- Email: [email protected]
The relativistic glass transition: A thought experiment
Journal of Non-Crystalline Solids: X 2 (2019) 100018
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids: X journal homepage: www.journals.elsevier.com/journal-of-non-crystalline-solids-x
The relativistic glass transition: A thought experiment a
a
b
a,⁎
Collin J. Wilkinson , Karan Doss , Greg Palmer , John C. Mauro a b
T
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, United States of America Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706, United States of America
ARTICLE INFO
ABSTRACT
Keywords: Glass transition Relativity Viscosity
In order to quantify the temporal effects on the glass transition at relativistic speeds, the Deborah number is analyzed incorporating the Maxwell relation for relaxation time. Using the time dilation coefficient proposed by Einstein for special relativity, the new observation times are calculated, and the corresponding shear modulus is used to determine the shift in the glass transition temperature. The new glass transition temperature is incorporated into the MYEGA expression to estimate the relativistic viscosity curve. B2O3 glass is used as an example to show the impact of relativistic time scales on viscosity and the glass transition.
1. Introduction The prophet Deborah sang in the book of Judges, “The mountains flowed before the Lord” [1,2]; however, the question the prophet failed to ask was what happens if the mountains are moving at relativistic speeds. The focus of this work is to highlight the temporal effects on ergodicity in the context of special relativity and the reference frame of the observer. In order to understand relativistic effects on materials, one must consider a well-characterized material that actively relaxes but appears solid on our timescale, viz., pure B2O3 glass [3,4]. Glass is a particularly good candidate to study the effects of time dilation on a material since it undergoes a kinetic transition known as the glass transition (Tg), which has multiple definitions that will be considered. The first definition is in terms of the aptly named “Deborah Number,” which was proposed by Reiner [5] to offer a view of fluidity and equilibrium mechanics. It was later expanded to explain the origin of the glass transition and the transition from an ergodic liquid system to a non-ergodic solid-like glassy state [6], which is the fundamental definition of the glass transition [6–8]. It is important to note here that the breakdown of ergodicity is relative to the observation timescale and is reflected in the Deborah number (D), which is defined as
D
t
(1)
where t is the (external) observation time and τ is the relaxation time of the material, which can be expressed through the Maxwell relation [9] as
=
⁎
G
(2)
In Eq. (2), η is the shear viscosity and G is the shear modulus. By definition, the Deborah Number equals 1 at the glass transition temperature, Tg [1,7]. Another definition of the glass transition temperature is due to Angell, who defined Tg as the temperature where the viscosity of the supercooled liquid is 1012 Pa s [10,11]. The Angell diagram is an important plot relevant to viscosity and the glass transition, where the abscissa is Tg/T and the ordinate axis is log10(η). In the Angell diagram, the value of the viscosity at Tg and in the limit of infinite temperature are fixed, and the difference in the temperature scaling of the Tg/Tnormalized viscosity is the slope of the log10(η) curve at Tg/T = 1, which Angell defined as the fragility (m) of the supercooled liquid. Using this definition, one can establish the consistency of the Angell plot at relativistic speeds. An example of the Angell plot is shown in Fig. 1. Using the two previous definitions we have,
DTg = 1 =
(Tg ) tG (Tg )
(3)
which can be rewritten as,
(Tg ) G (Tg )
=t
(4)
In order to solve this equation, one must know the shear modulus at the glass transition temperature. Recent improvements in topological constraint theory have led to the ability to predict properties with varying temperature and composition. Using a recent topological constraint model for elastic moduli proposed by Wilkinson et al. [15], one
Corresponding author. E-mail address: [email protected] (J.C. Mauro).
https://doi.org/10.1016/j.nocx.2019.100018 Received 8 January 2019; Received in revised form 5 March 2019; Accepted 11 March 2019 Available online 14 March 2019 2590-1591/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Journal of Non-Crystalline Solids: X 2 (2019) 100018
C.J. Wilkinson, et al.
Fig. 1. An Angell diagram created using the MYEGA expression [12] with a very strong glass (m~17), a highly fragile glass (m~100), and a pure borate glass (m~33) [3,4,13]. The infinite temperature limit is taken from the work of Zheng et al. [14], and the glass transition temperature is from the Angell definition.
Fig. 4. The equilibrium viscosity curves for a borate glass travelling at different fractions of light speed. All of the viscosities approaching the universal high temperature limit for viscosity.
Fig. 5. The modulus needed to satisfy the condition for the glass transition. Fig. 2. Temperature dependence of the shear modulus using the onsets previously published by Wilkinson et al. [15]; the shear modulus is then calculated to be 7.09 GPa at the Tg of 518 K [3] denoted by the red dashed line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
can predict the temperature dependence of the modulus,
G= =
dG d Fc
c
[ c nc qC (T )] (x ) NA M
dG ( F n q (T ) + F n q (T ) + F n q (T )) (x ) NA d Fc M
(5)
In Eq. (5), nc is the number of topological constraints (c), qc is the constraint onset function as described by Mauro et al. [16], ΔFc is the free energy of the constraint, ρ is the density of the system, M is the molar mass, and dG is a scaling factor. d Fc In the borate system there are three different constraints to consider: α denotes linear B-O constraint, β is the angular O-B-O constraint, and γ is the angular B-O-B constraint. The temperature dependence of this expression is shown in Fig. 2, where the shear modulus comes from Kodama et al. [4] and is equal to 8 GPa at room temperature. The B2O3 structure consists entirely of three-coordinated boron and bridging oxygen [17,18]. Using the modulus predicted in Fig. 2 and Eq. (4), the value of the observation time is: Fig. 3. The relativistic glass transition temperature for B2O3 glass.
t=
1012Pa s = 141 s 7.09 × 109Pa
(6)
With this description of the static (v/c = 0 where v is the speed of the glass and c is the speed of light) glass transition it is possible now to describe the relativistic glass transition for which we consider two separate cases. 2
Journal of Non-Crystalline Solids: X 2 (2019) 100018
C.J. Wilkinson, et al.
Fig. 6. (Left) The predicted glass transition temperature with the anomalous behavior occurring at v = 0.44c. (Right) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches the speed of light.
It is worth noting that viscosity has previously been studied in the field of relativistic fluid dynamics near large gravitational bodies (i.e., general relativity) [19]. Some work has also reported the relaxation time of an ideal fluid in a gravitational field and its relationship to the nonequilibrium entropy of a relativistic system [20,21]. Our current work emphasizes the theoretical effects of special relativity on liquids and their glass transition temperature, focusing on the key role of the observer.
log10 (T ) = log10 + (12
t0 (7)
v2 c2
)
(8)
(Tg )
= G (Tg )
t0
= G (Tg )
m log
Tg T
1
which when the relativistic shear modulus γ2 factor is included can be written as
(Tg ) (9)
t0
= G (Tg )
(13)
the same as Eq. (9). Solving in the same manner as before, it is shown which shear modulus is needed to satisfy the condition in Eq. (13). In Fig. 5, the shear modulus at the glass transition temperature is shown; however, the constraint theory mechanism implemented to calculate the shear modulus has a built-in upper limit when calculating the Tg. At sufficiently low temperatures, past the point at which all constraints are intact, no further increase in the modulus is present in the model since the modulus data below room temperature have not been measured. In turn, this lack of ability to fit the modulus below room temperature leads to a situation where it is not possible to predict the occurrence of the glass transition at speeds faster than 0.4c. Nonetheless, the calculated Tg over the available range is shown in Fig. 6, as well as the related viscosities. It is seen clearly that at higher velocities the glass transition shifts towards zero, more dramatically at the higher temperatures. This leads to the interesting result that if we take the limit of an observer moving past the earth at close to the speed of light, all glass would appear to be a liquid.
giving a condition that is temperature dependent to predict the new glass transition temperature as a function of γ. However, in order to describe the relativistic behavior accurately, density must be considered. Density will change by a factor of γ2, because both the mass and the volume will be affected. As such,
(Tg )
12
(12)
t = t0
Eq. (4) can be modified to account for relativistic effects by combining with Eq. (7),
t0
exp
In this case, we consider an observer moving past a sample of glass at relativistic speeds giving,
1
(1
T
3. Relativistic observer
where t0 = 141 s (as previously calculated) and γ is the Lorentz factor, 1 2
Tg
The MYEGA equation needs two other parameters besides Tg to generate the viscosity curve: fragility (m), which is the slope at Tg in the Angell plot, and the infinite temperature viscosity (η∞). The fragility will not change if the Angell plot is consistent at all relativistic values, and the infinite temperature viscosity will also not change since this is the minimum possible viscosity for a liquid. The new viscosity curves for various speeds are shown in Fig. 4. From this, we can see that any liquid moving past an observer at relativistic speeds will appear to be more solid and eventually appear as a glass.
The first case we consider is a sample of a glass-forming system moving past a stationary observer at a speed approaching light, giving the new observation time as
=
)
(11)
2. Relativistic liquid
t=
log10
(10)
The resulting relativistic glass transition is plotted in Fig. 3. In Fig. 3 the glass transition temperature increases monotonically with velocity, showing that any liquid will undergo an increase in the Tg and will cause any liquid to appear to be solid. Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature, which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-GuptaAllan (MYEGA) expression [12,22], 3
Journal of Non-Crystalline Solids: X 2 (2019) 100018
C.J. Wilkinson, et al.
4. Conclusion
[7] P.K. Gupta, J.C. Mauro, The laboratory glass transition, J. Chem. Phys. 126 (2007), https://doi.org/10.1063/1.2738471 2240504. [8] E.D. Zanotto, J.C. Mauro, The glassy state of matter: its definition and ultimate fate, J. Non-Cryst. Solids 471 (2017) 490–495, https://doi.org/10.1016/j.jnoncrysol. 2017.05.019. [9] C.J. Wilkinson, Y.Z. Mauro, J.C. Mauro, RelaxPy: Python code for modeling of glass relaxation behavior, SoftwareX. 7 (2018) 255–258, https://doi.org/10.1016/j. softx.2018.07.008. [10] C.A. Angell, Relaxation in liquids, polymers and plastic crystals - strong/fragile patterns and problems, J. Non-Cryst. Solids 13 (1991) 131–133, https://doi.org/10. 1016/0022-3093(91)90266-9. [11] R. Richert, C.A. Angell, Dynamics of glass-forming liquids. V. On the link between molecular dynamics and configurational entropy, J. Chem. Phys. 108 (1998) 9016–9026, https://doi.org/10.1063/1.476348. [12] J.C. Mauro, Y. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. 106 (2009) 19780–19784, https://doi.org/10.1073/ pnas.0911705106. [13] J.C. Mauro, P.K. Gupta, R.J. Loucks, Composition dependence of glass transition temperature and fragility. II. A topological model of alkali borate liquids, J. Chem. Phys. 130 (2009), https://doi.org/10.1063/1.3152432 234503. [14] Q. Zheng, J.C. Mauro, A.J. Ellison, M. Potuzak, Y. Yue, Universality of the hightemperature viscosity limit of silicate liquids, Phys. Rev. B - Condens. Matter Mater. Phys. 83 (2011), https://doi.org/10.1103/PhysRevB.83.212202 212202. [15] C.J. Wilkinson, Q. Zheng, L. Huang, J.C. Mauro, Topological constraint model for the elasticity of glass-forming systems, J. Non-Crystalline Solids. (15 Mar 2019), https://doi.org/10.1016/j.nocx.2019.100019. [16] J.C. Mauro, Topological constraint theory of glass, Am. Ceram. Soc. Bulletin 90 (4) (2011) 31. [17] S. Feller, S. Bista, A. O'Donovan-Zavada, T. Mullenbach, M. Franke, M. Affatigato, Packing in alkali and alkaline earth borosilicate glass systems, Phys. Chem. Glas. Eur. J. Glas. Sci. Technol. Part B. 50 (2009) 224–228. [18] P.J. Bray, S.A. Feller, G.E. Jellison, Y.H. Yun, B10NMR studies of the structure of borate glasses, J. Non-Cryst. Solids 38–39 (1980) 93–98, https://doi.org/10.1016/ 0022-3093(80)90400-7. [19] N. Andersson, G.L. Comer, Relativistic fluid dynamics: physics for many different scales living reviews in relativity, Living Rev. Relativ. 10 (2007) 1. [20] P. Ilg, H.C. Öttinger, Nonequilibrium relativistic thermodynamics in bulk viscous cosmology, Phys. Rev. D - Part. Fields Gravit. Cosmol. 61 (2000), https://doi.org/ 10.1103/PhysRevD.61.023510 023510. [21] P. Romatschke, Relativistic viscous fluid dynamics and non-equilibrium entropy, Class. Quantum Gravity 27 (2010), https://doi.org/10.1088/0264-9381/27/2/ 025006 025006. [22] D.C. Allan, Inverting the MYEGA equation for viscosity, J. Non-Cryst. Solids 358 (2012) 440–442, https://doi.org/10.1016/J.JNONCRYSOL.2011.09.036.
Using the mechanisms proposed by Einstein applied to the concept of the Deborah number, it has been shown that the glass transition temperature will change dramatically as a system approaches relativistic speeds. Specifically, any glass-forming system can appear either more or less fluid based on the speed in which they travel. This, in turn, implies that when discussing the role of the observer in governing the glass transition, one may further ask the velocity of both the sample and the observer with respect to the speed of light. This also serves to highlight the importance of the role of the observer on the glass transition. Acknowledgments The authors would like to thank Arron R. Potter and Rebecca S. Welch for ongoing insightful conversations. Conflict of interest We declare no competing interests. References [1] J.C. Mauro, P.K. Gupta, R.J. Loucks, Continuously broken ergodicity, J. Chem. Phys. 126 (2007) 184511, https://doi.org/10.1063/1.2731774. [2] J.C. Mauro, M.M. Smedskjaer, Statistical mechanics of glass, J. Non-Cryst. Solids 396–397 (2014) 41–53, https://doi.org/10.1016/j.jnoncrysol.2014.04.009. [3] M. Kodama, S. Kojima, S. Feller, M. Affatigato, The occurrence of minima in the borate anomaly, anharmonicity and fragility in lithium borate glasses, Phys. Chem. Glasses 46 (2005) 190–193. [4] Y. Fukawa, Y. Matsuda, Y. Ike, Y. Kondo, T. Kouyama, K. Ohno, M. Kawashima, Velocity of sound and elastic properties of Li2O-B2O3 glasses, Jpn. J. Appl. Phys. 34 (1995) 2570, https://doi.org/10.1143/JJAP.34.2570. [5] M. Reiner, The Deborah number, Phys. Today 17 (1964) 62, https://doi.org/10. 1063/1.3051374. [6] R.G. Palmer, Broken ergodicity, Adv. Phys. 31 (1982) 669–735, https://doi.org/10. 1080/00018738200101438.
4
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids: X journal homepage: www.journals.elsevier.com/journal-of-non-crystalline-solids-x
The relativistic glass transition: A thought experiment a
a
b
a,⁎
Collin J. Wilkinson , Karan Doss , Greg Palmer , John C. Mauro a b
T
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, United States of America Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706, United States of America
ARTICLE INFO
ABSTRACT
Keywords: Glass transition Relativity Viscosity
In order to quantify the temporal effects on the glass transition at relativistic speeds, the Deborah number is analyzed incorporating the Maxwell relation for relaxation time. Using the time dilation coefficient proposed by Einstein for special relativity, the new observation times are calculated, and the corresponding shear modulus is used to determine the shift in the glass transition temperature. The new glass transition temperature is incorporated into the MYEGA expression to estimate the relativistic viscosity curve. B2O3 glass is used as an example to show the impact of relativistic time scales on viscosity and the glass transition.
1. Introduction The prophet Deborah sang in the book of Judges, “The mountains flowed before the Lord” [1,2]; however, the question the prophet failed to ask was what happens if the mountains are moving at relativistic speeds. The focus of this work is to highlight the temporal effects on ergodicity in the context of special relativity and the reference frame of the observer. In order to understand relativistic effects on materials, one must consider a well-characterized material that actively relaxes but appears solid on our timescale, viz., pure B2O3 glass [3,4]. Glass is a particularly good candidate to study the effects of time dilation on a material since it undergoes a kinetic transition known as the glass transition (Tg), which has multiple definitions that will be considered. The first definition is in terms of the aptly named “Deborah Number,” which was proposed by Reiner [5] to offer a view of fluidity and equilibrium mechanics. It was later expanded to explain the origin of the glass transition and the transition from an ergodic liquid system to a non-ergodic solid-like glassy state [6], which is the fundamental definition of the glass transition [6–8]. It is important to note here that the breakdown of ergodicity is relative to the observation timescale and is reflected in the Deborah number (D), which is defined as
D
t
(1)
where t is the (external) observation time and τ is the relaxation time of the material, which can be expressed through the Maxwell relation [9] as
=
⁎
G
(2)
In Eq. (2), η is the shear viscosity and G is the shear modulus. By definition, the Deborah Number equals 1 at the glass transition temperature, Tg [1,7]. Another definition of the glass transition temperature is due to Angell, who defined Tg as the temperature where the viscosity of the supercooled liquid is 1012 Pa s [10,11]. The Angell diagram is an important plot relevant to viscosity and the glass transition, where the abscissa is Tg/T and the ordinate axis is log10(η). In the Angell diagram, the value of the viscosity at Tg and in the limit of infinite temperature are fixed, and the difference in the temperature scaling of the Tg/Tnormalized viscosity is the slope of the log10(η) curve at Tg/T = 1, which Angell defined as the fragility (m) of the supercooled liquid. Using this definition, one can establish the consistency of the Angell plot at relativistic speeds. An example of the Angell plot is shown in Fig. 1. Using the two previous definitions we have,
DTg = 1 =
(Tg ) tG (Tg )
(3)
which can be rewritten as,
(Tg ) G (Tg )
=t
(4)
In order to solve this equation, one must know the shear modulus at the glass transition temperature. Recent improvements in topological constraint theory have led to the ability to predict properties with varying temperature and composition. Using a recent topological constraint model for elastic moduli proposed by Wilkinson et al. [15], one
Corresponding author. E-mail address: [email protected] (J.C. Mauro).
https://doi.org/10.1016/j.nocx.2019.100018 Received 8 January 2019; Received in revised form 5 March 2019; Accepted 11 March 2019 Available online 14 March 2019 2590-1591/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Journal of Non-Crystalline Solids: X 2 (2019) 100018
C.J. Wilkinson, et al.
Fig. 1. An Angell diagram created using the MYEGA expression [12] with a very strong glass (m~17), a highly fragile glass (m~100), and a pure borate glass (m~33) [3,4,13]. The infinite temperature limit is taken from the work of Zheng et al. [14], and the glass transition temperature is from the Angell definition.
Fig. 4. The equilibrium viscosity curves for a borate glass travelling at different fractions of light speed. All of the viscosities approaching the universal high temperature limit for viscosity.
Fig. 5. The modulus needed to satisfy the condition for the glass transition. Fig. 2. Temperature dependence of the shear modulus using the onsets previously published by Wilkinson et al. [15]; the shear modulus is then calculated to be 7.09 GPa at the Tg of 518 K [3] denoted by the red dashed line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
can predict the temperature dependence of the modulus,
G= =
dG d Fc
c
[ c nc qC (T )] (x ) NA M
dG ( F n q (T ) + F n q (T ) + F n q (T )) (x ) NA d Fc M
(5)
In Eq. (5), nc is the number of topological constraints (c), qc is the constraint onset function as described by Mauro et al. [16], ΔFc is the free energy of the constraint, ρ is the density of the system, M is the molar mass, and dG is a scaling factor. d Fc In the borate system there are three different constraints to consider: α denotes linear B-O constraint, β is the angular O-B-O constraint, and γ is the angular B-O-B constraint. The temperature dependence of this expression is shown in Fig. 2, where the shear modulus comes from Kodama et al. [4] and is equal to 8 GPa at room temperature. The B2O3 structure consists entirely of three-coordinated boron and bridging oxygen [17,18]. Using the modulus predicted in Fig. 2 and Eq. (4), the value of the observation time is: Fig. 3. The relativistic glass transition temperature for B2O3 glass.
t=
1012Pa s = 141 s 7.09 × 109Pa
(6)
With this description of the static (v/c = 0 where v is the speed of the glass and c is the speed of light) glass transition it is possible now to describe the relativistic glass transition for which we consider two separate cases. 2
Journal of Non-Crystalline Solids: X 2 (2019) 100018
C.J. Wilkinson, et al.
Fig. 6. (Left) The predicted glass transition temperature with the anomalous behavior occurring at v = 0.44c. (Right) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches the speed of light.
It is worth noting that viscosity has previously been studied in the field of relativistic fluid dynamics near large gravitational bodies (i.e., general relativity) [19]. Some work has also reported the relaxation time of an ideal fluid in a gravitational field and its relationship to the nonequilibrium entropy of a relativistic system [20,21]. Our current work emphasizes the theoretical effects of special relativity on liquids and their glass transition temperature, focusing on the key role of the observer.
log10 (T ) = log10 + (12
t0 (7)
v2 c2
)
(8)
(Tg )
= G (Tg )
t0
= G (Tg )
m log
Tg T
1
which when the relativistic shear modulus γ2 factor is included can be written as
(Tg ) (9)
t0
= G (Tg )
(13)
the same as Eq. (9). Solving in the same manner as before, it is shown which shear modulus is needed to satisfy the condition in Eq. (13). In Fig. 5, the shear modulus at the glass transition temperature is shown; however, the constraint theory mechanism implemented to calculate the shear modulus has a built-in upper limit when calculating the Tg. At sufficiently low temperatures, past the point at which all constraints are intact, no further increase in the modulus is present in the model since the modulus data below room temperature have not been measured. In turn, this lack of ability to fit the modulus below room temperature leads to a situation where it is not possible to predict the occurrence of the glass transition at speeds faster than 0.4c. Nonetheless, the calculated Tg over the available range is shown in Fig. 6, as well as the related viscosities. It is seen clearly that at higher velocities the glass transition shifts towards zero, more dramatically at the higher temperatures. This leads to the interesting result that if we take the limit of an observer moving past the earth at close to the speed of light, all glass would appear to be a liquid.
giving a condition that is temperature dependent to predict the new glass transition temperature as a function of γ. However, in order to describe the relativistic behavior accurately, density must be considered. Density will change by a factor of γ2, because both the mass and the volume will be affected. As such,
(Tg )
12
(12)
t = t0
Eq. (4) can be modified to account for relativistic effects by combining with Eq. (7),
t0
exp
In this case, we consider an observer moving past a sample of glass at relativistic speeds giving,
1
(1
T
3. Relativistic observer
where t0 = 141 s (as previously calculated) and γ is the Lorentz factor, 1 2
Tg
The MYEGA equation needs two other parameters besides Tg to generate the viscosity curve: fragility (m), which is the slope at Tg in the Angell plot, and the infinite temperature viscosity (η∞). The fragility will not change if the Angell plot is consistent at all relativistic values, and the infinite temperature viscosity will also not change since this is the minimum possible viscosity for a liquid. The new viscosity curves for various speeds are shown in Fig. 4. From this, we can see that any liquid moving past an observer at relativistic speeds will appear to be more solid and eventually appear as a glass.
The first case we consider is a sample of a glass-forming system moving past a stationary observer at a speed approaching light, giving the new observation time as
=
)
(11)
2. Relativistic liquid
t=
log10
(10)
The resulting relativistic glass transition is plotted in Fig. 3. In Fig. 3 the glass transition temperature increases monotonically with velocity, showing that any liquid will undergo an increase in the Tg and will cause any liquid to appear to be solid. Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature, which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-GuptaAllan (MYEGA) expression [12,22], 3
Journal of Non-Crystalline Solids: X 2 (2019) 100018
C.J. Wilkinson, et al.
4. Conclusion
[7] P.K. Gupta, J.C. Mauro, The laboratory glass transition, J. Chem. Phys. 126 (2007), https://doi.org/10.1063/1.2738471 2240504. [8] E.D. Zanotto, J.C. Mauro, The glassy state of matter: its definition and ultimate fate, J. Non-Cryst. Solids 471 (2017) 490–495, https://doi.org/10.1016/j.jnoncrysol. 2017.05.019. [9] C.J. Wilkinson, Y.Z. Mauro, J.C. Mauro, RelaxPy: Python code for modeling of glass relaxation behavior, SoftwareX. 7 (2018) 255–258, https://doi.org/10.1016/j. softx.2018.07.008. [10] C.A. Angell, Relaxation in liquids, polymers and plastic crystals - strong/fragile patterns and problems, J. Non-Cryst. Solids 13 (1991) 131–133, https://doi.org/10. 1016/0022-3093(91)90266-9. [11] R. Richert, C.A. Angell, Dynamics of glass-forming liquids. V. On the link between molecular dynamics and configurational entropy, J. Chem. Phys. 108 (1998) 9016–9026, https://doi.org/10.1063/1.476348. [12] J.C. Mauro, Y. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. 106 (2009) 19780–19784, https://doi.org/10.1073/ pnas.0911705106. [13] J.C. Mauro, P.K. Gupta, R.J. Loucks, Composition dependence of glass transition temperature and fragility. II. A topological model of alkali borate liquids, J. Chem. Phys. 130 (2009), https://doi.org/10.1063/1.3152432 234503. [14] Q. Zheng, J.C. Mauro, A.J. Ellison, M. Potuzak, Y. Yue, Universality of the hightemperature viscosity limit of silicate liquids, Phys. Rev. B - Condens. Matter Mater. Phys. 83 (2011), https://doi.org/10.1103/PhysRevB.83.212202 212202. [15] C.J. Wilkinson, Q. Zheng, L. Huang, J.C. Mauro, Topological constraint model for the elasticity of glass-forming systems, J. Non-Crystalline Solids. (15 Mar 2019), https://doi.org/10.1016/j.nocx.2019.100019. [16] J.C. Mauro, Topological constraint theory of glass, Am. Ceram. Soc. Bulletin 90 (4) (2011) 31. [17] S. Feller, S. Bista, A. O'Donovan-Zavada, T. Mullenbach, M. Franke, M. Affatigato, Packing in alkali and alkaline earth borosilicate glass systems, Phys. Chem. Glas. Eur. J. Glas. Sci. Technol. Part B. 50 (2009) 224–228. [18] P.J. Bray, S.A. Feller, G.E. Jellison, Y.H. Yun, B10NMR studies of the structure of borate glasses, J. Non-Cryst. Solids 38–39 (1980) 93–98, https://doi.org/10.1016/ 0022-3093(80)90400-7. [19] N. Andersson, G.L. Comer, Relativistic fluid dynamics: physics for many different scales living reviews in relativity, Living Rev. Relativ. 10 (2007) 1. [20] P. Ilg, H.C. Öttinger, Nonequilibrium relativistic thermodynamics in bulk viscous cosmology, Phys. Rev. D - Part. Fields Gravit. Cosmol. 61 (2000), https://doi.org/ 10.1103/PhysRevD.61.023510 023510. [21] P. Romatschke, Relativistic viscous fluid dynamics and non-equilibrium entropy, Class. Quantum Gravity 27 (2010), https://doi.org/10.1088/0264-9381/27/2/ 025006 025006. [22] D.C. Allan, Inverting the MYEGA equation for viscosity, J. Non-Cryst. Solids 358 (2012) 440–442, https://doi.org/10.1016/J.JNONCRYSOL.2011.09.036.
Using the mechanisms proposed by Einstein applied to the concept of the Deborah number, it has been shown that the glass transition temperature will change dramatically as a system approaches relativistic speeds. Specifically, any glass-forming system can appear either more or less fluid based on the speed in which they travel. This, in turn, implies that when discussing the role of the observer in governing the glass transition, one may further ask the velocity of both the sample and the observer with respect to the speed of light. This also serves to highlight the importance of the role of the observer on the glass transition. Acknowledgments The authors would like to thank Arron R. Potter and Rebecca S. Welch for ongoing insightful conversations. Conflict of interest We declare no competing interests. References [1] J.C. Mauro, P.K. Gupta, R.J. Loucks, Continuously broken ergodicity, J. Chem. Phys. 126 (2007) 184511, https://doi.org/10.1063/1.2731774. [2] J.C. Mauro, M.M. Smedskjaer, Statistical mechanics of glass, J. Non-Cryst. Solids 396–397 (2014) 41–53, https://doi.org/10.1016/j.jnoncrysol.2014.04.009. [3] M. Kodama, S. Kojima, S. Feller, M. Affatigato, The occurrence of minima in the borate anomaly, anharmonicity and fragility in lithium borate glasses, Phys. Chem. Glasses 46 (2005) 190–193. [4] Y. Fukawa, Y. Matsuda, Y. Ike, Y. Kondo, T. Kouyama, K. Ohno, M. Kawashima, Velocity of sound and elastic properties of Li2O-B2O3 glasses, Jpn. J. Appl. Phys. 34 (1995) 2570, https://doi.org/10.1143/JJAP.34.2570. [5] M. Reiner, The Deborah number, Phys. Today 17 (1964) 62, https://doi.org/10. 1063/1.3051374. [6] R.G. Palmer, Broken ergodicity, Adv. Phys. 31 (1982) 669–735, https://doi.org/10. 1080/00018738200101438.
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