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Pure vs. mixed-gas sorption in glassy polymers: In a glassy polymer (below $T_g$), a single gas typically exhibits a concave sorption isotherm: uptake rises sublinearly with pressure, unlike the linear Henry’s-law behavior above $T_g$. This is attributed to “dual-mode” sorption: some penetrant dissolves in the dense polymer matrix (obeying Henry’s law) while additional penetrant saturates specific microvoid “sites” (a Langmuir-type adsorption in the polymer’s nonequilibrium free volume). By contrast, when two or more gases are present, they compete for those limited Langmuir sorption sites. As a result, each component’s sorbed concentration in a mixture is generally lower than it would be at the same partial pressure alone. In particular the less soluble (lower-affinity) gas tends to be more strongly suppressed, often causing the mixed-gas solubility selectivity (ratio of concentrations) to exceed the ideal value predicted from pure-gas isotherms. For example, in CO₂/CH₄ mixtures both components sorb less than in their pure-gas cases (especially CH₄), so the CO₂/CH₄ sorption selectivity is higher under mixture conditions. In extreme cases, mixed-gas effects can even reverse performance trends observed with single gases. (We note that unlike competitive sorption, synergistic sorption – where one gas increases the sorption of another via polymer swelling – is usually minor in rigid high-free-volume polymers under moderate pressures.)

Dual-mode sorption model and mixed-gas prediction: The dual-mode sorption (DMS) model quantitatively captures glassy polymer sorption by superposing a Henry’s-law term and a Langmuir term. For a single gas, the sorbed concentration $C$ (e.g. in cm³(STP) gas per cm³ polymer) is written as:

$C(p) = k_D\,p \;+\; \frac{C'_H\,b\,p}{\,1 + b\,p\,}\,.$

Here $k_D$ is the Henry’s law coefficient for dissolution, $C'_H$ is the Langmuir capacity constant (proportional to the polymer’s unrelaxed free-volume or “hole” capacity), and $b$ is the Langmuir affinity constant. This empirical formula yields the characteristic concave isotherm of glassy polymers and reduces to simple Henry’s law at the glass-transition (since $C'_H \to 0$ as $T \to T_g$). The same three parameters ($k_D, C'_H, b$) are obtained by fitting pure-gas sorption data and are then used to predict mixture sorption. For a binary mixture of gases A and B (with partial pressures $p_A, p_B$), the DMS model assumes each gas obeys its own Henry’s term and that both compete for the same Langmuir sites. One convenient form is to write the total sorbed concentration as the sum of contributions from each gas:

$C_{\text{tot}} = k_{D,A}p_A + k_{D,B}p_B \;+\; \frac{\,C'_{H,A}\,b_A\,p_A \;+\; C'_{H,B}\,b_B\,p_B\,}{\,1 + b_A p_A + b_B p_B\,}\,,$

which implicitly means the concentration of component i in the mixture is $C_i = k_{D,i} p_i + \frac{C'_{H,i} b_i p_i}{\,1+\sum_j b_j p_j\,}$. This reduces to the single-gas formula when the other component’s pressure is zero. In practice, partial pressure in these equations is often replaced by fugacity for accuracy at high pressures, though using fugacities or pressures gives nearly the same results if applied consistently. The DMS mixed-gas model predicts the competitive sorption effect observed experimentally – namely, each gas’s uptake is depressed relative to its pure-gas sorption isotherm.

Challenges and limitations: Although the dual-mode model is widely successful in correlating sorption data, it has notable limitations in predicting sorption behavior. First, the DMS model is empirical – its fitted parameters are not universal constants but can vary with polymer formation history, physical aging, and temperature/pressure range. This means a parameter set obtained under one set of conditions may lack predictive power outside its calibration range. For instance, sorption isotherms measured only over a limited pressure range can be fit well by DMS, but extrapolating beyond that (especially approaching plasticization pressures) may be unreliable. Indeed, multiple distinct combinations of $k_D$, $C'_H$, $b$ can sometimes fit the same pure-gas isotherm almost equally well – yet they yield markedly different mixed-gas predictions. This sensitivity undermines confidence in mixture forecasts, as one study showed that using different pure-gas fit sets led to only qualitative agreement with some mixed-CO₂/CH₄ data.

Furthermore, DMS inherently assumes a fixed population of Langmuir sites (microvoids) and ignores polymer relaxation or swelling induced by sorbate. In reality, penetrant-induced swelling (plasticization) can increase free volume and sorption capacity, especially for highly condensable gases at high activity – an effect the basic DMS model cannot capture. In binary sorption, this means any synergistic increase in sorption due to a second gas (or due to polymer dilation) is unaccounted for in the standard model. Conversely, the model also cannot describe sorption isotherm shapes that deviate from concave Langmuir-type behavior (e.g. sigmoidal isotherms caused by specific polymer-penetrant interactions or clustering). For example, glassy polymers absorbing polar molecules like alcohols can show sigmoidal sorption isotherms, which the simple dual-mode framework fails to represent. Finally, accurately measuring mixed-gas sorption itself is challenging, and discrepancies between idealized model assumptions (e.g. truly zero downstream pressure, no competitive diffusion limitations) and experimental realities can add to prediction error. Despite these challenges, the dual-mode sorption model – with appropriate care – remains a valuable first approximation for interpreting sorption in glassy polymers, and it highlights the key role of competitive site occupancy in mixed-gas environments. Researchers often turn to more elaborate models (e.g. non-equilibrium lattice fluid or thermodynamic theories) when the dual-mode model’s assumptions break down, but such models come at the cost of additional parameters or complexity. In summary, dual-mode sorption provides a useful baseline for pure- and mixed-gas sorption in glassy polymers, but one must be mindful of its empirical nature and the above limitations when predicting multicomponent sorption behavior in practice.

Sources: Dual-mode sorption model and mixed-gas extension; competitive sorption effects in mixtures; model equations and parameters; limitations of DMS model; parameter sensitivity and predictivity issues.