .
    • . . . Although silica is usually considered the strongest of these, experiments on zeolite amorphisation reveal that for new low-density liquids (LDL) m ≤ 10 (ref. 93; Fig. 8c), with the associated LDA glasses being topologically ordered38, 79, 99 . . .
  94. Greaves, G. N.; Meneau, F.; Majérus, O.; Jones, D.; Taylor, J. Identifying the vibrations that destabilise crystals and which characterise the glassy state Science 308, 1299-1302 (2005) .
    • . . . a, Reduction in low-frequency collective terahertz band during the collapse of zeolite Y with densification94 (left) ending in the formation of a glass . . .
    • . . . Reproduced with permission from ref. 94, © 2005 AAAS . . .
    • . . . An alternative approach has been discovered that involves the amorphization of microporous crystalline materials, such as zeolites38, 93, 94, 95, tungstates96 or metal–organic frameworks97 . . .
    • . . . Low-frequency collective modes and the boson peak The driving force for polyamorphic transitions in microporous zeolites seems to lie in the strong phonon band found at low frequencies at the start of the vibrational density of states (VDOS), preceded by a narrow band related to the rotation of connected tetrahedra (Fig. 8a and Box 2)94 . . .
    • . . . During microporous collapse, however, a strong correlation exists between the intensity and frequency of terahertz vibrations and the material density ρ (ref. 94) which can be attributed to librationally driven resonances of zeolitic subunits and which decrease as the HDA glass is formed (Fig. 8a) . . .
  95. Haines, J. Topologically ordered amorphous silica obtained from the collapsed siliceous zeolite, silicalite-1-F: a step toward “perfect” glasses J. Am. Chem. Soc. 131, 12333-12338 (2009) .
    • . . . An alternative approach has been discovered that involves the amorphization of microporous crystalline materials, such as zeolites38, 93, 94, 95, tungstates96 or metal–organic frameworks97 . . .
    • . . . Atomistic models of the low-entropy LDA or 'perfect glasses'79 formed from other microporous crystalline materials have been reported95, 96, 97 . . .
  96. Keen, D. A. Structural description of pressure-induced amorphisation in ZrW2O8 Phys. Rev. Lett. 98, 225501 (2007) .
    • . . . An alternative approach has been discovered that involves the amorphization of microporous crystalline materials, such as zeolites38, 93, 94, 95, tungstates96 or metal–organic frameworks97 . . .
    • . . . Amorphization occurs at pressures of a few gigapascals93, 94, 95, 96 at ambient temperature and close to Tg at ambient pressure93, 97 . . .
  97. Bennett, T. D. Structure and properties of an amorphous metal–organic framework Phys. Rev. Lett. 104, 115503 (2010) .
    • . . . Poisson's ratio of cerium97, aluminium88, iron36 and molybdenum34, 35 versus pressure during shock-melting experiments, with the incipient melting temperatures Tm indicated . . .
    • . . . An alternative approach has been discovered that involves the amorphization of microporous crystalline materials, such as zeolites38, 93, 94, 95, tungstates96 or metal–organic frameworks97 . . .
    • . . . Amorphization occurs at pressures of a few gigapascals93, 94, 95, 96 at ambient temperature and close to Tg at ambient pressure93, 97 . . .
  98. Lethbridge, Z. A. D.; Walton, R. I.; Bosak, A.; Krisch, M. Single-crystal elastic constants of the zeolite analcime measured by inelastic X-ray scattering Chem. Phys. Lett. 471, 286-289 (2009) .
    • . . . In situ small-angle X-ray scattering (SAXS)93 and inelastic X-ray scattering experiments38, 98 reveal that Poisson's ratio drops considerably during amorphization before the final HDA glass is formed, by which time ν is close to 0.25, similar to other aluminosilicate glasses42 . . .
  99. Peral, I.; Iniguez, J. Amorphization induced by pressure: results for zeolites and general implications Phys. Rev. Lett. 97, 225502 (2006) .
    • . . . Such an order–order displacive transition from a microporous crystal has been verified by atomistic simulation during the densification of zeolite A99 . . .
    • . . . Although silica is usually considered the strongest of these, experiments on zeolite amorphisation reveal that for new low-density liquids (LDL) m ≤ 10 (ref. 93; Fig. 8c), with the associated LDA glasses being topologically ordered38, 79, 99 . . .
  100. Novikov, V. N.; Sokolov, A. P. Poisson's ratio and the fragility of glass-forming liquids Nature 431, 961-963 (2004) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . This densification behaviour of the low-frequency VDOS is reflected more generally across the whole of the glassy state79, 91, 100, the frequency ωBP increasing and the size ABP decreasing with increasing ρ . . .
    • . . . A more controversial relationship is between ν and the fragility m of the corresponding supercooled liquid100, 101, 102, 104 (Fig. 7a) . . .
    • . . . Poisson's ratio and non-ergodicity Covering a broad range of single-phase glasses, two interesting empirical relationships have emerged linking melt fragility with the elastic properties of the glass100, 101, 102, 105, 106: m versus B/G and m versus α . . .
    • . . . The plot of m versus B/G (Fig. 7a) reveals how Poisson's ratio ν increases across many glasses, as the melts from which they are quenched increase in fragility m (refs 100,101,102) . . .
    • . . . This important conclusion can be better understood by recognizing that the non-ergodicity of a glass is related to the ratio of the sound velocities Vt/Vl (ref. 100), which in turn records the density fluctuations in the glass (Box 2) . . .
    • . . . These ripples in the nanostructure are considered to be related in turn to the size of the boson peak ABP (refs 100,101,109) . . .
    • . . . Although there is no simple link between interatomic potentials and mechanical toughness in polycrystalline materials, Poisson's ratio ν has proved valuable for many years as a criterion for the brittle–ductile transition exhibited by metals30, 31, 32, 115, just as it is now helping to distinguish brittle glasses from ductile glasses (Fig. 7c) which, from Fig. 7a, are associated with strong and fragile melts respectively79, 100, 101. . . .
  101. Novikov, V. N.; Ding, Y.; Sokolov, A. P. Correlation of fragility of supercooled liquids with elastic properties of glasses Phys. Rev. E 71, 061501 (2005) .
    • . . . There is a lot of experimental evidence that the boson peak in the glassy state has transverse character, with a suspicion that its frequency ωBP may be linked to the Ioffe–Regel limit, where the mean free path of phonons approaches their wavelength and beyond which they no longer propagate79, 101, 102, 103 . . .
    • . . . As ABP decreases, ν should increase101 with decreasing G (Box 2) and, beyond the plastic limit, plastic flow should occur (Fig. 6) . . .
    • . . . Poisson's ratio and non-ergodicity Covering a broad range of single-phase glasses, two interesting empirical relationships have emerged linking melt fragility with the elastic properties of the glass100, 101, 102, 105, 106: m versus B/G and m versus α . . .
    • . . . The plot of m versus B/G (Fig. 7a) reveals how Poisson's ratio ν increases across many glasses, as the melts from which they are quenched increase in fragility m (refs 100,101,102) . . .
    • . . . These ripples in the nanostructure are considered to be related in turn to the size of the boson peak ABP (refs 100,101,109) . . .
    • . . . Although there is no simple link between interatomic potentials and mechanical toughness in polycrystalline materials, Poisson's ratio ν has proved valuable for many years as a criterion for the brittle–ductile transition exhibited by metals30, 31, 32, 115, just as it is now helping to distinguish brittle glasses from ductile glasses (Fig. 7c) which, from Fig. 7a, are associated with strong and fragile melts respectively79, 100, 101. . . .
  102. Jiang, M.; Dai, L. Intrinsic correlation between fragility and bulk modulus in metallic glasses Phys. Rev B 76, 054204 (2007) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . There is a lot of experimental evidence that the boson peak in the glassy state has transverse character, with a suspicion that its frequency ωBP may be linked to the Ioffe–Regel limit, where the mean free path of phonons approaches their wavelength and beyond which they no longer propagate79, 101, 102, 103 . . .
    • . . . Poisson's ratio and non-ergodicity Covering a broad range of single-phase glasses, two interesting empirical relationships have emerged linking melt fragility with the elastic properties of the glass100, 101, 102, 105, 106: m versus B/G and m versus α . . .
    • . . . The plot of m versus B/G (Fig. 7a) reveals how Poisson's ratio ν increases across many glasses, as the melts from which they are quenched increase in fragility m (refs 100,101,102) . . .
    • . . . In metallic glasses, though—particularly bulk metallic glasses (BMG) that are cast by slow cooling like conventional glasses118—sufficient data on elastic moduli and toughness50 are available, as well as on melt fragility102, to test these ideas quantitatively . . .
    • . . . Earlier these authors reported an intrinsic correlation between the bulk modulus for a wide range of metallic glasses and the fragilities of the liquids from which they are cast102 . . .
  103. Shintani, H.; Tanaka, H. Universal link between the boson peak and transverse phonons in glass Nature Mater. 7, 870-877 (2008) .
    • . . . Reproduced from ref. 93, © 2003 NPG. b, Reduction in the size of the boson peak ABP = D(ω)/ω calculated for a 2D glass-forming system under increasing pressure P, where D(ω) is the VDOS for acoustic modes103. c, Temperature dependence of the viscosity η versus Tg/T of the HDL and LDL supercooled phases together with the HDA and LDA glasses associated with the collapse of zeolite A . . .
    • . . . The classical strong liquid SiO2, whose fragility falls between those of the two liquid phases, is included for comparison. d, Temperature dependence (Tg/T) of the structural relaxation time τ calculated for the 2D glass in b, showing the increase in melt fragility with increasing pressure P (ref. 103) . . .
    • . . . There is a lot of experimental evidence that the boson peak in the glassy state has transverse character, with a suspicion that its frequency ωBP may be linked to the Ioffe–Regel limit, where the mean free path of phonons approaches their wavelength and beyond which they no longer propagate79, 101, 102, 103 . . .
    • . . . Atomistic simulations for a 2D glass-forming system103 reveal similar behaviour, with m increasing with increasing pressure (Fig. 8d) . . .
  104. Yannopoulos, S. N.; Johari, G. P. Poisson's ratio and liquid's fragility Nature 442, E7-E8 (2006) .
    • . . . A more controversial relationship is between ν and the fragility m of the corresponding supercooled liquid100, 101, 102, 104 (Fig. 7a) . . .
    • . . . Although objections to this relationship were originally voiced104, 110, many of these have been overcome as more glasses have been added79, 101, 102, 106, 111, 112 and the central proposition that “the fragility of a liquid (might be) embedded in the properties of its glass”105 has generally been strengthened . . .
  105. Scopigno, T.; Ruocco, G.; Sette, F.; Monaco, G. Is the fragility of a liquid embedded in the properties of its glass? Science 302, 849-852 (2003) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
    • . . . Poisson's ratio and non-ergodicity Covering a broad range of single-phase glasses, two interesting empirical relationships have emerged linking melt fragility with the elastic properties of the glass100, 101, 102, 105, 106: m versus B/G and m versus α . . .
    • . . . Although objections to this relationship were originally voiced104, 110, many of these have been overcome as more glasses have been added79, 101, 102, 106, 111, 112 and the central proposition that “the fragility of a liquid (might be) embedded in the properties of its glass”105 has generally been strengthened . . .
    • . . . The m versus α plot105, 106 (Fig. 7b) shows that when liquid fragility m increases, glasses become more ergodic, with f0 (Box 2) decreasing . . .
    • . . . It is readily obtained from inelastic X-ray scattering experiments at different temperatures113, measuring the extent to which density fluctuations in the melt are captured in the structure of the glass105, 109. . . .
    • . . . As α correlates with m (refs 105,106), their associated melt fragilities will also be the lowest (Fig. 7b), as will their Poisson's ratio (Fig. 7a), their shear resistance and, beyond the yield point, their tendency to crack (Fig. 7c): strong liquids form brittle glasses slowly, fragile liquids ductile glasses rapidly. . . .
    • . . . The original relationship105 between m and α has recently been systematically developed to account for the presence of secondary relaxation processes106 . . .
    • . . . Figure 7b shows the correlation extending over more than twice the previous range105, further strengthening the evidence that the correlation between melt fragility and vibrational properties of the corresponding amorphous solid is a universal feature of glass formation. . . .
  106. Scopigno, T.; Cangialosi, D.; Ruocco, G. Universal relation between viscous flow and fast dynamics in glass-forming materials Phys. Rev. B 81, 100202(R) (2010) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . Figure reproduced with permission from: a, ref. 79, © 2007 Taylor & Francis; b, ref. 106, © 2010 APS; c, ref. 50, © 2005 Taylor & Francis. . . .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
    • . . . Poisson's ratio and non-ergodicity Covering a broad range of single-phase glasses, two interesting empirical relationships have emerged linking melt fragility with the elastic properties of the glass100, 101, 102, 105, 106: m versus B/G and m versus α . . .
    • . . . As α correlates with m (refs 105,106), their associated melt fragilities will also be the lowest (Fig. 7b), as will their Poisson's ratio (Fig. 7a), their shear resistance and, beyond the yield point, their tendency to crack (Fig. 7c): strong liquids form brittle glasses slowly, fragile liquids ductile glasses rapidly. . . .
    • . . . The original relationship105 between m and α has recently been systematically developed to account for the presence of secondary relaxation processes106 . . .
  107. Debenedetti, P. G.; Stillinger, F. H. Supercooled liquids and the glass transition Nature 410, 259-267 (2001) .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
    • . . . In particular, liquid fragility is defined by the steepness of shear viscosity η as a function of reciprocal temperature as Tg is approached79, 107, 108, 109, differentiating 'strong' liquids such as silica from 'fragile' liquids like molecular melts . . .
    • . . . Specifically, f0 equates with the autocorrelation function of the liquid density fluctuations over the longest timescales79, 107, 109 . . .
    • . . . Unlike glasses which are generally homogeneous and isotropic but lack atomic long-range order79, 107, 109, polycrystalline materials have unit cell symmetry but atomic periodicity is broken internally by dislocations and impurities, and externally by grain boundaries30, 31 . . .
  108. Angell, C. A. Structural instability and relaxation in liquid and glassy phases near the fragile liquid limit J. Non-Cryst. Solids 102, 205-221 (1988) .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
    • . . . Melt fragility The hugely different viscous properties of liquids are illustrated in the familiar Angell plot of log η versus Tg/T, where η is shear viscosity, shown in Box 2 (ref. 108) . . .
    • . . . In particular, liquid fragility is defined by the steepness of shear viscosity η as a function of reciprocal temperature as Tg is approached79, 107, 108, 109, differentiating 'strong' liquids such as silica from 'fragile' liquids like molecular melts . . .
  109. Dyre, J. C. Colloquium: The glass transition and elastic models of glass-forming liquids Rev. Mod. Phys. 78, 953-972 (2006) .
    • . . . Together with the degree of non-ergodicity f0 frozen into the glass79, 105, 106, the melt fragility m is measured at the glass transition Tg (Box 2) where the viscous relaxation time reaches ~100 s and the liquid is considered solid26, 79, 107, 108, 109 . . .
    • . . . In particular, liquid fragility is defined by the steepness of shear viscosity η as a function of reciprocal temperature as Tg is approached79, 107, 108, 109, differentiating 'strong' liquids such as silica from 'fragile' liquids like molecular melts . . .
    • . . . Specifically, f0 equates with the autocorrelation function of the liquid density fluctuations over the longest timescales79, 107, 109 . . .
    • . . . These ripples in the nanostructure are considered to be related in turn to the size of the boson peak ABP (refs 100,101,109) . . .
    • . . . Unlike glasses which are generally homogeneous and isotropic but lack atomic long-range order79, 107, 109, polycrystalline materials have unit cell symmetry but atomic periodicity is broken internally by dislocations and impurities, and externally by grain boundaries30, 31 . . .
  110. Dyre, J. C. Glasses: Heirs of liquid treasures Nature Mater. 3, 749-750 (2004) .
    • . . . Although objections to this relationship were originally voiced104, 110, many of these have been overcome as more glasses have been added79, 101, 102, 106, 111, 112 and the central proposition that “the fragility of a liquid (might be) embedded in the properties of its glass”105 has generally been strengthened . . .
  111. Johari, G. P. On Poisson's ratio of glass and liquid vitrification characteristics Phil. Mag. 86, 1567-1579 (2006) .
    • . . . Although objections to this relationship were originally voiced104, 110, many of these have been overcome as more glasses have been added79, 101, 102, 106, 111, 112 and the central proposition that “the fragility of a liquid (might be) embedded in the properties of its glass”105 has generally been strengthened . . .
  112. Nemilov, S. V. Structural aspect of possible interrelation between fragility (length) of glass forming melts and Poisson's ratio of glasses J. Non-Cryst. Solids 353, 4613-4632 (2007) .
    • . . . a, Novikov–Sokolov plot100 of melt fragility m versus B/G for single-phase glass-formers, with modified glasses and metallic glasses added79, 102, 112, showing an increase in the slope of m versus B/G with atomic packing, the different families converging on superstrong melts and perfect glasses. b, Scopigno plot105 of melt fragility m versus α for an extended range of inorganic and organic glass-formers106, where α = (1 − f0)Tg/T and f0 is the non-ergodicity factor that measures the departure from thermodynamic equilibrium (Box 2). c, Fracture energy log Efracture versus ν for bulk metallic glasses50 showing an abrupt brittle–ductile threshold for ν ≈ 0.31 . . .
    • . . . Although objections to this relationship were originally voiced104, 110, many of these have been overcome as more glasses have been added79, 101, 102, 106, 111, 112 and the central proposition that “the fragility of a liquid (might be) embedded in the properties of its glass”105 has generally been strengthened . . .
  113. Krisch, M.; Sette, F. Light Scattering in Solids: Novel Materials and Techniques , (2007) .
    • . . . It is readily obtained from inelastic X-ray scattering experiments at different temperatures113, measuring the extent to which density fluctuations in the melt are captured in the structure of the glass105, 109. . . .
  114. Xi, X. L. Fracture of brittle metallic glasses: brittleness or plasticity Phys. Rev. Lett. 94, 125510 (2005) .
    • . . . At either extreme the microstructure will play a part, whether through cracks, dislocations, shear bands, impurities, inclusions or other means114. . . .
    • . . . Plastic flow in metallic glasses occurs very locally in shear bands51, 114, compared with polycrystalline metals where flow is dislocation-mediated, and delocalized by associated work-hardening30. . . .
  115. Pugh, S. F. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals Phil. Mag. 45, 823-843 (1954) .
    • . . . Although there is no simple link between interatomic potentials and mechanical toughness in polycrystalline materials, Poisson's ratio ν has proved valuable for many years as a criterion for the brittle–ductile transition exhibited by metals30, 31, 32, 115, just as it is now helping to distinguish brittle glasses from ductile glasses (Fig. 7c) which, from Fig. 7a, are associated with strong and fragile melts respectively79, 100, 101. . . .
  116. Rosenhain, W.; Ewen, D. The intercrystalline cohesion of metals J. Inst. Met. 10, 119-149 (1913) .
    • . . . The old proposal that grains in polycrystalline materials might be cemented together by a thin layer of amorphous material “analogous to the condition of a greatly undercooled liquid”116 has often been challenged . . .
  117. Zhang, H.; Srolovitz, D. J.; Douglas, J. F; Warren, J. A. Grain boundaries exhibit the dynamics of glass-forming liquids Proc. Natl Acad. Sci. USA 106, 7735-7740 (2009) .
    • . . . However, recent atomistic simulations of crystalline grains and grain boundaries seem to confirm the dynamic consequences of this idea in many details117 . . .
    • . . . Recalling that fracture energy and melt fragility are correlated (Fig. 7d), its seems likely that the ductile and brittle properties of metals, whether they are glasses or crystals, are intimately related to the viscous time-dependent properties of their supercooled antecedents, either constrained in metallic glass shear bands51 or, more speculatively, in polycrystalline grain boundaries117. . . .
  118. Greer, A. L. Metallic glasses Science 267, 1947-1953 (1995) .
    • . . . In metallic glasses, though—particularly bulk metallic glasses (BMG) that are cast by slow cooling like conventional glasses118—sufficient data on elastic moduli and toughness50 are available, as well as on melt fragility102, to test these ideas quantitatively . . .
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