Abstract
The glass transition temperature $T_g$ is written as though it were a material constant. This paper asks whether it is one——the answer is that it is not a constant but a function of the measurement’s time scale. No new mathematical theorem and no new law is claimed.
First, what is called $T_g$ is where the relaxation time overtakes the observation time (Section 2).
Second, this is the core of the paper. Six decades of observation time take $T_g$ from $321.4286$ K to $300.0000$ K, a fall of $\mathbf{21.4286}$ K (Section 3).
Third, that is $\mathbf{4.1667}$ K per decade, inside the accepted 3--5 K (Section 3).
Fourth, the melting point does not move under the same treatment (Section 4).
Fifth, so a reported $T_g$ without its measurement conditions is a number without meaning (Section 4).
Sixth, the separator is thermodynamic equality against kinetic overtaking (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the VFT equation, the rate dependence of the glass transition, and the fragility classification are all standard. We do not build a theory of glasses——the origin of the glass transition (free volume, cooperatively rearranging regions, mode-coupling theory) is not entered. It is named and no more. We make no measurement——no $T_g$ of any particular substance is measured. The numbers are on VFT parameters stated explicitly. Fragility is not the subject——Angell’s strong/fragile classification is named and no more. We do not treat the detail of calorimetry——baseline construction in DSC and the midpoint/onset conventions are not the subject. We build no theory of phase transitions——whether the Ehrenfest classification applies is not entered. Relation to earlier papers: Paper 300 showed whether two things share a root is decidable——by that test, the melting point and the glass transition temperature have distinct roots. Paper 313 showed one word used in four senses——“transition temperature” has the same shape. Paper 351 showed one word “pumping” naming two mechanisms——here one phrase “transition temperature” names two. Paper 333 showed a constant ratio following from cancellation and not from measurement——there too the provenance of something that looked constant was the issue. What is added is giving the rate dependence of $T_g$ as a number from the VFT equation, checking the per-decade shift against the accepted range, setting the contrast with the melting point explicitly, and putting the separator on transition against rendezvous.
1. Notation and the Question
Setting: Write the relaxation time $\tau$ by the Vogel--Fulcher--Tammann (VFT) equation.
$$\log_{10}\tau = -14.0 + \frac{B}{T-T_0} ,\qquad B=1000\ \mathrm{K},\quad T_0=250\ \mathrm{K}$$Observation 1.1 $T_g$ is written as “the temperature at which something happens”. One writes that the $T_g$ of polystyrene is 100 degrees Celsius——the same format as a melting point.
Observation 1.2 But the definition of $T_g$ contains a measurement. $T_g$ is where the relaxation time overtakes the observation time——and “observation time” is not a quantity on the substance’s side.
Observation 1.3 The VFT equation diverges as $T\to T_0$. Because the denominator is $T-T_0$——$T_0=250$ K is called the ideal glass temperature and is not reachable experimentally.
Question: Is $T_g$ a property of the substance?
2. The Definition Contains an Observation Time
Observation 2.1 A glass is a liquid that flows too slowly to look like it is moving. The structure stays liquid and only the motion is slow——how slow counts as “stopped” is decided by how long one waits.
Observation 2.2 So the threshold is a time. By convention $\tau=100$ s is used to define $T_g$——the number 100 does not come out of the substance. It comes from how long a laboratory can wait.
Observation 2.3 Cooling rate and observation time are two faces of one thing. Cool faster and less time is available at each temperature——which is why faster cooling raises $T_g$.
3. $21.4286$ K over Six Decades
Calculation 3.1: $T_g$ against observation time (computed in this paper).
| Observation time [s] | $\log_{10}t$ | $T_g$ [K] | Change from above |
|---|---|---|---|
| $10^{0}$ | $0$ | $\mathbf{321.4286}$ | — |
| $10^{1}$ | $1$ | $316.6667$ | $-4.7619$ K |
| $10^{2}$ | $2$ | $312.5000$ | $-4.1667$ K |
| $10^{3}$ | $3$ | $308.8235$ | $-3.6765$ K |
| $10^{4}$ | $4$ | $305.5556$ | $-3.2680$ K |
| $10^{5}$ | $5$ | $302.6316$ | $-2.9240$ K |
| $10^{6}$ | $6$ | $\mathbf{300.0000}$ | $-2.6316$ K |
Observation 3.2 This is the core of the paper. Six decades move it by $\mathbf{21.4286}$ K——an average of $\mathbf{4.1667}$ K per decade, inside the accepted “3 to 5 K per decade”.
Observation 3.3 $21.4286$ K is not small. In Celsius it is the difference between $48.28$ and $26.85$ degrees——a span across which the answer to “is it above room temperature” changes.
Observation 3.4 The steps are not equal; they shrink. From $-4.7619$ to $-2.6316$——because VFT is not an exponential. An Arrhenius law would give equal steps.
Observation 3.5 It can be checked backwards. Putting the resulting $T_g$ back into VFT gives $\log_{10}\tau=2.0000000000$——matching the defined observation time of $10^2$ s. The formula closes.
Observation 3.6 Changing $B$ changes the size of the shift. A smaller $B$ gives a smaller shift per decade——so the strength of the rate dependence is itself material-specific. That is the content of fragility.
4. The Melting Point Does Not Move
Observation 4.1 The same treatment at the melting point gives nothing. Warm ice six decades more slowly and $273.15$ K is still $273.15$ K——the melting point is where two free energies are equal, and it contains no waiting time.
Observation 4.2 Here the criterion of Paper 300 does the work. Both are written as “the temperature at which something changes”——but one is fixed by a thermodynamic equality and the other by a kinetic overtaking. Distinct roots.
Observation 4.3 So reporting $T_g$ requires reporting conditions. Writing “$T_g$ is 320 K” does not say whether it is the $10^0$ s value or the $10^6$ s one——and on the numbers of this paper those two differ by $21.4286$ K.
Observation 4.4 The melting point has no such need. One may report it without a cooling rate because writing one would not change the answer——whether the condition may be omitted is what separates the character of the two temperatures.
Observation 4.5 To be stated honestly——supercooling and superheating do affect melting. But those are departures from equilibrium, and the equilibrium value itself does not move——$T_g$ has no counterpart to an “equilibrium value that does not move”.
5. The Separator: A Transition, or a Rendezvous?
Observation 5.1 The separator——whether the temperature is fixed by a thermodynamic equality or by one time scale overtaking another.
| What fixes it | Moves with conditions | Latent heat | |
|---|---|---|---|
| Melting point | equality of free energies | no | yes |
| Glass transition | relaxation overtaking observation | yes ($4.1667$ K/decade) | no |
Observation 5.2 The rightmost column is an independent check. That the glass transition has no latent heat has long been known——the classification of this paper agrees with that independently known fact.
Observation 5.3 The separator can be moved. Setting $T_0=0$ turns VFT into Arrhenius, and the rate dependence remains——what the move removes is only the shrinking of the steps, not the rate dependence itself.
Observation 5.4 It can be moved so as to abolish $T_g$. Let $B\to0$ and the relaxation time stops depending on temperature, so $T_g$ cannot be defined——a check that the formula is doing work.
Observation 5.5 To be stated honestly——$4.1667$ K and $21.4286$ K are values on a model with $B=1000$ K and $T_0=250$ K stated explicitly. Another substance gives other numbers——what does not move is only the direction: a longer observation time lowers $T_g$.
Observation 5.6 One last thing——some of the numbers written down as “properties of a substance” are outcomes of a rendezvous between substance and measurement. $T_g$ is the clearest example——when a quantity wears the face of a constant, it is worth looking for a time inside its definition.
6. The Census and the Close
- An observation time is in the definition (Observation 1.2): not on the substance’s side.
- The convention is $\tau=100$ s (Observation 2.2): how long a laboratory can wait.
- $321.4286$ K at $10^0$ s (Calculation 3.1): a fast measurement.
- $300.0000$ K at $10^6$ s (Calculation 3.1): a slow one.
- $21.4286$ K over six decades (Observation 3.2): a span across room temperature.
- $4.1667$ K per decade (Observation 3.2): inside the accepted 3--5 K.
- Steps shrink from $-4.7619$ to $-2.6316$ (Observation 3.4): Arrhenius would be equal.
- Back-substitution gives $\log_{10}\tau=2.0000000000$ (Observation 3.5): the formula closes.
- The melting point does not move (Observation 4.1): no waiting time in it.
- The glass transition has no latent heat (Observation 5.2): an independent check.
Closing: The glass transition temperature $T_g$ is reported in the same format as a melting point——write “the $T_g$ of polystyrene is 100 degrees Celsius” and it wears the face of a material constant. But the definition of $T_g$ contains an observation time——a glass is a liquid that flows too slowly to look like it is moving, and how slow counts as “stopped” is decided by how long one waits. The conventional $\tau=100$ s does not come out of the substance; it comes from how long a laboratory can wait. So the number moves with the speed of measurement——on the VFT equation, stretching the observation time six decades from $10^0$ s to $10^6$ s takes $T_g$ from $\mathbf{321.4286}$ K to $\mathbf{300.0000}$ K, a fall of $\mathbf{21.4286}$ K. That is $\mathbf{4.1667}$ K per decade, inside the accepted “3 to 5 K per decade”. In Celsius it is the difference between $48.28$ and $26.85$ degrees — a span across which the answer to “is it above room temperature” changes. The same treatment at the melting point gives nothing——warm ice six decades more slowly and $273.15$ K is still $273.15$ K, because the melting point is where two free energies are equal and contains no waiting time. By the criterion of Paper 300 the two have distinct roots: one is fixed by a thermodynamic equality, the other by a kinetic overtaking. The practical consequence shows in how one reports——writing “$T_g$ is 320 K” does not say whether it is the $10^0$ s value or the $10^6$ s one, and on the numbers of this paper those two differ by $21.4286$ K. One may report a melting point without a cooling rate because writing one would not change the answer; whether the condition may be omitted is what separates the character of the two temperatures. One thing separates them——whether the temperature is fixed by a thermodynamic equality or by one time scale overtaking another. There is an independent check——that the glass transition has no latent heat has long been known, and the classification here agrees with that independently known fact. To be said honestly——$4.1667$ K and $21.4286$ K are values on a model with $B=1000$ K and $T_0=250$ K, and another substance gives other numbers. What does not move is only the direction: a longer observation time lowers $T_g$. One last thing——some of the numbers written down as “properties of a substance” are outcomes of a rendezvous between substance and measurement. $T_g$ is the clearest example, and when a quantity wears the face of a constant it is worth looking for a time inside its definition.
References
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- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). This Corpus Has Used “Synchrony” in Four Senses, and Only the Fifth Has a Threshold [Paper 313]. Zenodo. 10.5281/zenodo.22174953
- Yamagishi, Y. (2026). That the Ratio of Osmotic Pressure to Freezing-Point Depression Is Constant Follows from an Assumption, Not from a Measurement [Paper 333]. Zenodo. 10.5281/zenodo.22681740
- Yamagishi, Y. (2026). Standing and Sitting on a Swing Are Two Mechanisms [Paper 351]. Zenodo. 10.5281/zenodo.22695649
Reference
Yamagishi, Y. (2026). The Glass Transition Temperature Moves with the Speed of Measurement [Paper 352]. YOI: https://www.yamagishiyuuki.com/yoi/352
DOI: 10.5281/zenodo.22695723
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 352. This paper is a preprint and has not been peer reviewed.