Find the chapter you need
Choose the mass-fraction arithmetic, the Kelvin calculation or the experimental decision you need. The open chapters and grouped index are meant to make the worked checks easy to revisit without mistaking the formula for an instrument reading.
Understand the first distinction
Chapters 1–2: Distinguish a molecular mobility transition from melting and identify the claim being tested.
Define the evidence and model
Chapters 3–5: List component fractions, temperature units and the model assumptions before calculating.
Calculate and check the result
Chapters 6–7: Calculate a blend prediction, audit its units and compare it with a tempting linear average.
Change the conditions
Chapters 8–10: Change the composition and test whether one observed transition really settles miscibility.
Practise and continue
Chapters 11–12: Repair common errors, plan an evidence check and follow related polymer guides.
Two polymers soften over different temperature ranges. A learner blends them and calculates a temperature of about 29°C. Is that the temperature at which the actual blend will change from glassy to rubbery? The arithmetic can be sound and the conclusion still be premature.
The Fox equation estimates a blend glass-transition temperature from component mass fractions and component transition temperatures in Kelvin. We will calculate it, change the blend ratio and separate a numerical prediction from evidence of miscibility, molecular interaction and a measured thermal transition.
Chapter index
Understand the first distinction · Chapters 1–2
1. What a glass transition does—and does not—measure
2. Define the Fox prediction and its domain
Define the evidence and model · Chapters 3–5
3. Turn a mixing recipe into fractions
4. Write down the input audit
5. Work the reciprocal-temperature estimate
Calculate and check the result · Chapters 6–7
6. Why the simple arithmetic mean gives a different answer
7. Change the recipe and make a fresh prediction
Change the conditions · Chapters 8–10
8. Place all calculations beside their changed conditions
9. What does one DSC glass-transition step actually establish?
10. Check thermal history and interactions before diagnosing the blend
Practise and continue · Chapters 11–12
11. Practice cases and the misconceptions that follow them
12. Design a measurement and build a connected reading route
CHAPTER 1 OF 12 · Understand the first distinction
1. What a glass transition does—and does not—measure
The glass transition Tg marks a temperature region where motions of an amorphous polymer phase change significantly on the timescale of a measurement. It is not the same as crystalline melting, and one polymer may display both a glass transition and a separate melting endotherm.
Because a measured Tg depends on thermal history, scan rate, method and composition, it is better to describe an interval and measurement protocol than to assume every material has one immutable switch temperature.
When two polymers are mixed, investigators often use the number and positions of glass transitions as clues about miscibility. Those clues need supporting evidence: a single apparent transition is not automatic proof that every chain is uniformly mixed at the molecular scale.
CHAPTER 2 OF 12 · Understand the first distinction
2. Define the Fox prediction and its domain
For a proposed miscible binary blend, a common empirical approximation is 1/Tg,mix = wA/Tg,A + wB/Tg,B. The symbols w represent mass fractions, and they must add to one for a two-component blend in this form.
All three temperature quantities in this reciprocal equation must be absolute temperatures in Kelvin. Using negative Celsius values in denominators is not a cosmetic units mistake: it can produce a nonsensical mathematical answer.
The equation is a model for the single-transition behaviour of a sufficiently miscible blend under appropriate conditions. It is not a chemical mixing-energy calculation, a phase diagram, a guarantee of homogeneity or an alternative to DSC and other material-characterisation evidence.
CHAPTER 3 OF 12 · Define the evidence and model
3. Turn a mixing recipe into fractions
Suppose a fictional dry blend contains 60 g of polymer A and 40 g of polymer B, with no other constituents. Then total polymer mass is 100 g and the fractions are wA=0.60 and wB=0.40.
Let the independently measured pure-component glass transitions for the relevant protocols be Tg,A=350 K (76.85°C) and Tg,B=250 K (−23.15°C). These are deliberately illustrative material values, not identifiers of commercial grades.
Because the Fox form is based on weight fractions, substituting 60% and 40% as the numbers 60 and 40 would scale the reciprocal sum by a factor of one hundred. Fractions must be dimensionless decimals adding to one.
The input table labels masses, fractions and absolute temperatures separately. A useful scientific calculation is reproducible: another person can locate each given value, convert it if necessary and obtain the same result.
Glass transitions stated in °C should be converted by adding 273.15 before the reciprocal calculation. Only after solving for Tg,mix in Kelvin do we convert back to °C for an accessible report.
The equation treats composition as an input, not as evidence of mixing quality. It tells us how to calculate a conditional prediction, not whether the mixture is in fact one homogeneous amorphous phase.
| Component | Dry mass | Mass fraction | Pure-component Tg |
|---|---|---|---|
| A | 60 g | 0.60 | 350 K / 76.85°C |
| B | 40 g | 0.40 | 250 K / −23.15°C |
| Total | 100 g | 1.00 | No measured blend Tg |
CHAPTER 5 OF 12 · Define the evidence and model
5. Work the reciprocal-temperature estimate
Substitute the given values: 1/Tg,mix = 0.60/350 + 0.40/250 K−1. The two terms are 0.001714286 K−1 and 0.001600000 K−1. Their sum is 0.003314286 K−1.
Invert the sum: Tg,mix≈301.724 K. Convert to Celsius: 301.724−273.15≈28.57°C. That is the Fox-predicted transition for the stated model and recipe.
The result sits between the two pure-component transition temperatures. This is one basic sense check for a positive-temperature reciprocal weighted average. A prediction outside the endpoints would make us review the given data, fractions, units or chosen model.
CHAPTER 6 OF 12 · Calculate and check the result
6. Why the simple arithmetic mean gives a different answer
An ordinary mass-weighted arithmetic average would be 0.60×350+0.40×250=310 K, equivalent to 36.85°C. It differs from the Fox prediction of approximately 301.72 K by 8.28 K.
The difference comes from averaging reciprocal absolute temperatures rather than temperatures themselves. Replacing the formula because the arithmetic mean feels more intuitive silently changes the model.
This does not mean Fox is always the most accurate physical model. Blend-specific interactions, morphology and thermal history can create systematic deviations. The legitimate comparison is between predictions and measurements under specified conditions, not between a remembered shortcut and a measured value.
CHAPTER 7 OF 12 · Calculate and check the result
7. Change the recipe and make a fresh prediction
Now use 25 g of A and 75 g of B. The new fractions are wA=0.25 and wB=0.75. Nothing about the pure-component inputs has changed, but the blend contains more of the low-Tg component.
The reciprocal prediction becomes 1/Tg,mix=0.25/350+0.75/250≈0.003714286 K−1. Inverting gives 269.231 K, or approximately −3.92°C.
An arithmetic mean would instead give 275 K or 1.85°C. Record which equation produced each number; do not report them as two independent observations of the material. Both are predictions, and only the experimental thermal response could test which is useful.
CHAPTER 8 OF 12 · Change the conditions
8. Place all calculations beside their changed conditions
The table compares two recipes with the same two fictional component transition temperatures. All amounts are dry polymer mass fractions, the Fox calculation uses Kelvin, and no phase state or mechanical performance has been measured.
The smaller Fox prediction for the 25:75 mixture is expected because the blend gives greater weight to component B with Tg=250 K. The formula cannot tell us whether the mixture will retain one smooth DSC transition, separate into phases or form a composition-dependent microstructure.
When presenting these results to a materials team, include a sentence such as “Fox estimates for a hypothetical miscible blend under the stated dry-mass composition.” That prevents a calculation from quietly becoming a product claim.
| A:B mass recipe | Fox reciprocal prediction | In Celsius | Arithmetic mean (for comparison) |
|---|---|---|---|
| 60:40 | 301.724 K | 28.57°C | 310 K / 36.85°C |
| 25:75 | 269.231 K | −3.92°C | 275 K / 1.85°C |
| 50:50 | 291.667 K | 18.52°C | 300 K / 26.85°C |
CHAPTER 9 OF 12 · Change the conditions
9. What does one DSC glass-transition step actually establish?
A differential scanning calorimeter may show a heat-capacity step near a transition. A miscible polymer blend often exhibits one composition-dependent transition, while sufficiently immiscible components can exhibit separate transitions. But detection limits, overlapping values and broadening complicate that shorthand.
If component Tg values are close, two transitions may overlap into one apparent step. A partially miscible specimen can show a broad or composition-dependent transition; incompatible mixtures may also change with annealing or processing.
Therefore one apparent Tg, even if it resembles the Fox prediction, is supportive evidence at most. Add microscopy, scattering, spectroscopy, mechanical response or another appropriate probe if molecular-scale homogeneity matters to the decision.
CHAPTER 10 OF 12 · Change the conditions
10. Check thermal history and interactions before diagnosing the blend
Thermal processing can change crystallinity, solvent content, molecular organisation, degradation and residual stresses. Record first heating, cooling history and second heating separately when appropriate; a change between runs may reflect erased thermal history rather than a new equilibrium composition.
Strong specific interactions, hydrogen bonding, molecular-weight distribution, chain ends and plasticisation can shift measured transition behaviour. A different empirical blend equation, such as Gordon–Taylor, may describe some measured systems better than the basic Fox rule.
Never adjust an experimental temperature by forcing it to the Fox estimate. Identify the claimed composition, actual mass fractions, moisture or plasticizer content, instrument calibration, scan rate, transition definition and uncertainty before interpreting a mismatch.
CHAPTER 11 OF 12 · Practise and continue
11. Practice cases and the misconceptions that follow them
Exercise: using the original 350 K and 250 K components at equal 50:50 mass fractions, find Tg,mix = 1/(0.5/350+0.5/250)≈291.667 K, or 18.52°C. Compare with the arithmetic prediction of 300 K.
FAQ: Is the Fox equation only a Celsius interpolation rule? No; use Kelvin. Does a single DSC transition prove all chains are mixed on every length scale? No. Can an immiscible blend show two Tg values? Yes, when the transitions are detectable and the phases behave sufficiently separately.
Other recurring errors are treating mass and volume fractions as interchangeable, using the model on a semicrystalline melting peak, and claiming that a successful numerical fit proves a particular interpolymer interaction. Each error confuses an input or observation with an inference.
CHAPTER 12 OF 12 · Practise and continue
12. Design a measurement and build a connected reading route
Prepare pure-component controls and multiple blend compositions. Record actual component masses, additives, preparation method, thermal history, calibration, scan rate, repeat measurements and how the Tg midpoint or onset was assigned. Compare the trend, not just one convenient data point.
Begin at The Gold Standard of Polymer Science, then visit first and second DSC heating, why averages can hide populations, and density-derived crystallinity limitations.
Read a peer-reviewed polymer-blend DSC example with the Fox equation, the polymer-blend Tg modelling study, and Steven Abbott’s interactive Fox-equation calculation. The transferable habit is to separate a conditional numerical estimate from evidence for its physical assumptions.
Choose the next question
Thermal history, composition and bulk evidence
- The Gold Standard of Polymer Science | DSC First Heating, Second Heating and Crystallinity
- The Gold Standard of Polymer Science | Density-Derived Crystallinity, Voids and Filler Bias
- The Gold Standard of Polymer Science | Extractable Additives and Incomplete Mass Recovery
- The Gold Standard of Polymer Science | Branching, Gel Points and Network Formation
Molecular populations and measurement
- The Gold Standard of Polymer Science | Matching Molar-Mass Averages, Different Chain Populations
- The Gold Standard of Polymer Science | Branching and the Molar-Mass Ambiguity Behind One SEC Peak
- The Gold Standard of Polymer Science | Detector Response Factors and Mass Fractions in Copolymer SEC
- The Gold Standard of Polymer Science | Size-Exclusion Calibration and Distorted Molar-Mass Distributions
- The Gold Standard of Polymer Science | Chain Scission, Molecule Counts and the Fall in Number-Average Molar Mass
Keep exploring The Gold Standard of Polymer Science, The Gold Standard of Education and How X Works. Original worked teaching examples use only their stated assumptions; primary references and product descriptions checked 22 October 2026.