Glass Physics and Chemistry

, Volume 39, Issue 6, pp 609–623 | Cite as

Maxwell equation and classical theories of glass transition as a basis for direct calculation of viscosity at glass transition temperture



The main relaxation theories of glass transition (Leontovich-Mandelstam and Volkenstein-Ptitsyn) variously formulate the kinetic criteria of glass transition. Both of the approaches are shown to be equivalent in physical sense and provide a single expression that connects the rate of change of a melt temperature and the structure relaxtion time. The theory characterizes the width of a temperature band δT g in which the glass transition occurs. The values of δT g can either be obtained from the experiment or be accepted to conform to the value of a temperature step under the change of viscosity by an order of magnitude (direct method for finding by the Volkenstein-Ptitsyn theory). Using the Maxwell equation, a new equation for the calculation of the viscosity for viscoelastic relaxation was suggested, which is based on a shear modulus of glass and the cooling rate. The theory was verified basing on the published data for oxide glass. The average difference between the calculated and measured values of lgη upon glass transition comprises 0 ± 0.30. These results correspond to the cooling rate of 3 K/min and log(η, Pa s) = 12.76 ± 0.26 (for all glass considered). It is shown that the most probable cooling rate which provides the viscosity upon glass transition of ∼1012 Pa s is close to 20 K/min (oxide melts). The theory predetermines the dependence of viscosity upon glass transition on the nature of a glass-forming liquid. The disadvantages of other approaches to the problem under consideration are demonstrated.


mechanical properties relaxation glass transition viscosity internal friction Maxwell equation 


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  1. 1.
    Scherer, G.W., Use of the Adam-Gibbs equation in the analysis of structural relaxation, J. Am. Ceram. Soc., 1984, vol. 67, no. 7, pp. 504–511.CrossRefGoogle Scholar
  2. 2.
    Binder, K., Baschnagel, J., and Paul, W., Glass transition of polymer melts: Tests of theoretical concepts by computer simulation, Prog. Polym. Sci., 2003, vol. 28, no. 1, pp. 115–172.CrossRefGoogle Scholar
  3. 3.
    Lyulin, A.V., Balabaev, N.K., and Micheles, M.A.J., Molecular-weight and cooling-rate dependence of stimulated Tg of amorphous polystyrene, Macromolecules, 2003, vol. 36, no. 22, pp. 8574–8575.CrossRefGoogle Scholar
  4. 4.
    Vollmayer, K., Kob, W., and Binder, K., How do the properties of a glass depend on the cooling rate? A computer simulation study of a Lennard-Jones system, J. Chem. Phys., 1996, vol. 105, no. 11, pp. 4714–4728.CrossRefGoogle Scholar
  5. 5.
    Simon, F., Über den Zustand der Unterkühlten Flüssigkeiten und Gläser, Z. Anorg. Allg. Chem., 1931, vol. 203, nos. 1–2, pp. 219–227.CrossRefGoogle Scholar
  6. 6.
    Prigogine, I. and Defay, R., Chemical Thermodynamics (Treatise on Thermodynamics), New York: Longmans Green, 1950.Google Scholar
  7. 7.
    Davies, R.O. and Jones, G.O., The irreversible approach to equilibrium in glasses, Adv. Phys., 1953, vol. 2, no. 7, pp. 370–410.CrossRefGoogle Scholar
  8. 8.
    Mazurin, O.V., Steklovanie (Glass Transition), Leningrad: Nauka, 1986.Google Scholar
  9. 9.
    Nemilov, S.V., Thermodynamic and Kinetic Aspects of the Vitreous State, Boca Raton, Florida, United States: CRC Press, 1995.Google Scholar
  10. 10.
    Ehrenfest, P., Phasenumwandlungen im üblichen und erweiterten Sinn, klassifiziert nach den entsprechenden Singularitäten des thermodynamischen Potentials, Proc. R. Neth. Acad. Arts Sci. (Amsterdam), 1933, vol. 36, no. 2, pp. 153–157.Google Scholar
  11. 11.
    Schmelzer, J.W.P. and Gutzov, I., The Prigogine-Defay ratio revisited, J. Chem. Phys., 2006, vol. 125, no. 18, p. 184511 (11 pages)CrossRefGoogle Scholar
  12. 12.
    Troipin, T.V., Schmelzer, J.W.P., Gutzov, I., and Schick, C., On the theoretical determination of the Prigogine-Defay ratio in glass transition, J. Chem. Phys., 2012, vol. 136, no. 14, p. 124502 (14 pages).CrossRefGoogle Scholar
  13. 13.
    Kobeko, P.P., Amorfnye veshchestva (Amorphous Materials), Moscow: Academy of Sciences of the Soviet Union, 1952.Google Scholar
  14. 14.
    Maxwell, J.C., On the dynamical theory of gases, Philos. Trans., 1867, vol. 157, pp. 49–88.CrossRefGoogle Scholar
  15. 15.
    Lillie, H.R., Stress release in glass: A phenomenon involving viscosity as a variable with time, J. Amer. Ceram. Soc., 1936, vol. 19, no. 1, pp. 45–54.CrossRefGoogle Scholar
  16. 16.
    Ojovan, M.I., Viscosity and glass transition in amorphous oxides, Adv. Condens. Matter Phys., 2008, vol. 2008, article ID 817829 (23 pages).Google Scholar
  17. 17.
    Angell, C.A., Ngai, K.L., McKenna, G.B., McMillan, P.F., and Martin, S.W., Relaxation in glass forming liquids and amorphous solids, J. Appl. Phys., 2000, vol. 88, no. 6, pp. 3113–3157.CrossRefGoogle Scholar
  18. 18.
    Ediger, M.D., Angell, C.A., and Nagel, S.R., Supercooled liquids and glasses, J. Phys. Chem., 1996, vol. 100, no. 31, pp. 13200–13212.CrossRefGoogle Scholar
  19. 19.
    Richer, R., Heterogeneous dynamics in liquids: Fluctuations in space and time, J. Phys.: Condens. Matter, 2002, vol. 14, no. 23, pp. R703–R738.Google Scholar
  20. 20.
    Debenedetti, P.G. and Stillinger, F.H., Supercooled liquids and the glass transition, Nature (London), 2001, vol. 410, no. 6825, pp. 259–267.CrossRefGoogle Scholar
  21. 21.
    Scholze, H., Glas: Natur, Struktur und Eigenschaften, Berlin: Springer-Verlag, 1977.CrossRefGoogle Scholar
  22. 22.
    Mazurin, O.V., Problems of the compatibility of the values of glass transition temperatures published in the world literature, Glass Phys. Chem., 2007, vol. 33, no. 1, pp. 22–36.CrossRefGoogle Scholar
  23. 23.
    ISO 11357-2: 1999. Plastics-Differential Scanning Calorimetry (DSC)-Part 2: Determination of Glass Transition Temperature, Washington, DC, United States: American National Standards Institute (ANSI), 2007.Google Scholar
  24. 24.
    IUPAC Compendium of Chemical Terminology-IUPAC Recommendations (IUPAC Chemical Nomenclature Series), McNaught, A.D. and Wilkinson, A., Eds., Oxford: Blackwell Scientific Publications, 1997, 2nd ed.Google Scholar
  25. 25.
    Leontovich, M.A., Some problems of sound absorption in polyatomic gases, Izv. Akad. Nauk SSSR, Ser. Fiz., 1936, vol. 5, pp. 633–642.Google Scholar
  26. 26.
    Leontovich, M.A., Comments to the theory of sound absorption in gases, Zh. Eksp. Teor. Fiz., 1936, vol. 6, no. 6, pp. 561–576.Google Scholar
  27. 27.
    Mandelshtam, L.I. and Leontovich, M.A., On the theory of sound absorption in liquids, Zh. Eksp. Teor. Fiz., 1937, vol. 7, no. 3, pp. 438–449.Google Scholar
  28. 28.
    De Donder, Th., L’affiniteÌ (Series: MeÌmoires de la Classe des sciences, AcadeÌmie royale de Belgique), Bruxelles: M. Lamertin: 1927.Google Scholar
  29. 29.
    Solov’ev, V.A., The thermodynamic theory of relaxation, in Relaksatsionnye yavleniya v polimerakh (Relaxation Phenomena in Polymers), Bartenev, G.M. and Zelenev, Yu.V., Eds., Leningrad: Khimiya, 1972, pp. 129–139.Google Scholar
  30. 30.
    Solov’ev, V.A., The general thermodynamic theory of relaxation processes in an acoustic wave (the theory by L.I. Mandelshtam and M.A. Leontovich), in Osnovy molekulyarnoi akustiki (Principles of Molecular Acoustics), Mikhailov, I.G., Solov’ev, V.A., and Syrnikov, Yu.P., Eds., Moscow: Nauka, 1964, pp. 236–282.Google Scholar
  31. 31.
    Nemilov, S.V., Relationship between the reduced thermodynamic functions of vitreous systems at 0 K, Fiz. Khim. Stekla, 1981, vol. 7, no. 5, pp. 575–583.Google Scholar
  32. 32.
    Nemilov, S.V., Thermodynamic functions of nonequilibrium disordered systems at absolute zero and the nature of the vitreous state, Fiz. Khim. Stekla, 1982, vol. 8, no. 1, pp. 11–24.Google Scholar
  33. 33.
    Nemilov, S.V., Zero-point entropy of glasses as physical reality, J. Non-Cryst. Solids, 2009, vol. 355, nos. 10–12, pp. 607–616.CrossRefGoogle Scholar
  34. 34.
    Volkenstein, M.V. and Ptitsyn, O.B., The relaxation theory of glass transition, Dokl. Akad. Nauk SSSR, 1955, vol. 103, no. 5, pp. 795–798.Google Scholar
  35. 35.
    Volkenstein, M.V. and Ptitsyn, O.B., The relaxation theory of glass transition: I. The solution of the basic equation and its investigation, Zh. Tekh. Fiz., 1956, vol. 26, no. 10, pp. 2204–2222.Google Scholar
  36. 36.
    Nemilov, S.V., Interrelation between the shear modulus and the molecular parameters of the viscous flow for glass-forming liquids, J. Non-Cryst. Solids, 2006, vol. 352, nos. 26–27, pp. 2715–2725.CrossRefGoogle Scholar
  37. 37.
    Nemilov, S.V., The review of possible interrelations between the ionic conductivity, internal friction, and the viscosity of glasses and glass-forming melts within the framework of Maxwell equations, J. Non-Cryst. Solids, 2011, vol. 357, no. 4, pp. 1243–1263.CrossRefGoogle Scholar
  38. 38.
    Nemilov, S.V., Maxwell equation for electrical conductivity of dielectrics as the basis for direct relationship between the ionic electrical conductivity and mechanical losses in glasses: New problems of the physical chemistry of glass, Glass Phys. Chem., 2012, vol. 38, no. 1, pp. 27–40.CrossRefGoogle Scholar
  39. 39.
    Narayanaswamy, O.S., Model of structural relaxation in glass, J. Am. Ceram. Soc., 1971, vol. 54, no. 10, pp. 491–498.CrossRefGoogle Scholar
  40. 40.
    Moynihan, C.T., Eastel, A.J., DeBolt, M.A., and Tucker, J., Dependence of the glass transition temperature on cooling rate, J. Am. Ceram. Soc., 1976, vol. 59, nos. 1–2, pp. 12–16.CrossRefGoogle Scholar
  41. 41.
    Jordanov, N., Wondraczek, L., and Gutzow, I., Thermodynamic properties of amorphous solids: The electrochemical approach, J. Non-Cryst. Solids, 2012, vol. 358, no. 10, pp. 1239–1256.CrossRefGoogle Scholar
  42. 42.
    Schmelzer, J.W.P., Kinetic criteria of glass formation and the pressure dependence of the glass transition temperature, J. Chem. Phys., 2012, vol. 136, no. 7, p. 074512 (11 pages)CrossRefGoogle Scholar
  43. 43.
    Moynihan, C.T., Esteal, A.J., Wilder, J., and Tucker, J., Dependence of the glass transition temperature on heating and cooling rate, J. Phys. Chem., 1974, vol. 78, no. 26, pp. 2673–2677.CrossRefGoogle Scholar
  44. 44.
    Bartenev, G.M., On the relationship between the glass transition temperature of silicate glass and the rate of cooling and heating, Dokl. Akad. Nauk SSSR, 1951, vol. 76, no. 2, pp. 227–230.Google Scholar
  45. 45.
    Bartenev, G.M., Stroenie i mekhanicheskie svoistva neorganicheskikh stekol (Structure and Mechanical Properties of Inorganic Glasses), Moscow: Stroiizdat, 1966.Google Scholar
  46. 46.
    Bragg, W.L. and Williams, E.J., The effect of thermal agitation on atomic arrangement in alloys, Proc. R. Soc. London, Ser. A, 1934, vol. 145, no. 855, pp. 699–730.CrossRefGoogle Scholar
  47. 47.
    Möller, J., Gutzow, I., and Schmelzer, J.W.P., Freezing-in and production of entropy in vitrification, J. Chem. Phys., 2006, vol. 125, no. 9, p. 094505 (13 pages).CrossRefGoogle Scholar
  48. 48.
    Bartenev, G.M., On the glass transition of high-molecular polymers under periodic deformation, Dokl. Akad. Nauk SSSR, 1949, vol. 69, no. 3, pp. 373–376.Google Scholar
  49. 49.
    Nemilov, S.V., Genesis of the vitreous state: Three variants of the approach to the solution of the problem, Fiz. Khim. Stekla, 1992, vol. 18, no. 5, pp. 1–24.Google Scholar
  50. 50.
    Nemilov, S.V., Structural aspect of possible interrelation between fragility (length) of glass-forming melts and Poisson’ ratio of glasses, J. Non-Cryst. Solids, 2007, vol. 353, nos. 52–54, pp. 4613–4632.CrossRefGoogle Scholar
  51. 51.
    Startsev, Yu.K., Klyuev, V.P., and Vostrikova, M.S., Determination of the glass transition temperatures from the simultaneously measured dependences of the thermal expansion and electrical conductivity, Fiz. Khim. Stekla, 1978, vol. 4, no. 3, pp. 278–288.Google Scholar
  52. 52.
    Mazurin, O.V. and Stolyar, S.V., On the relationship between the glass transition and liquidus temperatures for eutectic compositions in the sodium borosilicate system, Fiz. Khim. Stekla, 1984, vol. 10, no. 2, pp. 163–166.Google Scholar
  53. 53.
    Mazurin, O.V. and Klyuev, V.P., Investigation of the structural genesis in some multicomponent glasses by the dilatometric method, Fiz. Khim. Stekla, 1975, vol. 1, no. 3, pp. 245–250.Google Scholar
  54. 54.
    Klyuev, V.P. and Bulaeva, A.V., Viscosity and thermal expansion of lead borate glasses in the glass transition range, Fiz. Khim. Stekla, 1980, vol. 6, no. 6, pp. 674–678.Google Scholar
  55. 55.
    MDL®SciGlass-7.8, Shrewsbury, Massachusetts, United States: Institute of Theoretical Chemistry, 2012.Google Scholar
  56. 56.
    Vinogradov, G.V. and Malkin, A.Ya., Reologiya polimerov (Rheology of Polymers), Moscow: Khimiya, 1977.Google Scholar
  57. 57.
    Tropin, T.V., Schmelzer, J.W.P., and Schick, C., On the dependence of the properties of glasses on cooling and heating rates: II. Prigogine-Defay ratio, fictive temperature, and fictive pressure, J. Non-Cryst. Solids, 2011, vol. 357, no. 23, pp. 1303–1309.CrossRefGoogle Scholar
  58. 58.
    Hess, S. and Evans, D.J., Computation of the viscosity of a liquid from time averages of stress fluctuations, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2001, vol. 64E, no. 1, p. 011207 (6 pages)CrossRefGoogle Scholar
  59. 59.
    Tropin, T.V., Schmelzer, J.W.P., and Schick, C., On the dependence of the properties of glasses on cooling and heating rates: I. Entropy, entropy production, and glass transition temperature, J. Non-Cryst. Solids, 2011, vol. 357, no. 23, pp. 1291–1302.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.St. Petersburg National Research University of Information Technologies, Mechanics and OpticsSt. PetersburgRussia

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