glasspy.viscosity package

Submodules

glasspy.viscosity.diffusion module

Equations for computing the effective diffusion coefficient from viscosity.

glasspy.viscosity.diffusion.diff_coeff_eyring(T, viscosity, diameter)

Computes the viscosity diffusion coefficient using Eyring equation

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • viscosity – float or array_like with same lenght as T Viscosity at temperature T.

  • diameter – float or array_like with same lenght as T The diameter of the structural unit that is moving due to viscous flow.

Returns

Returns the effective diffusion coefficient computed using the Eyring equation. This equation is similar to the Stokes-Einstein equation, but they were obtained by different routes.

References

[1] Eyring, H. (1936). Viscosity, plasticity, and diffusion as examples of

absolute reaction rates. The Journal of Chemical Physics 4, 283–291.

glasspy.viscosity.diffusion.diff_coeff_stokeseinstein(T, viscosity, diameter)

Computes the viscosity diffusion coefficient using Stokes-Einstein equation

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • viscosity – float or array_like with same lenght as T Viscosity at temperature T.

  • diameter – float or array_like with same lenght as T The diameter of the structural unit that is moving due to viscous flow.

Returns

Returns the effective diffusion coefficient computed using the Stokes-Einstein equation. This equation is similar to the Eyring equation, but they were obtained by different routes.

References

[1] Einstein, A. (1905). On the movement of small particles suspended in

stationary liquids required by the molecular-kinetic theory of heat. Annalen Der Physik 17, 549–560.

[2] Einstein, A. (1905). Über die von der molekularkinetischen Theorie der

Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen Der Physik 322, 549–560.

[3] Stokes, G.G. (1851). On the effect of the internal friction of fluids

on the motion of pendulums. Transactions of the Cambridge Philosophical Society 9, 8–106.

glasspy.viscosity.equilibrium module

Equations for equilibrium viscosity.

glasspy.viscosity.equilibrium.ag(T, eta_inf, B, S_conf_fun)

Computes the viscosity using the Adam & Gibbs equation.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • B – float Adjustable parameter related to the potential energy hindering the cooperative rearrangement per monomer segment.

  • S_conf_fun – callable Function that computes the configurational entropy. This function accepts one argument, which is the absolute temperature.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

References

[1] Adam, G., and Gibbs, J.H. (1965). On the temperature dependence of

cooperative relaxation properties in glass-forming liquids. The Journal of Chemical Physics 43, 139–146.

glasspy.viscosity.equilibrium.am(T, eta_inf, alpha, beta)

Computes the viscosity using the Avramov & Milchev equation.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • alpha – float Adjustable parameter, see original reference. Unitless.

  • beta – float Adjustable parameter with unit of Kelvin.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

References

[1] Avramov, I., and Milchev, A. (1988). Effect of disorder on diffusion

and viscosity in condensed systems. Journal of Non-Crystalline Solids 104, 253–260.

[2] Cornelissen, J., and Waterman, H.I. (1955). The viscosity temperature

relationship of liquids. Chemical Engineering Science 4, 238–246.

glasspy.viscosity.equilibrium.am_alt(T, eta_inf, T12, m)

Computes the viscosity using the Avramov & Milchev equation.

This is the rewriten AM equation found in ref. [3].

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • T12 – float Temperature were the viscosity is 10**12 Pa.s. Unit: Kelvin.

  • m – float Fragility index as defined by Angell, see ref. [4]. Unitless.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

References

[1] Avramov, I., and Milchev, A. (1988). Effect of disorder on diffusion

and viscosity in condensed systems. Journal of Non-Crystalline Solids 104, 253–260.

[2] Cornelissen, J., and Waterman, H.I. (1955). The viscosity temperature

relationship of liquids. Chemical Engineering Science 4, 238–246.

[3] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[4] Angell, C.A. (1985). Strong and fragile liquids. In Relaxation in

Complex Systems, K.L. Ngai, and G.B. Wright, eds. (Springfield: Naval Research Laboratory), pp. 3–12.

glasspy.viscosity.equilibrium.myega(T, eta_inf, K, C)

Computes the viscosity using the MYEGA equation.

Mathematicaly, this equation is the same as that proposed in ref. [2] (see page 250), however the physical considerations are different.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • K – float See the original reference for the meaning. Unit: Kelvin.

  • C – float See the original reference for the meaning. Unit: Kelvin.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

Notes

In the original reference the equation is in base-10 logarithm, see Eq. (6) in [1].

References

[1] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[2] Waterton, S.C. (1932). The viscosity-temperature relationship and some

inferences on the nature of molten and of plastic glass. J Soc Glass Technol 16, 244–249.

glasspy.viscosity.equilibrium.myega_alt(T, eta_inf, T12, m)

Computes the viscosity using the MYEGA equation.

This is an alternate form of the MYEGA equation found in [1]

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • T12 – float Temperature were the viscosity is 10**12 Pa.s. Unit: Kelvin.

  • m – float Fragility index as defined by Angell, see ref. [2]. Unitless.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

References

[1] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[2] Angell, C.A. (1985). Strong and fragile liquids. In Relaxation in

Complex Systems, K.L. Ngai, and G.B. Wright, eds. (Springfield: Naval Research Laboratory), pp. 3–12.

glasspy.viscosity.equilibrium.vft(T, eta_inf, A, T0)

Computes the viscosity using the empirical Vogel-Fulcher-Tammann eq.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • A – float Adjustable parameter inside the exponential. Unit: Kelvin.

  • T0 – float Divergence temperature. Unit: Kelvin.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

References

[1] Vogel, H. (1921). Das Temperatureabhängigketsgesetz der Viskosität von

Flüssigkeiten. Physikalische Zeitschrift 22, 645–646.

[2] Fulcher, G.S. (1925). Analysis of recent measurements of the viscosity

of glasses. Journal of the American Ceramic Society 8, 339–355.

[3] Tammann, G., and Hesse, W. (1926). Die Abhängigkeit der Viscosität von

der Temperatur bie unterkühlten Flüssigkeiten. Z. Anorg. Allg. Chem. 156, 245–257.

glasspy.viscosity.equilibrium.vft_alt(T, eta_inf, T12, m)

Computes the viscosity using the Vogel-Fulcher-Tammann eq.

This is the rewriten VFT equation found in ref. [4].

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • T12 – float Temperature were the viscosity is 10**12 Pa.s. Unit: Kelvin.

  • m – float Fragility index as defined by Angell, see ref. [5]. Unitless.

Returns

it is not the logarithm of viscosity.

Return type

Returns the viscosity in the units of eta_inf. Note

References

[1] Vogel, H. (1921). Das Temperatureabhängigketsgesetz der Viskosität von

Flüssigkeiten. Physikalische Zeitschrift 22, 645–646.

[2] Fulcher, G.S. (1925). Analysis of recent measurements of the viscosity

of glasses. Journal of the American Ceramic Society 8, 339–355.

[3] Tammann, G., and Hesse, W. (1926). Die Abhängigkeit der Viscosität von

der Temperatur bie unterkühlten Flüssigkeiten. Z. Anorg. Allg. Chem. 156, 245–257.

[4] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[5] Angell, C.A. (1985). Strong and fragile liquids. In Relaxation in

Complex Systems, K.L. Ngai, and G.B. Wright, eds. (Springfield: Naval Research Laboratory), pp. 3–12.

glasspy.viscosity.equilibrium_log module

Equations for the base-10 logarithm of equilibrium viscosity.

glasspy.viscosity.equilibrium_log.ag(T, eta_inf, B, S_conf_fun)

Computes the viscosity using the Adam & Gibbs equation.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • B – float Adjustable parameter related to the potential energy hindering the cooperative rearrangement per monomer segment.

  • S_conf_fun – callable Function that computes the configurational entropy. This function accepts one argument, which is the absolute temperature.

Returns

Returns the base-10 logarithm of viscosity.

References

[1] Adam, G., and Gibbs, J.H. (1965). On the temperature dependence of

cooperative relaxation properties in glass-forming liquids. The Journal of Chemical Physics 43, 139–146.

glasspy.viscosity.equilibrium_log.am(T, log_eta_inf, alpha, beta)

Computes the viscosity using the Avramov & Milchev equation.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • alpha – float Adjustable parameter, see original reference. Unitless.

  • beta – float Adjustable parameter with unit of Kelvin.

Returns

Returns the base-10 logarithm of viscosity.

References

[1] Avramov, I., and Milchev, A. (1988). Effect of disorder on diffusion

and viscosity in condensed systems. Journal of Non-Crystalline Solids 104, 253–260.

[2] Cornelissen, J., and Waterman, H.I. (1955). The viscosity temperature

relationship of liquids. Chemical Engineering Science 4, 238–246.

glasspy.viscosity.equilibrium_log.am_alt(T, log_eta_inf, T12, m)

Computes the viscosity using the Avramov & Milchev equation.

This is the rewriten AM equation found in ref. [3].

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • T12 – float Temperature were the viscosity is 10**12 Pa.s. Unit: Kelvin.

  • m – float Fragility index as defined by Angell, see ref. [4]. Unitless.

Returns

Returns the base-10 logarithm of viscosity.

References

[1] Avramov, I., and Milchev, A. (1988). Effect of disorder on diffusion

and viscosity in condensed systems. Journal of Non-Crystalline Solids 104, 253–260.

[2] Cornelissen, J., and Waterman, H.I. (1955). The viscosity temperature

relationship of liquids. Chemical Engineering Science 4, 238–246.

[3] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[4] Angell, C.A. (1985). Strong and fragile liquids. In Relaxation in

Complex Systems, K.L. Ngai, and G.B. Wright, eds. (Springfield: Naval Research Laboratory), pp. 3–12.

glasspy.viscosity.equilibrium_log.myega(T, log_eta_inf, K, C)

Computes the viscosity using the MYEGA equation.

Mathematicaly, this equation is the same as that proposed in ref. [2] (see page 250), however the physical considerations are different.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • K – float See the original reference for the meaning. Unit: Kelvin.

  • C – float See the original reference for the meaning. Unit: Kelvin.

Returns

Returns the base-10 logarithm of viscosity.

Notes

In the original reference the equation is in base-10 logarithm, see Eq. (6) in [1].

References

[1] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[2] Waterton, S.C. (1932). The viscosity-temperature relationship and some

inferences on the nature of molten and of plastic glass. J Soc Glass Technol 16, 244–249.

glasspy.viscosity.equilibrium_log.myega_alt(T, log_eta_inf, T12, m)

Computes the viscosity using the MYEGA equation.

This is an alternate form of the MYEGA equation found in [1]

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • T12 – float Temperature were the viscosity is 10**12 Pa.s. Unit: Kelvin.

  • m – float Fragility index as defined by Angell, see ref. [2]. Unitless.

Returns

Returns the base-10 logarithm of viscosity.

References

[1] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[2] Angell, C.A. (1985). Strong and fragile liquids. In Relaxation in

Complex Systems, K.L. Ngai, and G.B. Wright, eds. (Springfield: Naval Research Laboratory), pp. 3–12.

glasspy.viscosity.equilibrium_log.vft(T, log_eta_inf, A, T0)

Computes the viscosity using the empirical Vogel-Fulcher-Tammann eq.

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • A – float Adjustable parameter inside the exponential. Unit: Kelvin.

  • T0 – float Divergence temperature. Unit: Kelvin.

Returns

Returns the base-10 logarithm of viscosity.

References

[1] Vogel, H. (1921). Das Temperatureabhängigketsgesetz der Viskosität von

Flüssigkeiten. Physikalische Zeitschrift 22, 645–646.

[2] Fulcher, G.S. (1925). Analysis of recent measurements of the viscosity

of glasses. Journal of the American Ceramic Society 8, 339–355.

[3] Tammann, G., and Hesse, W. (1926). Die Abhängigkeit der Viscosität von

der Temperatur bie unterkühlten Flüssigkeiten. Z. Anorg. Allg. Chem. 156, 245–257.

glasspy.viscosity.equilibrium_log.vft_alt(T, log_eta_inf, T12, m)

Computes the viscosity using the Vogel-Fulcher-Tammann eq.

This is the rewriten VFT equation found in ref. [4].

Parameters
  • T – float or array_like Temperature. Unit: Kelvin.

  • eta_inf – float Asymptotic viscosity at the limit of infinite temperature.

  • T12 – float Temperature were the viscosity is 10**12 Pa.s. Unit: Kelvin.

  • m – float Fragility index as defined by Angell, see ref. [5]. Unitless.

Returns

Returns the base-10 logarithm of viscosity.

References

[1] Vogel, H. (1921). Das Temperatureabhängigketsgesetz der Viskosität von

Flüssigkeiten. Physikalische Zeitschrift 22, 645–646.

[2] Fulcher, G.S. (1925). Analysis of recent measurements of the viscosity

of glasses. Journal of the American Ceramic Society 8, 339–355.

[3] Tammann, G., and Hesse, W. (1926). Die Abhängigkeit der Viscosität von

der Temperatur bie unterkühlten Flüssigkeiten. Z. Anorg. Allg. Chem. 156, 245–257.

[4] Mauro, J.C., Yue, Y., Ellison, A.J., Gupta, P.K., and Allan, D.C.

(2009). Viscosity of glass-forming liquids. Proceedings of the National Academy of Sciences of the United States of America 106, 19780–19784.

[5] Angell, C.A. (1985). Strong and fragile liquids. In Relaxation in

Complex Systems, K.L. Ngai, and G.B. Wright, eds. (Springfield: Naval Research Laboratory), pp. 3–12.

Module contents