Pochetinno’s²⁶ viscosity vs temperature data from 1914 for a pitch material he postulated was a model for the glassy materials in magma. Data are from those tabulated in the original paper. Here we also apply the VFT fit to the data as well as estimate the T_g from the extrapolation to the temperature at which the viscosity is 10¹³ P (10¹² Pa-s).

Rendition of Tammann’s view of the viscosity–temperature behavior of a glass-forming supercooled liquid (curve A) vs that at the liquid–solid transition upon crystallization at the melting point and the crystal viscosity (curve B). Adapted from Tammann.¹⁷

Length-change dilatometry for a glass from a Jena59^III thermometer. One sample measured in length direction and another radially. The range of “usefulness” of the thermometer would have been 773.2 K,  and the two results are linear below that temperature. Above that temperature, one begins to see the rapid increase in the expansion coefficient that we now find typical in glass-forming substances. The data also show evidence of hysteresis upon cooling. Data digitized from Peters and Cragoe.³⁷

Depiction of early calorimetric data (as galvanometric deflection from differential thermocouple measurements upon heating and cooling of a borosilicate glass). Data digitized from Tool and Valasek.³⁶

Specific heat vs temperature for glycerol. Adapted with permission from ref 49. Copyright 2006 John Wiley & Sons.

Early calorimetric data showing “anomalous” heat capacity changes in amorphous liquids at low temperatures. (a) Specific heat vs temperature for ethyl and propyl alcohols and their equal molar mixture. Adapted with permission from ref 52. Copyright 1920 American Chemical Society. (b) Heat capacity of glycerol for amorphous form and crystal form. Adapted with permission from ref 53. Copyright 1923 American Chemical Society.

Volume–temperature dependence of NR by fluid confinement dilatometry (solid lines) and length-change dilatometry (dashed lines). Digitized from Bekkedahl.⁶⁰

Coefficient of thermal expansion vs temperature for amorphous and semicrystalline rubber. Adapted from Bekkedahl.⁶⁰

Schematic of the meaning of T_f. Upon cooling below the melting point, the supercooled liquid will form a glass when crystallization is slow enough. Some systems cannot crystallize in which case the metastable liquid line is an equilibrium liquid. The three situations depicted are the cooling path typical of an “ordinary glass” and two different “arbitrary paths” that show how the fictive temperature is defined. A line of slope equivalent to the glassy line is extrapolated to the intersection with the equilibrium (metastable) line to obtain T_f. From the drawing, then T_f,1 is the fictive temperature for the ordinary glass; T_f,2 is the fictive temperature for a glass formed by some arbitrary history to achieve the volume or enthalpy depicted; and, in this case, T < T_f. T_f,3 is the fictive temperature for a different path and shows the case for which the temperature is greater than the fictive temperature, that is, T > T_f. Note that we do not define the T_g here because this had not been defined in the time of Tool.

Reproduction of the Kauzmann data of the entropy differences ΔS between supercooled liquids and the stable crystal forms at one atmosphere pressure. ΔS_m is the entropy of fusion. The short dashed lines represent the glassy state response, and the intermediate dashed lines shows the entropy extrapolation for lactic acid (green on-line), glycerol (red on-line), and ethanol (blue on-line). Also shown are the points where the entropy of the liquid and the crystal become the same (ΔS/ΔS_m = 0), which seems to violate the third law at the notional T_Ks for the ethanol (T_K,EA), glycerol (T_K,GLY), and lactic acid (T_K,LA). Adapted with permission from ref 89. Copyright 1948 American Chemical Society.

Extensional creep responses of a PVC at 40 °C following a quench from above the glass transition at the aging times indicated. Curve shows how the creep response evolves as the aging time increases due to the changing structure of the nonequilibrium glass. Data are digitized from Struik.¹³²

Double log representation of the aging time shift factor vs aging time for the PVC creep data shown in Figure 11. The data are scaled to the reference aging time of $t_{\text{e},\text{ref}}=2600 \text{s}$ (0.03 d). The plot shows that at 40 °C for this material one is far enough below the glass transition that even after nearly 3 yr of aging, the response continues to evolve, in this case approximately proportionally to the aging time itself. Data obtained by analysis of Figure 11. Adapted with permission from ref 138. Copyright 2012 Society of Rheology.

Volume recovery data for a glucose glass-forming liquid. Quench was from 40 °C to the temperatures indicated. Data taken by digitizing data points in figure 3 from Kovacs.¹³²

Double logarithmic representation of a_te vs t_e for a polycarbonate glass quenched from 145 °C to the temperatures indicated where the relaxation responses were measured. The scaled relaxation times are all referenced to the response at t_e=1800 s at the temperature of interest. Data from O’Connell and McKenna.¹²⁵

Temperature and reciprocal temperature dependence of the relaxation time for polycarbonate that has been aged into equilibrium compared with the VFT extrapolation from above T_g measurements. The relaxation times calculated assuming that the relaxation time at 142 °C is 100 s to give sense to the range of relaxation times accessed using time–aging time and time–temperature superposition methods of data treatment. VFT curve for segmental relaxation modified from Mercier and Groeninckx.¹⁴¹ (a) Exponential temperature dependence. (b) Arrhenius temperature dependence. Data come from O’Connell and McKenna.¹²⁵

Logarithm of the scaled relaxation time vs 1000/(T−T_∞) in “VFT” plot for polystyrene. Linear behavior is consistent with VFT description, whereas the deviation toward shorter relaxation times shows weaker than the VFT extrapolation below the T_g. Data come from Simon et al.¹⁵²

Logarithm of viscosity (P) vs (a) temperature and (b) reciprocal temperature for α-phenyl-o-cresol showing how this material seems to follow the VFT equation very well at high temperatures, but deviates strongly at low temperatures, well below the nominal glass temperature of 10¹³ P. Data reported by Laughlin and Uhlmann,¹⁵⁵ and VFT curves calculated from their given VFT equation parameters.

Comparison of down-jump and up-jump structural recovery of a poly(vinyl acetate) glass showing the asymmetry of approach that leads to the conclusion that when δ < 0 or T>T_f the relaxation or retardation times will be longer than those in equilibrium where δ=0 or T=T_f. Data provided to me by A. J. Kovacs and is the same as he used in ref 133.

Example of logarithm of retardation time vs δ for two different conditions of down-jump and up-jumps through the glass transition regime. Both final conditions are for T=60 °C and relative humidity [RH]=50%. The temperature-jumps were from either 55 °C or 65 °C to 60 °C  and the RH Jumps were from 35 and 65% RH to 50% RH. Data show that when δ < 0 (or T>T_f) the retardation time is longer than that at equilibrium where δ=0. Data are from Zheng and McKenna.¹⁸⁵

DSC heating traces for a pristine amber (first heating) and subsequent heating scans for the sample after cooling at 10 °C/min from above the T_g All heating rates were at 10 °C/min. Data reported by Zhao et al.¹²

Master curve of the relaxation modulus vs reduced time for a 20-million-yr-old Dominican amber. T_ref is 135 °C (note that the DSC-determined T_g=136.2 °C). The fictive temperature for the amber was 92.6 °C; thus, the data encompass the fictive temperature. Data reported by Zhao et al.¹²

Relaxation time vs temperature for 20-million-yr-old Dominican amber. The lowest temperature for the upper bound data is 368.2 K (95 °C), which is 2.6 K above the fictive temperature of the glass. Data reported by Zhao et al.¹²

Relaxation time vs reciprocal temperature for 20-million-yr-old Dominican amber. The lowest temperature for the upper bound data is 368.2 K (95 °C), which is 2.6 K above the fictive temperature of the glass. Data reported by Zhao et al.¹²

Flash DSC scans for a 300-nm-thick perfluoropolymer film formed by vapor deposition showing the very large enthalpy “overshoot” typical of the stable glass. The rejuvenated scans (second and third heatings) show the normal glass after cooling from above the T_g. Data reported by Yoon et al.¹⁷¹

Logarithm of the relaxation time vs temperature for a vapor-deposited amorphous perfluoropolymer in the upper bound regime where T_meas>T_f=348.2 K compared with equilibrium data and the VFT extrapolation from the equilibrium data fitted to Eq. 2. The exponential fit to the upper bound data shows that depth below the T_g is insufficient to distinguish between exponential dependence of times on temperature vs Arrhenius-type dependence on reciprocal temperature. Data reported by Yoon and McKenna.¹⁷²

Logarithm of the relaxation time vs reciprocal temperature for a vapor-deposited amorphous perfluoropolymer in the upper bound regime where T_meas>T_f=348.2 K compared with equilibrium data and the VFT extrapolation from the equilibrium data fitted to Eq. 2. The Arrhenius fit to the upper bound data shows that depth below the T_g is insufficient to distinguish between exponential dependence of times on temperature vs Arrhenius-type dependence on reciprocal temperature. Data reported by Yoon and McKenna.¹⁷²

Thickness measurements for a set of vapor-deposited films of the amorphous perfluoropolymer CYTOP® showing the fictive temperature of 269.5 K, which is some 27.7 K below the T_∞ of the material. The ultrastable glass data shown are during a stepwise heating and illustrate the devitrification of the glass. The equilibrium data (liquid line) were obtained from the last two points of the devitrification, the points on stepwise cooling of the rejuvenated glass and data obtained upon structural recovery of the films by aging into equilibrium (hexagonal data points). The data are from El Banna and McKenna.¹⁷⁴

Molar volume of vapor-deposited ethylbenzene from Ishii et al.²⁰⁶ showing the fictive temperature for the system deposited at 104.7 K and the calorimetric T_K=100.3 K as proposed by Beasley et al.²⁰⁷ as a correction to the value of 101 K given by Tatsumi et al.²⁰⁹ The interpretation of the data was originally given by McKenna.¹⁰¹

Photograph of a “stone” of amber from Fushun, China. Scale is in centimeters. Adapted with permission from ref 170. Copyright 2023 Elsevier.

Normalized thickness of the Fushun amber vs temperature during a heating (rejuvenation or devitrification) scan and for the subsequent cooling to create a normal glass. Converting thickness change to volume follows $v(T)=v(T_0){\left[\frac{h(T)}{h_0}\right]}^3$. At 20 °C this gives a densification of the pristine amber of 13.7% relative to the rejuvenated glass. Adapted with permission from ref 170. Copyright 2023 Elsevier.

Logarithm of the relaxation time vs temperature for the Fushun amber comparing two conditions. First, the rejuvenated or “normal” glass and second, a stable glass with fictive temperature of 278.2 K (5.0 °C). Data reported in Kong et al.¹⁶⁹ The blue dashed line shows the VFT curve for a Dominican amber,¹² with T_∞=338.35 K shifted to reflect the differences in glass transitions between the two different types of amber. Note that the lowest temperature data point at T=268.2 K is below the fictive temperature and would nominally be a lower bound regime, which does not test the VFT extrapolation.

Arrhenius representation of the data presented in Figure 31.

Figure showing de-aging (orange arrows) followed by cooling along the glass line. The starred points represent the case for the line along which the data points for a given fictive temperature are found. The blue star-like symbols are the points for which $T=T_{\text{meas}}=T_{\text{f}}$. Thus, both upper bound and nominally equilibrium relaxation times can be obtained at the relevant temperatures and fictive temperatures. Adapted with permission from ref 170. Copyright 2023 Elsevier.

Logarithm of the relaxation time for the 50-million-yr-old Fushun amber in a condition of T=T_f, which is nominally the condition of the equilibrium relaxation times. The straight line suggests an exponential dependence of the relaxation times on temperature. Data from Kong et al.¹⁷⁰

Logarithm of the relaxation time vs 1/T for the 50-million-yr-old Fushun amber in a condition of T=T_f, which is nominally the condition of the equilibrium relaxation times. The line comes from the fit of the response as exponential in temperature as determined from the data in Figure 34. Data from Kong et al.¹⁷⁰

Logarithm of the relaxation time (referenced to that in equilibrium at 478.2 K) for the 50-million-yr-old Fushun amber in a condition of T=T_f, which is nominally the condition of the equilibrium relaxation times. The straight lines suggest that the relaxation times at constant fictive temperature are exponential in the specific volume. Unfilled data points are from $T_{\text{meas}}=T<T_{\text{f}}$ condition. Adapted with permission from ref 170. Copyright 2023 Elsevier.

Logarithm of the scaled relaxation time for a bitumen sample aged into equilibrium showing strong deviation from the VFT or WLF types of extrapolation below the T_g. The $T_{\text{ref}}=T_{\text{g}}=253.2\text{ K}$ in this case. Data from Laukkanen et al.²¹⁶

Logarithm of the relaxation time (nominally at equilibrium) vs T for ultrastable indomethacin glass formed by vapor deposition. Results show strong deviation from the VFT extrapolation as well as an exponential dependence on temperature (straight line through data with $\text{log}(\text{τ}/\text{s})=72.247-0.22271T$. Data reported by Pogna et al.²¹⁷

Logarithm of the relaxation time (nominally at equilibrium) vs 1/T for ultrastable indomethacin glass formed by vapor deposition. Results show strong deviation from the VFT extrapolation as well as a slight curvature to the data that is well represented by the expression $\text{log}(\text{τ}/\text{s})=72.247-0.22271T$ determined from the regression to the data in Figure 38. Data reported by Pogna et al.²¹⁷

Heat capacity of mixtures of poly(α-methyl styrene) and its pentamer showing that the equilibrium liquid line and the glass line are both independent of concentration. Reprinted with permission from ref 220. Copyright 2009 Elsevier.

Heat capacity of mixtures of poly(α-methyl styrene) and its pentamer as a function of mass fraction of the polymer at representative temperatures from above T_g=444 K to below the T_∞=394.2 K. Data from Huang et al.²¹⁹

Liquid entropy [as the difference between $S\left(T\right)\text{and }S(T_{\text{g}})$] for poly(α-methyl styrene) polymer showing near-linear dependence on T from above T_g=444 K to over 100 K below the T_∞=394.2 K. Data from Huang et al.²¹⁹

Excess configurational entropy [as the difference between $S_{\text{c},\text{ex}}\left(T\right)\text{and }{S}_{\text{c},\text{ex}}(T_{\text{g}})$ for poly(α-methyl styrene) polymer showing no evidence of a second order transition from above T_g=444 K to over 100 K below the T∞=394.2 K. Data from Huang et al.²¹⁹