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Predicting the density-scaling exponent of a glass-forming liquid from Prigogine–Defay ratio measurements

Abstract

Understanding the origin of the dramatic temperature and density dependence of the relaxation time of glass-forming liquids is a fundamental challenge in glass science. The recently established ‘density-scaling’ relation quantifies the relative importance of temperature and density for the relaxation time in terms of a material-dependent exponent. We show that this exponent for approximate single-parameter liquids can be calculated from thermoviscoelastic linear-response data at a single state point, for instance an ambient-pressure state point. This prediction is confirmed for the van der Waals liquid tetramethyl-tetraphenyl-trisiloxane. Consistent with this, a compilation of literature data for the Prigogine–Defay ratio shows that van der Waals liquids and polymers are approximate single-parameter systems, whereas associated and network-forming liquids are not.

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Figure 1: The dielectric relaxation time τ measured along different isotherms and along the atmospheric pressure isobar for the silicone oil DC704.
Figure 2: Examples of frequency-dependent real and imaginary parts of the four required thermoviscoelastic response functions of DC704, illustrating the experimental challenges associated with checking the isomorph prediction γscale=γisom.
Figure 3: Literature values of the classical (NpT) Prigogine–Defay (PD) ratios ΠpTclassic of 22 glass formers.

References

  1. Debenedetti, P. G. Supercooled glassy water. J. Phys. Condens. Matter 15, R1669–R1726 (2003).

    ADS  Article  Google Scholar 

  2. Angell, C. A. Insights into phases of liquid water from study of its unusual glass-forming properties. Science 319, 582–587 (2008).

    Article  Google Scholar 

  3. Pedersen, U. R., Bailey, N. P., Schrøder, T. B. & Dyre, J. C. Strong pressure–energy correlations in van der Waals liquids. Phys. Rev. Lett. 100, 015701 (2008).

    ADS  Article  Google Scholar 

  4. Schrøder, T. B., Bailey, N. P., Pedersen, U. R., Gnan, N. & Dyre, J. C. Pressure–energy correlations in liquids. III. Statistical mechanics and thermodynamics of liquids with hidden scale invariance. J. Chem. Phys. 131, 234503 (2009).

    ADS  Article  Google Scholar 

  5. Gnan, N., Schrøder, T. B., Pedersen, U. R., Bailey, N. P. & Dyre, J. C. Pressure–energy correlations in liquids. IV. ”Isomorphs” in liquid phase diagrams. J. Chem. Phys. 131, 234504 (2009).

    ADS  Article  Google Scholar 

  6. Ellegaard, N. L. et al. Single-order-parameter description of glass-forming liquids: A one-frequency test. J. Chem. Phys. 126, 074502 (2007).

    ADS  Article  Google Scholar 

  7. Davies, R. O. & Jones, G. O. Thermodynamic and kinetic properties of glasses. Adv. Phys. 2, 370–410 (1953).

    ADS  Article  Google Scholar 

  8. Prigogine, I. & Defay, R. Chemical Thermodynamics (Longmans Green and Co., 1954).

    Google Scholar 

  9. Goldstein, M. Some thermodynamic aspects of the glass transition: Free volume, entropy, and enthalpy theories. J. Chem. Phys. 39, 3369–3374 (1963).

    ADS  Article  Google Scholar 

  10. Moynihan, C. T. & Gupta, P. K. Order parameter model for structural relaxation in glass. J. Non-Cryst. Solids 29, 143–158 (1978).

    ADS  Article  Google Scholar 

  11. Berg, J. I. & Cooper, A. R. Linear non-equilibrium thermodynamic theory of glass-transition kinetics. J. Chem. Phys. 68, 4481–4485 (1978).

    ADS  Article  Google Scholar 

  12. Hodge, I. M. Enthalpy relaxation and recovery in amorphous materials. J. Non-Cryst. Solids 169, 211–266 (1994).

    ADS  Article  Google Scholar 

  13. Nieuwenhuizen, Th. M. Ehrenfest relations at the glass transition: Solution to an old paradox. Phys. Rev. Lett. 79, 1317–1320 (1997).

    ADS  Article  Google Scholar 

  14. Wondraczek, L., Krolikowski, S. & Behrens, H. Relaxation and Prigogine–Defay ratio of compressed glasses with negative viscosity-pressure dependence. J. Chem. Phys. 130, 204506 (2009).

    ADS  Article  Google Scholar 

  15. Lion, A. & Peters, J. Coupling effects in dynamic calorimetry: Frequency-dependent relations for specific heat and thermomechanical responses—a one-dimensional approach based on thermodynamics with internal state variables. Thermochim. Acta 500, 76–78 (2010).

    Article  Google Scholar 

  16. Angell, C. A., Ngai, K. L., McKenna, G. B., McMillan, P. F. & Martin, S. W. Relaxation in glassforming liquids and amorphous solids. J. Appl. Phys. 88, 3113–3157 (2000).

    ADS  Article  Google Scholar 

  17. Dyre, J. C. The glass transition and elastic models of glass-forming liquids. Rev. Mod. Phys. 78, 953–972 (2006).

    ADS  Article  Google Scholar 

  18. Cavagna, A. Supercooled liquids for pedestrians. Phys. Rep. 476, 51–124 (2009).

    ADS  Article  Google Scholar 

  19. Floudas, G., Paluch, M., Grzybowski, A. & Ngai, K. L. Molecular Dynamics of Glass-Forming Systems (Advances in Dielectrics, Springer, 2010).

    Google Scholar 

  20. Berthier, L., Biroli, G., Bouchaud, J-P., Cipelletti, L. & van Saarloos, W. (eds). Dynamical heterogeneities in glasses, colloids, and granular media. Oxford Univ. Press (in the press).

  21. Alba-Simionesco, C., Cailliaux, A., Alegria, A. & Tarjus, G. Scaling out the density dependence of the α relaxation in glass-forming polymers. Europhys. Lett. 68, 58–64 (2004).

    ADS  Article  Google Scholar 

  22. Dreyfus, C., Le Grand, A., Gapinski, J., Steffen, W. & Patkowski, A. Scaling the α-relaxation time of supercooled fragile organic liquids. Eur. Phys. J. B 42, 309–319 (2004).

    ADS  Article  Google Scholar 

  23. Casalini, R. & Roland, C. M. Thermodynamical scaling of the glass transition dynamics. Phys. Rev. E 69, 062501 (2004).

    ADS  Article  Google Scholar 

  24. Roland, C. M., Hensel-Bielowka, S., Paluch, M. & Casalini, R. Supercooled dynamics of glass-forming liquids and polymers under hydrostatic pressure. Rep. Prog. Phys. 68, 1405–1478 (2005).

    ADS  Article  Google Scholar 

  25. Ferrer, M. L. et al. Supercooled liquids and the glass transition: Temperature as the control variable. J. Chem. Phys. 109, 8010–8015 (1998).

    ADS  Article  Google Scholar 

  26. Roland, C. M. Relaxation phenomena in vitrifying polymers and molecular liquids. Macromol 43, 7875–7890 (2010).

    Article  Google Scholar 

  27. Bridgman, P. W. Viscosities to 30,000 kg/cm2. Proc. Am. Acad. Arts Sci. 77, 117–128 (1949).

    Google Scholar 

  28. Roland, C. M., Casalini, R., Bergman, R. & Mattsson, J. Role of hydrogen bonds in the supercooled dynamics of glass-forming liquids at high pressures. Phys. Rev. B 77, 012201 (2008).

    ADS  Article  Google Scholar 

  29. Coslovich, D. & Roland, C. M. Density scaling in viscous liquids: From relaxation times to four-point susceptibilities. J. Chem. Phys. 131, 151103 (2009).

    ADS  Article  Google Scholar 

  30. Schrøder, T. B., Pedersen, U. R., Bailey, N. P., Toxvaerd, S. & Dyre, J. C. Hidden scale invariance in molecular van der Waals liquids: A simulation study. Phys. Rev. E 80, 041502 (2009).

    ADS  Article  Google Scholar 

  31. Casalini, R., Mohanty, U. & Roland, C. M. Thermodynamic interpretation of the scaling of the dynamics of supercooled liquids. J. Chem. Phys. 125, 014505 (2006).

    ADS  Article  Google Scholar 

  32. de Groot, S. R. & Mazur, P. Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962).

    MATH  Google Scholar 

  33. Christensen, T., Olsen, N. B. & Dyre, J. C. Conventional methods fail to measure c p(ω) of glass-forming liquids. Phys. Rev. E 75, 041502 (2007).

    ADS  Article  Google Scholar 

  34. Niss, K., Christensen, T. & Dyre, J. C. Measuring the dynamic thermal expansivity of molecular liquids near the glass transition. Preprint at http://arXiv.org/abs/1103.4104 (2011).

  35. Christensen, T. & Olsen, N. B. A rheometer for the measurement of a high shear modulus covering more than seven decades of frequency below 50 kHz. Rev. Sci. Instrum. 66, 5019–5031 (1995).

    ADS  Article  Google Scholar 

  36. Christensen, T. & Olsen, N. B. Determination of the frequency-dependent bulk modulus of glycerol using a piezoelectric spherical shell. Phys. Rev. B 49, 15396–15399 (1994).

    ADS  Article  Google Scholar 

  37. Birge, N. O. & Nagel, S. R. Specific-heat spectroscopy of the glass-transition. Phys. Rev. Lett. 54, 2674–2677 (1985).

    ADS  Article  Google Scholar 

  38. Jakobsen, B., Olsen, N. B. & Christensen, T. Frequency-dependent specific heat from thermal effusion in spherical geometry. Phys. Rev. E 81, 061505 (2010).

    ADS  Article  Google Scholar 

  39. Bauer, C. et al. Capacitive scanning dilatometry and frequency-dependent thermal expansion of polymer films. Phys. Rev. E 61, 1755–1764 (2000).

    ADS  Article  Google Scholar 

  40. Roe, R. J. Thermodynamics of glassy state with multiple order parameters. J. Appl. Phys. 48, 4085–4091 (1977).

    ADS  Article  Google Scholar 

  41. Moynihan, C. T. & Lesikar, A. V. Comparison and analysis of relaxation processes at the glass-transition temperature. Ann. N.Y. Acad. Sci. 371, 151–169 (1981).

    ADS  Article  Google Scholar 

  42. Takahara, S., Ishikawa, M., Yamamuro, O. & Matsuo, T. Structural relaxations of glassy polystyrene and o-terphenyl studied by simultaneous measurement of enthalpy and volume under high pressure. J. Phys. Chem. B 103, 792–796 (1999).

    Article  Google Scholar 

  43. Schmelzer, J. W. P. & Gutzow, I. The Prigogine–Defay ratio revisited. J. Chem. Phys. 125, 184511 (2006).

    ADS  Article  Google Scholar 

  44. Banerjee, R., Modak, S. K. & Samanta, S. Glassy phase transition and stability in black holes. Eur. Phys. J. C 70, 317–328 (2010).

    ADS  Article  Google Scholar 

  45. Pick, R. M. The Prigogine–Defay ratio and the microscopic theory of supercooled liquids. J. Chem. Phys. 129, 124115 (2008).

    ADS  Article  Google Scholar 

  46. Liebl, C., Lion, A., Kolmeder, S. & Peters, J. Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables. J. Mech. Phys. Solids 58, 1338–1360 (2010).

    ADS  MathSciNet  Article  Google Scholar 

  47. Javaheri, M. R. H & Chamberlin, R. V. A free-energy landscape picture and Landau theory for the dynamics of disordered materials. J. Chem. Phys. 125, 154503 (2006).

    ADS  Article  Google Scholar 

  48. Lesikar, A. V. & Moynihan, C. T. Some relations connecting volume and enthalpy relaxation in the order parameter model of liquids and glasses. J. Chem. Phys. 72, 6422–6423 (1980).

    ADS  Article  Google Scholar 

  49. Gnan, N., Maggi, C., Schrøder, T. B. & Dyre, J. C. Predicting the effective temperature of a glass. Phys. Rev. Lett. 104, 125902 (2010).

    ADS  Article  Google Scholar 

  50. Igarashi, B. et al. A cryostat and temperature control system optimized for measuring relaxations of glass-forming liquids. Rev. Sci. Instrum. 79, 045105 (2008).

    ADS  Article  Google Scholar 

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Acknowledgements

The Centre for Viscous Liquid Dynamics ‘Glass and Time’ is sponsored by the Danish National Research Foundation (DNRF). Work at NRL is supported by Office of Naval Research. URP is supported by The Danish Council for Independent Research in Natural Sciences.

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U.R.P. and K.N. conceived the project. D.G., U.R.P., B.J., J.C.D. and K.N. wrote the paper with input from C.M.R. D.G., U.R.P., and K.N. did the main data analysis. T.H. measured the shear modulus and compressibility and did the raw data analysis. B.J. and T.C. measured the heat capacity and did the raw data analysis. K.N. measured the expansion coefficient and did the raw data analysis. T.C. and N.B.O. conceived and developed the four thermoviscoelastic measuring techniques used. D.G., D.F. and R.C. measured the high-pressure data and did the scaling data analysis. U.R.P., N.P.B., T.C., T.B.S. and J.C.D. supplied the theoretical background for the project, which was coordinated by K.N.

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Correspondence to Kristine Niss.

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Gundermann, D., Pedersen, U., Hecksher, T. et al. Predicting the density-scaling exponent of a glass-forming liquid from Prigogine–Defay ratio measurements. Nature Phys 7, 816–821 (2011). https://doi.org/10.1038/nphys2031

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