When a gas-loading curve refuses to stay linear
A practical route from sorption isotherms to concentration-dependent transport, with one complete numerical model and explicit limits.
Choose the question you need to answer
Name the quantities
Calculate the curve
Connect sorption to transport
Change assumptions and test evidence
1. One permeability value can hide a changing material response
A barrier-film datasheet may present one permeability coefficient as if the polymer were a fixed resistor. That is a useful first model, but it can fail when the amount of gas sorbed by the polymer changes nonlinearly with pressure or concentration.
Permeation joins at least two questions. How much penetrant enters the polymer at a given driving condition? How rapidly can the penetrant move through the polymer once it is inside? Sorption belongs to the first question; diffusion belongs to the second. Their combined effect is often summarised as permeability.
This article extends the constant-permeability multilayer model. The earlier series-resistance calculation remains valuable, but its coefficient must not be carried into a new pressure range without evidence.
2. Separate concentration, solubility, diffusivity and permeability
Let C be penetrant concentration in the polymer and p the gas pressure or, more generally, f the fugacity. A linear Henry model writes C=kD p. The ratio C/p is then constant and can be called a solubility coefficient under that model.
For a simple solution–diffusion picture with constant diffusivity D and linear sorption, permeability P is D times the solubility coefficient. Permeance is P divided by membrane thickness. Flux is permeance multiplied by the pressure difference when the assumptions are satisfied.
When C is not proportional to p, C/p depends on pressure and dC/dp need not equal C/p. A single P can then be an apparent value tied to the test range, direction, conditioning history and calculation convention.
3. Use a dual-mode sorption equation as a teaching model
A common model for glassy polymers combines a dissolved mode with a saturable hole-filling mode: C(p)=kD p + C′H bp/(1+bp). The first term is linear. The second approaches the capacity C′H as pressure grows.
Here kD has concentration-per-pressure units, C′H is a capacity and b has inverse-pressure units. The equation describes a curve, not a universal law for every polymer–penetrant pair. It is useful because each parameter changes a visible part of the curve.
At low p, the initial slope is kD+C′H b. At high p, the saturable contribution approaches C′H while its incremental slope falls. That is why the secant quantity C/p can decline even though total concentration continues to rise.
4. Work the sorption curve at three pressures
Use an original teaching set: kD=0.04 concentration units per bar, C′H=1.20 concentration units and b=0.50 per bar. Calculate both terms before adding them.
| Pressure p (bar) | Henry term | Saturable term | Total C | Secant C/p |
|---|---|---|---|---|
| 1 | 0.040 | 0.400 | 0.440 | 0.440 |
| 5 | 0.200 | 0.857 | 1.057 | 0.211 |
| 10 | 0.400 | 1.000 | 1.400 | 0.140 |
| 20 | 0.800 | 1.091 | 1.891 | 0.095 |
From 1 to 5 bar, concentration rises from 0.440 to about 1.057, yet C/p falls by more than half. Calling the first ratio a permanent solubility constant would misread the curve.
5. Compare the secant with the local slope
The local slope is dC/dp=kD+C′H b/(1+bp)². At 1 bar it is 0.04+0.60/2.25≈0.307 concentration units per bar. At 5 bar it is 0.04+0.60/12.25≈0.089.
Neither local slope equals the secant C/p except in the linear case. The secant connects the origin to a point on the curve; the derivative describes a small change around that point. A report must state which one enters its transport approximation.
This is a first-principles check: if a fitted property changes when the operating point changes, the model cannot be represented faithfully by one number without naming the range.
6. Translate the sorption curve into an ideal transport calculation
Suppose, only for this educational model, that diffusivity is constant at 2×10⁻⁸ cm²/s and that pressure is a suitable driving coordinate. Across a 0.01 cm film from 5 bar to 1 bar, the concentration difference is 1.057−0.440=0.617 concentration units.
Fick’s one-dimensional steady expression gives a flux magnitude D(C5−C1)/l. That is 2×10⁻⁸×0.617/0.01≈1.23×10⁻⁶ concentration-length units per second.
The calculation integrates the nonlinear sorption response through its endpoint concentrations. It is not a claim that real gas transport always has constant D, ideal fugacity, no swelling and no interfacial resistance. Those assumptions are deliberately exposed.
7. Show why a single apparent P depends on the test window
If one forms D(C/p), the apparent product is 8.8×10⁻⁹ at 1 bar, 4.23×10⁻⁹ at 5 bar and 2.8×10⁻⁹ at 10 bar in the model’s corresponding units. The decline arises here from saturation of the second sorption mode.
Using the 1-bar apparent value across the 5-to-1 bar interval would overstate the idealised flux; using only the 5-bar secant would also fail to reproduce the integrated curve exactly. The error is not repaired by quoting more decimal places.
Fit and report the full relation, or restrict a local coefficient to a narrow, stated range. A number without its operating window is incomplete evidence.
8. Change one assumption and the trend can reverse
| Change | Likely model consequence | Evidence needed |
|---|---|---|
| Remove saturable capacity C′H | C becomes linear in p | Sorption isotherm across range |
| Let D fall with concentration | Permeability may fall faster | Time-lag or transient transport tests |
| Let D rise through plasticisation | Permeability can turn upward | Repeated pressure steps and hysteresis |
| Add a second gas | Competitive sorption may change both species | Mixed-gas measurements |
| Add defects or pinholes | Bypass flux can dominate the polymer response | Integrity and thickness mapping |
Real glassy polymers can show concentration-dependent diffusivity, plasticisation, physical ageing and hysteresis. Therefore a falling C/p curve does not guarantee that measured permeability falls monotonically.
The changed-context transfer check is simple: keep the sorption parameters but let D double above 5 bar. The product D(C/p) at 10 bar becomes 5.6×10⁻⁹ rather than 2.8×10⁻⁹. The trend is less steep because a second mechanism changed.
9. Design measurements that can distinguish the mechanisms
Measure a sorption isotherm rather than inferring sorption solely from steady flux. Measure permeation at several upstream pressures, with downstream conditions and temperature controlled. Record film thickness and whether values are normalised by pressure, fugacity or concentration.
Use pressure steps in both directions to look for hysteresis and plasticisation. Repeat after conditioning and ageing. A time-lag analysis can offer diffusion information under its assumptions, while independent thickness and integrity checks guard against a pinhole being mistaken for bulk transport.
For mixtures, single-gas values are not enough. Competitive sorption and composition-dependent transport can change selectivity. Report feed and permeate compositions, not just the nominal upstream pressure.
10. Limits and misconceptions
Nonlinear sorption is not automatically dual-mode sorption. Other models may fit a system, and fitted parameters are not unique evidence of a microscopic mechanism.
Permeability, permeance and flux are not interchangeable. Thickness belongs explicitly in permeance. A pressure-normalised coefficient does not prove that pressure is the correct thermodynamic driving force.
A strong fit inside the calibration range does not license extrapolation. At high loading the polymer can swell or plasticise; at low loading uncertainty and leak corrections can dominate. Temperature changes both sorption and mobility.
11. Questions readers should ask before trusting a number
Was the coefficient measured or calculated from separate D and S values? At what temperature, pressure or fugacity range? Was the polymer glassy, rubbery, semicrystalline, blended or filled? Was the specimen dry, conditioned or aged?
Does the reported value use a secant, local slope or interval average? Were both pressure directions tested? Was thickness measured on the same specimen? Were mixed gases involved?
A useful answer keeps the specimen, penetrant, state history and range attached to the number. Removing those details turns a material property into a floating label.
12. Connect the model to the wider polymer map
Begin with the Polymer Science root. Compare the multilayer barrier calculation, composition-basis lesson and blend and miscibility limits. The polymer chemistry subject guide supplies the wider molecular map.
Authoritative reading includes the research paper Limit-Behavior Investigations of Dual-Mode Sorption Theory and the scholarly overview Gas Transport in Glassy Polymers. Use them for model scope, not as a substitute for specimen-specific data.
Return to the Gold Standard of Education and the How X Works hub. The transferable habit is to replace one convenient coefficient with the full relation whenever the operating condition changes it.