A polymer can show a different reported glass-transition temperature when the heating programme changes, even when its chemical composition stays the same. The reading reflects molecular response, sample history, temperature control and the rule used to mark a transition on the curve. A temperature without its method is an incomplete comparison.
The calculations below use an explicitly invented relaxation-timescale model and a separate thermal-lag model. Their numbers are teaching examples, not fitted polymer parameters or predictions of a real DSC midpoint. The models help separate explanations that one shifted thermogram cannot identify by itself.
Find the question you need
Choose a route below or read in order. All teaching chapters remain open; the index groups let you jump directly to a worked example or check.
Route 1 · Name the measured transition
Chapters 1–2: Name the measured transition.
Route 2 · Connect the scan with a response time
Chapters 3–4: Connect the scan with a response time.
Route 3 · Separate kinetics from thermal lag
Chapters 5–6: Separate kinetics from thermal lag.
Route 4 · Compare curves and sample histories
Chapters 7–8: Compare curves and sample histories.
Route 5 · Practise and follow the evidence
Chapters 9–12: Practise and follow the evidence.
Full chapter index · Try the transfer checks · Polymer Science subject root
Chapter contents
Name the measured transition · Chapters 1–2
Connect the scan with a response time · Chapters 3–4
Separate kinetics from thermal lag · Chapters 5–6
Compare curves and sample histories · Chapters 7–8
Practise and follow the evidence · Chapters 9–12
Differential scanning calorimetry, DSC, records a differential heat-flow response during a temperature programme. A glass transition can appear as a heat-capacity step associated with amorphous material. It is distinct from the melting of crystalline regions.
A reported onset, half-height midpoint, inflection temperature and enthalpy-based fictive temperature are not interchangeable labels. Each is obtained by a specified analysis. Two laboratories can report different values from a broadly similar curve if their definitions or baselines differ.
The existing Fox-equation and blend-miscibility guide asks how composition enters an approximate blend prediction. This branch holds chemistry fixed and asks what changes when the measurement clock, thermal history or thermal response changes.
Begin a comparison by naming the measurand and the protocol. “The half-height midpoint from a second heating at ten kelvin per minute after a stated cooling programme” is more inspectable than “the polymer’s Tg is this number”.
CHAPTER 2 OF 12 · Name the measured transition
2. Translate scan rate into an available time
Let the positive heating rate q be measured in kelvin per minute. During a chosen temperature interval ΔT, the programme spends Δt = ΔT/q minutes. Increments of one kelvin and one degree Celsius have the same size, although absolute temperature in an exponential or reciprocal relation must use kelvin.
For an illustrative five-kelvin interval, q = 5 K/min provides one minute. At 10 K/min the same interval lasts half a minute; at 20 K/min it lasts a quarter of a minute. The temperature range is unchanged, but the time available for a response is different.
A molecular relaxation time τ describes a characteristic response timescale in a chosen model. If it is long compared with the observation time, the structure cannot keep up in the same way as when it is short. Real polymer relaxations commonly need more than one timescale and can depend on history.
The operational crossover used here is qτ(T*) = ΔT. This is our transparent teaching criterion, not a definition of every DSC glass-transition temperature. It lets us examine the direction and size of a clock effect before discussing what a real instrument establishes.
CHAPTER 3 OF 12 · Connect the scan with a response time
3. Specify an invented relaxation model completely
Take τ(T) = τ₀ exp[B(1/T − 1/T₀)], with τ₀ = 1 minute, T₀ = 350 K and B = 12,000 K. The exponent is dimensionless. At 350 K the relaxation time is one minute; at higher T it is shorter in this model.
These parameters belong to the example. They are not universal constants, not a fit to a named polymer and not a substitute for a full glass-relaxation model. A local Arrhenius-shaped relation is being used to make the arithmetic and its limitations visible.
Insert the five-kelvin observation window into the criterion. Then τ(T*) must equal 5/q minutes. For q = 5, τ must be one minute, immediately giving T* = 350 K. Faster scans require a shorter τ under the same criterion.
Keep the constructed crossover T* separate from an experimentally selected DSC midpoint. Establishing a relation between them would require a model for heat capacity and structural recovery, a temperature account and actual data. A convenient timescale equation alone does not supply those missing steps.
CHAPTER 4 OF 12 · Connect the scan with a response time
4. Solve the crossover and verify its direction
Taking natural logarithms gives B(1/T* − 1/T₀) = ln[ΔT/(qτ₀)]. Therefore 1/T* = 1/T₀ + ln[ΔT/(qτ₀)]/B. The logarithm takes a dimensionless ratio because qτ₀ and ΔT are both temperature intervals.
At q = 10 K/min, the ratio is 0.5. Thus 1/T* = 1/350 + ln(0.5)/12,000, giving T* ≈ 357.22 K, or 84.07°C. At q = 1, the ratio is five and T* ≈ 334.31 K.
The direction is consistent: increasing q reduces the available time, so the crossover requires the shorter relaxation time found at a higher temperature. The calculation changes no molecular composition or individual-chain mass.
This is an explanatory result under a chosen operational criterion. It does not establish that a real polymer’s reported midpoint will shift by 7.22 K when a laboratory doubles its rate. A real rate dependence has to be measured with a consistent analysis and thermal history.
CHAPTER 5 OF 12 · Separate kinetics from thermal lag
5. Model temperature lag as a separate mechanism
Now imagine an indicated programme temperature Tp rising at rate q, while a representative sample temperature Ts responds through dTs/dt = (Tp − Ts)/τinst. Here τinst is an invented thermal response time, separate from the molecular τ(T).
For a long enough constant ramp in this simple model, the lag Tp − Ts approaches qτinst. This follows by setting the time derivative of the lag to zero: d(Tp − Ts)/dt = q − (Tp − Ts)/τinst. The initial transient is excluded from the steady-ramp estimate.
Take τinst = 0.10 minute. At 5 K/min the indicated temperature leads the representative sample by 0.5 K; at 10 K/min it leads by 1 K; at 20 K/min it leads by 2 K. The sign matters during heating: the lagging sample is cooler.
Actual DSC temperature sensing and correction are instrument-specific. Do not simply subtract qτinst from a real instrument reading without a validated response model. This calculation isolates a physical possibility, not an operating calibration procedure.
CHAPTER 6 OF 12 · Separate kinetics from thermal lag
6. Combine the two toy effects without confusing them
If the model sample crosses T* while the programme is ahead by qτinst, the corresponding indicated temperature is Tind = T* + qτinst. At 5 K/min it is 350.50 K; at 10 K/min it is 358.22 K.
The 7.72 K difference separates into about 7.22 K from our timescale criterion and 0.50 K from the additional instrument lag. Looking only at the combined shift cannot determine those two contributions without further information.
| q, K/min | Required τ, min | T*, K | Thermal lag, K | Indicated model value, K |
|---|---|---|---|---|
| 1 | 5 | 334.31 | 0.10 | 334.41 |
| 5 | 1 | 350.00 | 0.50 | 350.50 |
| 10 | 0.5 | 357.22 | 1.00 | 358.22 |
| 20 | 0.25 | 364.75 | 2.00 | 366.75 |
A shift can also contain changes in history, moisture, plasticiser content, chemistry or the selected analysis region. The two-effect calculation is not an exhaustive decomposition of every thermogram. It demonstrates why attributing the whole change to composition is an inverse problem with missing evidence.
Nor does our heating calculation identify a unique thermodynamic transition or fictive temperature. Those terms describe different constructions and state information. Keep the operational reading and the underlying explanatory model explicitly attached to one another.
CHAPTER 7 OF 12 · Compare curves and sample histories
7. Compare heat flow on a consistent rate and mass basis
Away from reactions and recovery features, a simple baseline relation is heat flow ≈ mCpq. Here m is sample mass and Cp is specific heat capacity. Doubling q doubles this baseline heat flow even if Cp is unchanged.
For m = 0.010 g and Cp = 1.2 J/(g K), heating at 5 K/min requires 0.060 J/min, equal to 1 mW. At 10 K/min the value is 0.120 J/min, or 2 mW. Dividing by the appropriate mass and rate recovers the same Cp.
A taller raw heat-flow baseline after a faster scan is therefore not by itself evidence of a larger heat capacity or more amorphous material. The instrument’s calibration, baseline subtraction and reporting convention determine the appropriate conversion.
Within a transition, structural recovery or other overlapping processes can add to the response, so this baseline arithmetic is not a complete transition analysis. The general DSC mechanism guide explains the wider heat-flow and integration context.
CHAPTER 8 OF 12 · Compare curves and sample histories
8. Keep thermal history attached to the comparison
A first heating can retain evidence of manufacture, cooling, storage and physical ageing. A second heating follows an imposed thermal programme that may have changed that history. It is a different sample state, even if it comes from the same piece of material.
An enthalpic recovery feature near a glass transition can complicate a midpoint or inflection assignment. Fictive-temperature and equal-area analyses use enthalpy-related constructions; they should not be relabelled as an arbitrary graphical midpoint.
Follow first heating, second heating and crystallinity for the separate crystals-and-enthalpy calculation. Its melting and cold-crystallisation corrections do not become a glass-transition-temperature correction merely because both use DSC.
For an investigation of rate, hold the comparison protocol as consistent as the scientific question requires. For an investigation of original processing history, erasing that history before every comparison could remove the effect being studied. Experimental preparation follows the measurand.
CHAPTER 9 OF 12 · Practise and follow the evidence
9. Identify which extra comparison tests which explanation
A rate series with controlled history examines whether the chosen transition reading changes with the programme. A sample-mass or geometry comparison can probe whether thermal response contributes. Neither automatically isolates one cause, because more than one factor can change together.
Repeat preparations help distinguish specimen variability from repeated instrument analysis of one preparation. Reference materials and appropriate calibration address the measurement system. Consistent tangent and midpoint constructions address a separate analysis-choice contribution.
| Evidence check | What it examines | What it does not establish alone |
|---|---|---|
| Rate series with controlled history | Programme dependence | Unique molecular mechanism |
| Mass/geometry comparison | Possible temperature-response contribution | Complete instrument correction |
| Matched midpoint analysis | Analysis-definition consistency | Identical sample history |
| Independent preparation | Specimen and preparation reproducibility | Correctness of every model assumption |
If a composition comparison is the goal, verify actual composition and additive or moisture state through suitable evidence. A rate-sensitive mismatch with a Fox prediction is not proof of phase separation, degradation or incorrect mixing. Those hypotheses need observations capable of distinguishing them.
Present uncertainty and the raw curves needed to inspect the conclusion. A reported difference smaller than the relevant reproducibility or method uncertainty should not be given a detailed molecular explanation simply because the software prints extra decimal places.
CHAPTER 10 OF 12 · Practise and follow the evidence
10. Know where these simple equations stop
Our τ(T) contains a single timescale and no explicit structural-history variable. A real glass can need a distribution of relaxation times and a description of how its structure evolves. An Arrhenius-shaped local model should not be extrapolated across an arbitrary temperature range.
Our temperature-lag equation treats the sample as one representative temperature with a constant response time. Real specimens can have gradients and the instrument can have more than one thermal pathway. Sample geometry, contact and thermal properties affect how useful that approximation is.
Our crossover criterion uses a selected five-kelvin window. Choosing another window changes the calculated T*. That dependence is intentional and visible, illustrating why an operational definition belongs beside a reported result.
The practical value is discrimination: molecular response, thermal lag, thermal history and curve analysis are distinct possible influences. A model earns a more specific conclusion only when independent data identify its parameters and show that its predictions fit the intended conditions.
CHAPTER 11 OF 12 · Practise and follow the evidence
11. Practise changed clocks and test an inference
Check one: at 20 K/min in the stated model, τ(T*) = 0.25 minute, T* ≈ 364.75 K and the indicated value is about 366.75 K after adding the two-kelvin lag. The molecular and instrument contributions must be calculated separately.
Check two: halve τinst from 0.10 to 0.05 minute while keeping q = 10 and the molecular model fixed. T* stays 357.22 K, while the indicated model reading falls from 358.22 to 357.72 K. Better thermal response changes the indication without changing the molecular crossover.
Check three: two samples show the same reported midpoint but were measured at different rates and histories. Does that prove identical amorphous behaviour? No. Several differences may cancel. Match the protocols or explicitly model and test the changed conditions.
For transfer, review an unfamiliar thermal claim in four columns: state of the specimen, imposed programme, observed signal and analysis definition. A useful question follows from the column that lacks evidence, rather than from the desire to attach a polymer name to a temperature.
CHAPTER 12 OF 12 · Practise and follow the evidence
12. Questions, reading routes and sources
Is glass transition the same as melting?
No. This article concerns the glass-transition response of amorphous material. A semicrystalline polymer can show additional melting and crystallisation features requiring separate analysis.
Are the calculated values actual polymer Tg data?
No. T* is defined by our invented timescale criterion, and Tind adds an invented thermal response. Their role is to make competing effects calculable, not to replace experimental measurements.
Should every experiment use the slowest scan?
No universal rate is established here. Sensitivity, equilibration, overlapping processes, sample stability, apparatus and the intended method all matter. Use a justified protocol and report it.
Choose the next question
Connect composition and thermal history
- Differential Scanning Calorimetry, Heat Flow and Phase-Transition Evidence
- DSC First Heating, Second Heating and Crystallinity
- Fox Equation, Glass Transition and Blend Miscibility
- Density-Derived Crystallinity, Voids and Filler Bias
Test material identity and recovered populations
- Extractable Additives and Incomplete Mass Recovery
- Chain Scission, Molecule Counts and the Fall in Number-Average Molar Mass
- Matching Molar-Mass Averages, Different Chain Populations
- Trace Aggregates and the Molar Mass Seen by Light Scattering
- Detector Response Factors and Mass Fractions in Copolymer SEC
- Size-Exclusion Calibration and Distorted Molar-Mass Distributions
- Branching and the Molar-Mass Ambiguity Behind One SEC Peak
Sources and scope
TA Instruments’ glass-transition analysis note describes alternative midpoint and enthalpic constructions. The University of Utah laboratory’s heating-rate discussion examines temperature gradients and rate effects. Chang and Weeks’ NIST polymer calorimetry study supplies original research on thermal history and glass-related response. The parameters, timescale criterion, lag calculations and practice cases here are original teaching models.
Sources checked 22 October 2026. Return through The Gold Standard of Polymer Science, The Gold Standard of Education and How X Works to practise separating a measured signal from the claim it supports.