In my last post I covered the classic debunking: cathedral windows aren't thicker at the bottom because glass flows, and window glass would need something like 10²³–10³² years to sag visibly at room temperature. Which raises the obvious question:
Where does a number like 10³² years actually come from? Nobody waited 800 years with a ruler, and the room-temperature viscosity of glass is far too high to measure directly.
So this post walks through Edgar Zanotto's 1998 paper (Am. J. Phys. 66, 392) step by step: which physical quantities go in, which equations they feed, what the actual parameter values are, and how the conclusion falls out. It's the most equation-heavy post in this series, but you only need three ingredients and one logarithm.
The whole calculation: ① a glass recipe (chemical composition) ② empirical formulas turn it into three parameters ③ those define a viscosity–temperature curve ④ extrapolate to room temperature and read off a cosmic timescale. Paper and pencil — no experiment.
Ingredient 1: Defining "flow" as a number — the relaxation time
First, "the glass flows" has to become a quantity. Put a material under stress (a windowpane under its own weight) and it gradually yields. The timescale of that yielding is the relaxation time τ — roughly, the time for about half (40–60%) of an imposed disturbance to dissipate. If τ is a second, you have a liquid. If τ is 10³² years… that's glass.
The key is that τ is expressible through two measurable properties — the Maxwell relation:
τ = η / G∞ (relaxation time = viscosity ÷ instantaneous shear modulus)
Intuition: viscosity η measures reluctance to flow; G∞ measures stiffness when pushed. The units make it obvious — Pa·s divided by Pa leaves seconds.
(If "stiffer means faster relaxation" feels backwards — it doesn't mean stiff materials are liquid-like. Under a given stress, a stiffer material stores less elastic strain, so there's simply less to unwind. What decides solid vs liquid in practice is viscosity alone: G∞ is nearly constant for glasses, while η swings over fifty orders of magnitude.)
One parameter is now fixed: for window-glass compositions, G∞ ≈ 30 GPa, roughly constant from absolute zero up to the glass transition. Which leaves a single problem: what is η at room temperature?
Ingredient 2: How viscosity depends on temperature
Viscosity is brutally temperature-sensitive: syrup-like around the 1400–1500 °C melting range, but 10¹² Pa·s — a quadrillion times water — by the glass transition (~550–600 °C). Two laws describe this:
① Arrhenius: η(T) = η₀·exp(E/kT). The standard form for thermally activated processes. Taking the log gives a straight line in 1/T — ideal for extrapolating beyond measured data. The catch: it only holds for materials whose structure doesn't change with temperature. That means a handful of network-forming glasses — pure SiO₂, GeO₂, P₂O₅ — oxides whose atoms form a self-supporting 3D network.
The same data replotted against 1/T reveals the heart of the calculation. GeO₂ (blue) is a straight line — extendable with a ruler all the way to room temperature (red point). Window glass (orange) bends upward, offering no guarantee of where it goes beyond the measured range. This is why Zanotto brought in GeO₂ as the stand-in.
② Vogel–Fulcher–Tammann (VFT): log η = A + B/(T − T₀). Ordinary glasses (window glass included) restructure internally as temperature changes, so an empirical law with three composition-dependent parameters is used instead. Each has a geometric meaning:
- A — the limit of log η as T → ∞: the "floor" of the curve.
- B — how steeply viscosity climbs as temperature falls (the activation-energy role).
- T₀ — the temperature where the denominator hits zero and viscosity diverges to infinity: a vertical wall. Free-volume theory reads T₀ as the point where the empty space molecules need for jumping vanishes entirely.
The three VFT parameters, plotted with the actual values for the Gatien Cathedral glass (A = −4.22, B = 5460.9, T₀ = 196.3 °C). A sets the floor (blue dotted), B the steepness, T₀ the vertical wall (red dashed). Axes: temperature (°C) vs log₁₀ viscosity (Pa·s).
Ingredient 3: From recipe to parameters — no experiment needed
But where do you get A, B, T₀ for an 800-year-old windowpane? You can't melt down a cathedral.
Here glass offers a gift: it has no microstructure — no crystal grains, no grain boundaries, no pores — so bulk properties like viscosity are functions of chemical composition alone. The glass industry has long exploited this with empirical formulas; Zanotto used the Lakatos relations (1972), which map the K₂O, Na₂O, CaO, MgO, Al₂O₃, SiO₂ content onto the three VFT parameters. Medieval compositions are well documented (~350 analyzed samples; potassium-rich, heavy on iron and manganese impurities). For his worked example — the yellow potash glass of Gatien Cathedral in Tours, France — the formulas give:
A = −4.22, B = 5460.9, T₀ = 196.3 °C
Three numbers, and the full viscosity curve of an 800-year-old glass is reconstructed on paper.
The wall: VFT cannot reach room temperature
Now just plug in 25 °C? No — and Zanotto is upfront about why. T₀ for this glass is 196.3 °C, and room temperature is below it. The VFT expression diverges at T₀; below that, it returns mathematical nonsense. (Physically, some other flow mechanism — individual atomic diffusion, perhaps — would operate down there, but no validated equation describes it.)
Measurement impossible, formula divergent. Zanotto's response is the best part of the paper: two detours.
Both curves computed from the paper's actual parameters. Orange (medieval glass, VFT) slams into the vertical wall at T₀ = 196.3 °C and can't reach room temperature. Blue (GeO₂, Arrhenius) extrapolates cleanly to 25 °C, where log η ≈ 50 (red point) gives τ ≈ 10³² years. The purple point is detour 1's answer: 414 °C.
Detour 1 — invert the question: "How hot for 800 years to suffice?"
If you can't compute the time at room temperature, compute the temperature at which flow would show up within the cathedral's 800-year lifetime — that answer lies inside VFT's valid range. Work it through: τ = 800 yr ≈ 2.5×10¹⁰ s requires η = τ·G∞ ≈ 7.6×10²⁰ Pa·s; solving the VFT equation for that viscosity gives about 414 °C. A medieval window would need to sit at 414 °C — permanently — to sag noticeably in 800 years.
Detour 2 — send in a stand-in: GeO₂ glass
To get an actual room-temperature number, you need a glass whose law doesn't diverge. GeO₂ is Arrhenius (structure independent of temperature), so extrapolating its straight line to room temperature is legitimate — and its glass transition temperature is similar to window glass. Literature values: A = −9.94, B = 17962 (η in Pa·s, T in kelvin). At T = 298 K:
log η = −9.94 + 17962/298 ≈ 50 → η ≈ 10⁵⁰ Pa·s
τ = η/G∞ = 10⁵⁰ ÷ 3×10¹⁰ ≈ 3×10³⁹ s ≈ 10³² years
One logarithm, one division. And crucially, this is a lower bound: on cooling, window glass's viscosity rises far faster than GeO₂'s (compare the curves above). The stand-in flows more readily than the real thing — so if the stand-in doesn't flow, the cathedral certainly doesn't.
Attacking his own result
Zanotto then stress-tests the conclusion himself. Medieval impurities (iron, manganese) that the Lakatos formulas ignore? Even a generous 1–2 order-of-magnitude viscosity reduction turns 10³² into 10³⁰ — the conclusion doesn't twitch. That is the power of order-of-magnitude reasoning. Centuries of weathering? Surface chemistry only; it dulls the shine but leaves bulk viscosity untouched. And the standing empirical check: millennia-old glass vases sit undeformed in museums worldwide.
The later refinements
The estimate was sharpened twice. In 1999, Zanotto & Gupta noted that below Tg the structure is frozen, so the extrapolation should use the isostructural (structure-held-fixed) viscosity — revising window glass to ~10²³ years. In 2018, Gulbiten et al. measured a synthetic glass reproducing a medieval Westminster Abbey composition: relaxation far faster than Zanotto's bound, but the flow still amounts to ~1 nm per billion years. The number moved by nine orders of magnitude between methods; the conclusion never moved at all, because every method lands dozens of orders of magnitude beyond human time.
| Step | Input | Tool | Output |
|---|---|---|---|
| ① | Medieval glass composition (literature, ~350 samples) | Lakatos empirical formulas | VFT parameters A, B, T₀ |
| ② | A=−4.22, B=5460.9, T₀=196.3 °C | VFT, inverted | "800-year flow needs 414 °C" |
| ③ | GeO₂: A=−9.94, B=17962 | Arrhenius extrapolation (298 K) | η ≈ 10⁵⁰ Pa·s |
| ④ | η and G∞ = 30 GPa | Maxwell τ = η/G∞ | τ ≈ 10³² yr (lower bound) |
What stayed with me
The most instructive thing in this paper isn't the conclusion — it's the conduct when every direct route is blocked. Measurement: impossible (too slow). The governing equation: divergent below T₀. Zanotto didn't stop; he inverted the question (414 °C), then brought in an extrapolatable stand-in to secure a bound, then attacked his own assumptions to check the conclusion survives any plausible error.
Trading a precise answer for a bulletproof bound is exactly how real engineering problems get closed, too. When you can't pin down the exact number behind an equipment fault, showing that even the worst case stays inside spec is what lets the line keep running. A calculation that no error can overturn beats a precise one every time.
References
- E. D. Zanotto, "Do cathedral glasses flow?," Am. J. Phys. 66, 392–395 (1998). DOI: 10.1119/1.19026 — all equations, parameters, and figures in this post follow this paper; the logarithms were re-derived by the author
- E. D. Zanotto & P. K. Gupta, "Do cathedral glasses flow? — Additional remarks," Am. J. Phys. 67, 260 (1999). DOI: 10.1119/1.19236
- O. Gulbiten, J. C. Mauro, X. Guo, O. N. Boratav, "Viscous flow of medieval cathedral glass," J. Am. Ceram. Soc. 101, 5 (2018). DOI: 10.1111/jace.15092
- T. Lakatos, L.-G. Johansson, B. Simmingsköld, Glass Technology 13, 88 (1972)
Part of an ongoing series on solid-state physics in everyday life. The Korean version of this post is on my Naver blog.
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