The Gold Standard of Polymer Science | Multilayer Barrier Films and Permeability in Series

Consider a laminate made of two polymer films bonded together. The first is only 20 micrometres thick and resists a chosen gas strongly; the second is 80 micrometres thick and allows that gas to pass more readily. An observer focused on material quantity may assume the thick layer controls transport. A steady diffusing molecule, however, must cross the thin layer as well.

In a series path, the same molar or volume flux passes through each layer at steady state when there are no sinks, reactions or leaks along the route. The partial-pressure drop across the laminate is distributed among its constituent resistances. A high-resistance thin layer can take most of that drop, just as a small but restrictive pipe section can govern a whole connected path.

Packaging claims require more than a permeability coefficient in isolation. Relevant variables include film thickness, gas identity, temperature, humidity, layer bonding, pressure difference, pinholes and test conditions. We begin by distinguishing what belongs to a material from what belongs to a measured multilayer specimen.

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For a dense polymer in an ideal steady solution–diffusion regime, permeability P connects gas flux J and partial-pressure difference Δp across a thickness l by J = (P/l)Δp. Here P represents a material property at the specified gas, temperature, humidity and pressure regime. The quantity P/l is permeance: a transport property of the particular thickness of material.

In laboratory and industry literature, P can be expressed in Barrer, while permeance may be given in GPU or equivalent consistent units. We will not mix Barrer, GPU and micrometres without conversion. Instead we compare ratios using P values in the same permeability unit and every thickness in micrometres; the resulting expressions preserve the correct dimensions of permeability divided by thickness.

It is a common error to say that a 20-micrometre film has twice the permeability of a 40-micrometre film made from the same uniform polymer. Under the same valid material conditions P is unchanged, while permeance P/l doubles when thickness is halved. That distinction matters whenever different specimens are compared.

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Our fictitious laminate has layer A of thickness l_A=20 μm and permeability P_A=2 Barrer. Layer B has thickness l_B=80 μm and P_B=20 Barrer for the same gas under the same reference conditions. The total thickness is L=100 μm.

Assume the layers are dense, continuous and planar with common area; transport is one-dimensional; their permeability coefficients are constant over the imposed partial-pressure range; gas flux has reached steady state; each interface maintains the appropriate equilibrium for the solution–diffusion model; and there is no additional interfacial resistance or gas-consuming reaction.

These assumptions are a teaching simplification. Many practical membranes contain porous supports, gutter layers, gradients, swelling or pressure-dependent sorption. A research model with explicit additional resistances is needed for those cases. Our calculation isolates the mathematical logic of two simple dense layers in series before considering the limitations.

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Under the assumptions, a layer’s partial-pressure drop is Δp_i = J(l_i/P_i). The factor l_i/P_i is its resistance per unit area in consistent units. The flux J is common to both layers because material is not accumulating between them in steady state.

For layer A, R_A=20/2=10 μm/Barrer. For layer B, R_B=80/20=4 μm/Barrer. Although B is four times thicker, its resistance is less than half that of A because its permeability is ten times larger. The total resistance is 10+4=14 μm/Barrer.

Both values can be checked by a ratio: R_A/R_B=10/4=2.5. At a given total pressure drop, the smaller material layer therefore carries 2.5 times as much partial-pressure drop as the thicker one. That is why the thin barrier layer cannot be treated as a negligible correction.

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Add the two layer pressure drops: Δp = Δp_A + Δp_B = J(l_A/P_A + l_B/P_B). Rearranging yields the whole-film permeance J/Δp = 1/(l_A/P_A + l_B/P_B). This reciprocal is the appropriate series combination.

If we define an effective permeability P_eff as that of a hypothetical uniform film of the same total thickness L, then J/Δp = P_eff/L. Therefore P_eff=L/(l_A/P_A + l_B/P_B). Insert our figures: P_eff=100/(10+4)=7.142857 Barrer.

Notice where the total thickness appears. The series resistance sum alone gives permeance when inverted; dividing by it and multiplying by L gives a permeability that can be compared with an equal-thickness homogeneous film. Skipping that final L creates a quantity with the wrong units and an erroneous materials claim.

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Since the same steady flux passes through both layers, each layer’s share of the total pressure drop is proportional to its resistance. A takes 10/14, or approximately 71.4 percent. B takes 4/14, or 28.6 percent. A has only 20 percent of the thickness but is responsible for over 70 percent of the transport resistance.

For a concrete normalized check, let the applied partial-pressure difference be one arbitrary consistent pressure unit. The steady flux is 1/14 in the associated Barrer-per-micrometre scale. Layer A consumes 10/14 of that unit; layer B consumes 4/14. The two fractions add to exactly one, so the bookkeeping has not invented or lost driving force.

This pressure split does not imply that the two layers experience an identical concentration drop. A gas’s solubility can differ between polymers, and concentrations can change discontinuously across interfaces at equilibrium even though the proper chemical-potential or partial-pressure accounting remains consistent. That is another reason not to combine concentration gradients without a stated constitutive model.

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Make a comparison film of thickness 100 μm entirely from polymer B, with P=20 Barrer. At the same gas and external partial-pressure difference, its permeance would be 20/100=0.2 in the same scaled units. Our laminate has 1/14≈0.07143. The laminate therefore passes only 0.07143/0.2≈35.7 percent as much gas at steady state under the ideal assumptions.

Expressed another way, adding 20 μm of the low-permeability material while replacing 20 μm of B cuts ideal flux by about 64.3 percent at constant overall thickness. The arithmetic is a relative comparison; a real oxygen transmission rate would still require correctly converted gas units and the actual pressure driving force.

The important mechanism is not that polymer A is intrinsically thick enough to be a free-standing barrier. It is that a continuous, defect-free layer contributes substantial resistance in a series path. Thin coatings must still be demonstrated to be intact after forming, handling, bending and environmental exposure.

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A naive calculation takes the permeability values 2 and 20 Barrer, weights them by their thickness fractions 0.2 and 0.8, and reports 0.2×2 + 0.8×20 = 16.4 Barrer. That looks like a sensible average but predicts much greater flux through this stack than the proper 7.14 Barrer result.

The error is geometric. The arithmetic average corresponds to an idealized parallel-area arrangement: 20 percent of an equal-thickness membrane area offers a 2-Barrer path and 80 percent offers a 20-Barrer path, both seeing the same external pressure difference independently. That is not the same as forcing every gas molecule to traverse both layers.

Both formulas can be right in their proper architectures. The engineer must draw the path before choosing which variables add. A laminated barrier is a sequential path; distinct side-by-side permeable channels are parallel paths. Calling both a “blend” without specifying geometry hides the crucial decision.

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Change only A’s thickness from 20 μm to 10 μm, while increasing B from 80 μm to 90 μm so the total remains 100 μm. At unchanged permeabilities, the resistances become R_A=10/2=5 and R_B=90/20=4.5 μm/Barrer. The new P_eff=100/9.5≈10.526 Barrer.

The effective permeability increased from 7.14 to 10.53 Barrer even though the stronger-barrier material lost only 10 μm of thickness. A now takes 5/9.5≈52.6 percent of the total resistance rather than 71.4 percent. Its thinness makes manufacturing tolerance and continuity much more important to final performance.

Conversely, make A 40 μm thick and B 60 μm. Then R_A=40/2=20, R_B=60/20=3, giving P_eff=100/23≈4.348 Barrer. This prediction assumes permeability itself does not change with thickness, processing or ageing. A measured specimen can depart from the formula if those assumptions fail.

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Suppose 99 percent of a film area behaves like our ideal stack with P_eff≈7.143 Barrer, but a distinct, simplified one-percent bypass path has an equivalent P=200 Barrer over the same 100 μm thickness. The paths are now parallel by area. Their effective permeability under a common pressure difference is 0.99×7.143 + 0.01×200≈9.071 Barrer.

That is an increase of about 27 percent despite the bypass taking only one percent of the area. This is an intentionally idealized numerical sensitivity test, not a physical model for the complex geometry of every pinhole. Actual leakage may depend on defect shape, overlap, sealing and whether the bypass traverses all layers.

The lesson transfers to polymer barrier design: a well-fitted series model can coexist with poor real-world results when the laminate contains defects, edge leaks, damaged coatings or interfacial transport paths. Characterisation must examine film continuity, not merely intrinsic gas permeability of neat materials.

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A credible report should name the gas species, measurement technique, temperature, relative humidity, pressure difference, conditioning history and permeating direction. Measure or independently validate each layer’s thickness and composition. State whether values refer to intrinsic permeability, permeance or transmission rate, and confirm the unit conversion before comparing suppliers.

Check the steady-state assumption. Many polymers show initial uptake, lag times and time-dependent physical ageing, and different layers can store gas. A multilayer film may approach a steady flux only after the transient concentration profile develops. Interfacial layers, porous supports and pressure-dependent sorption call for models more involved than our two-resistor sum.

FAQ: Does the order A–B versus B–A matter? In this specific linear steady model with constant P and no interface effects, the series sum is unchanged by reversal. Can order matter experimentally? Yes, when humidity, swelling, asymmetric exposures or transient effects change layer properties. Is one neat-material P measurement proof of packaging shelf life? No; shelf life depends on the complete application.

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For a research-grounded multilayer model, read “Characteristics of Gas Permeation Behaviour in Multilayer Thin Film Composite Membranes for CO₂ Separation”. It explicitly develops a model for contributions from different layers and discusses porous support and dense-layer transport. Our calculation is a narrower ideal case, not a reproduction of the study’s material data.

Continue with the Polymer Science subject root, density-derived crystallinity and void bias, and first and second DSC heating. These describe other ways a polymer’s measured properties can be misunderstood.

The useful practice is to sketch the flow path, mark the variables that remain common across layers, compute resistance with dimensional consistency and test the assumptions against a real specimen. An attractive arithmetic average should never replace the physical arrangement of the material.

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