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# Differences Between Pure- and Mixed-Gas Sorption in Glassy Polymers: Insights for Laboratory Implementation

### Introduction to Gas Sorption in Glassy Polymers

Gas sorption in glassy polymeric materials plays a crucial role in membrane separation processes for applications such as natural gas sweetening, biogas upgrading, carbon capture, and hydrogen recovery ((321)). These applications rely on the ability of polymers to selectively absorb and transport gases through their structure ((11)). Glassy polymers exhibit a unique non-equilibrium state with excess free volume that significantly influences their gas sorption capabilities ((13)). Understanding the fundamental differences between pure-gas and mixed-gas sorption behaviors is essential for accurately predicting membrane performance in real-world applications ((12)).

### Fundamental Differences Between Pure-Gas and Mixed-Gas Sorption

The sorption mechanisms for pure gases and gas mixtures in glassy polymers differ significantly ((319)). In pure-gas sorption, the gas molecules dissolve within the polymer matrix based solely on their interaction with the polymer, without interference from other gas species ((321)). This behavior can typically be characterized by Henry's law for the dissolved portion and by Langmuir adsorption for the portion absorbed in the microvoids ((16)). By contrast, in mixed-gas scenarios, multiple gas species compete for the same sorption sites within the polymer matrix, leading to complex interactions that alter the sorption behavior of each component ((202)).

The presence of one gas can significantly affect the sorption of another, typically reducing the amount of gas absorbed compared to what would be expected under pure-gas conditions ((223)). This competitive effect is particularly pronounced in glassy polymers due to their limited unrelaxed free volume ((229)). For instance, experimental studies have shown that the presence of CO₂ strongly influences the solubility of CH₄, while CH₄ also affects CO₂ solubility, albeit to a lesser extent ((346)).

Mixed-gas sorption typically yields lower solubility coefficients for each gas compared to their pure-gas values at the same partial pressure or fugacity ((344)). This reduction in sorption capacity is due to the competition between different gas species for the available sorption sites, particularly those associated with the Langmuir mode ((183)). The magnitude of this reduction depends on factors such as the relative affinities of the gases for the polymer and their partial pressures ((224)).

### Solution-Diffusion Model Framework

The transport of small molecules in dense polymeric membranes is described by the solution-diffusion model ((18)). According to this model, permeability (P) is the product of solubility (S) and diffusion (D) coefficients ((32)). The relationship can be expressed as:

\\[ P = S \times D \\]

This framework allows for the separation of thermodynamic (solubility) and kinetic (diffusion) contributions to gas transport ((33)). In the context of gas mixtures, this becomes particularly important as both the solubility and diffusivity of gases can be affected differently by the presence of other components ((37)).

The selectivity of a polymer membrane for one gas over another (permselectivity) contains contributions from both solubility-selectivity and diffusivity-selectivity ((34)):

\\[ \alpha_{A/B} = \frac{P_A}{P_B} = \frac{S_A}{S_B} \times \frac{D_A}{D_B} = \alpha_{S} \times \alpha_{D} \\]

This approach enables researchers to analyze how mixed-gas conditions specifically affect each component of selectivity ((36)).

### Dual-Mode Sorption Model: Theoretical Framework

The Dual-Mode Sorption (DMS) model has been widely used to describe gas sorption in glassy polymers ((53)). This model postulates the existence of two different gas populations at equilibrium within the polymer matrix ((122)). The first population dissolves in the dense portion of the material according to Henry's law, while the second population saturates the non-equilibrium excess free volume of the polymer, described by a Langmuir isotherm ((122)).

For pure gases, the total sorbed concentration (C) can be expressed as the sum of these two contributions ((221)):

\\[ C = C_H + C_L = k_H \times p + \frac{C'_H \times b \times p}{1 + b \times p} \\]

Where:
- \\(C_H\\) represents the Henry's law contribution, with \\(k_H\\) as Henry's law constant
- \\(C_L\\) represents the Langmuir contribution
- \\(C'_H\\) is the Langmuir capacity constant, related to the sorption capacity in the low-pressure region
- \\(b\\) is the Langmuir affinity constant, representing the ratio of sorption and desorption rate constants
- \\(p\\) is the gas pressure or fugacity ((222))

The three parameters (\\(k_H\\), \\(C'_H\\), and \\(b\\)) are typically determined through non-linear least-square fitting of pure-gas sorption data ((230)).

### Extension of Dual-Mode Sorption Model to Mixed Gases

The extension of the DMS model to mixed-gas sorption is based on phenomenological arguments derived from competitive sorption theories for catalysts ((223)). Since the amount of unrelaxed free volume in a glassy polymer is limited, different gas species compete to occupy this volume ((224)). The competition is controlled by the relative values of the product of the affinity constant and partial pressure (or fugacity) of each penetrant ((224)).

Under the assumptions that the affinity parameter (b), Henry's constant (\\(k_H\\)), and the molar density of components sorbed inside the Langmuir sites are independent of the presence of other penetrants, the concentration of component i in the presence of a second component j can be expressed as ((224)):

\\[ C_i = k_{H,i} \times p_i + \frac{C'_{H,i} \times b_i \times p_i}{1 + b_i \times p_i + b_j \times p_j} \\]

For systems with more than two penetrants, the general expression becomes ((126)):

\\[ C_i = k_{H,i} \times p_i + \frac{C'_{H,i} \times b_i \times p_i}{1 + \sum_j b_j \times p_j} \\]

This multicomponent form of the DMS model captures the competitive effect where the presence of one gas reduces the sorption of others ((229)). The parameters used in these equations are the same as those retrieved from pure-gas sorption experiments ((225)).

### Solubility-Selectivity in Mixed-Gas Conditions

The solubility-selectivity of a polymer for one gas over another is an important parameter in membrane separation processes ((227)). It can be calculated using solubility coefficients (S) of pure gases (ideal case) or those obtained from mixed-gas sorption (multicomponent case) ((227)):

\\[ \alpha_{S,A/B} = \frac{S_A}{S_B} \\]

Where the solubility coefficient (S) is defined as the ratio of gas concentration (c) to gas fugacity (f) ((228)).

Experimental studies have shown that the mixed-gas solubility-selectivity can differ significantly from the ideal selectivity ((265)). For instance, in CO₂/CH₄ mixtures in polymers of intrinsic microporosity (PIMs), the multicomponent CO₂/CH₄ solubility-selectivity is generally higher than the ideal value calculated from pure-gas behavior ((175)). This is because the reduction in CH₄ solubility due to the presence of CO₂ is typically more significant than the reduction in CO₂ solubility ((175)).

### Experimental Approaches for Mixed-Gas Sorption

To perform mixed-gas sorption experiments, specialized equipment is necessary ((327)). A commonly used setup is the barometric pressure-decay apparatus equipped with a gas chromatograph for analyzing gas compositions ((328)). The apparatus typically consists of a feed chamber, a sample chamber containing the polymer film, and a calibration chamber ((328)). Gas uptakes are calculated via mass balance using pressure transducer readings and gas chromatography data ((337)).

Mixed-gas sorption experiments often follow a constant feed concentration procedure, where custom-made gas mixtures are introduced to the feed chamber and then expanded to the sample chamber ((337)). At sorption equilibrium, gas samples from both chambers are analyzed by gas chromatography to determine their compositions ((337)). Mixed-gas uptakes are then calculated using mass balance principles and the gas composition data ((338)).

The availability of experimental data for mixed-gas sorption is limited compared to pure-gas data ((37)). Most studies have focused on binary mixtures such as CO₂/CH₄, CO₂/C₂H₆, and CO₂/N₂, with some limited work on ternary mixtures ((37)). This scarcity of mixed-gas data highlights the need for reliable models that can predict mixed-gas behavior from pure-gas measurements ((204)).

### Challenges in Predicting Pure-Gas Sorption with the Dual-Mode Model

Despite its widespread use, the DMS model faces several challenges in accurately predicting pure-gas sorption in glassy polymers ((128)). One major issue is that the DMS parameters are sensitive to the polymer's physical state, thermal history, and processing conditions ((211)). This sensitivity means that parameters may vary significantly based on how the polymer sample was prepared and treated ((211)).

Another challenge is that the DMS model cannot represent all types of sorption isotherms encountered in glassy polymers ((210)). For example, it cannot adequately describe the sigmoidal shape of sorption isotherms observed for alcohols in some glassy polymers ((128)). This limitation has led to the development of modified models incorporating multilayer sorption theory to better represent these complex isotherms ((210)).

The parameters obtained from the DMS model are also pressure-range dependent ((249)). Studies have shown that \\(C'_H\\) tends to decrease and b tends to increase if a broader pressure range is considered during regression ((249)). This dependence means that extrapolation outside the derivation range should be avoided to prevent significant prediction errors ((249)).

Temperature dependency of the DMS parameters adds another layer of complexity ((248)). Generally, \\(k_H\\), \\(C'_H\\), and b are expected to decrease as temperature increases, consistent with their physical meaning ((248)). However, when regression is performed independently at each temperature, fluctuations in parameter values may occur, leading to inconsistent trends ((248)).

### Challenges in Predicting Mixed-Gas Sorption with the Dual-Mode Model

The prediction of mixed-gas sorption using the DMS model presents additional challenges beyond those associated with pure-gas sorption ((202)). A fundamental issue is that the model does not explicitly account for polymer swelling that may occur during gas sorption ((229)). While this limitation is less problematic for ultra-high free volume glassy polymers with limited swelling tendency, it can lead to significant errors for polymers that exhibit substantial swelling ((229)).

A sensitivity analysis of the DMS model has revealed that pure-gas data can be represented with the same accuracy by several different parameter sets, which, however, yield markedly different mixed-gas predictions ((203)). This lack of robustness means that even if the model accurately fits pure-gas data, it may only qualitatively agree with experimental mixed-gas data in some cases ((203)).

The DMS model assumes that the affinity parameter, Henry's constant, and the molar density of a component sorbed inside the Langmuir sites are independent of the presence of other penetrants ((224)). This assumption may not hold in real systems, especially when there are specific interactions between the penetrants or when one penetrant causes significant swelling of the polymer matrix ((269)).

The competitive sorption effect, which is the primary multicomponent phenomenon captured by the DMS model, may not be the only factor influencing mixed-gas sorption ((229)). Synergistic effects, where the presence of one gas enhances the sorption of another, cannot be represented by the standard DMS model. These effects might be particularly important in cases where one gas induces significant polymer swelling, creating additional free volume for other gases ((50)).

### Parameter Determination and Sensitivity Issues

The determination of DMS parameters involves non-linear regression of experimental sorption data ((231)). The quality of the parameter estimation depends on the quantity and quality of the experimental data, as well as the regression method used ((231)). Different regression approaches, such as minimizing the sum of squared residuals or weighted residuals that account for experimental error, can yield different parameter sets ((271)).

A comprehensive search in the parameter space has shown that multiple sets of DMS parameters can provide equally satisfactory representations of pure-gas sorption data ((307)). This non-uniqueness in parameter determination poses a significant challenge for predicting mixed-gas behavior ((307)). It means that even with excellent fitting of pure-gas data, there is no guarantee that the resulting parameters will accurately predict mixed-gas sorption ((287)).

Imposing constraints on parameters to ensure consistency with theoretical expectations, such as enforcing a specific temperature dependence, can improve the internal consistency of the parameter set but may result in slightly less accurate representation of pure-gas sorption data ((304)). This trade-off between accuracy in fitting pure-gas data and reliability in predicting mixed-gas behavior highlights the complexity of the parameter determination process ((305)).

### Alternative Approaches and Model Modifications

Several alternative approaches have been proposed to address the limitations of the standard DMS model ((132)). The Guggenheim-Anderson-de Boer (GAB) model, which considers multilayer adsorption of small molecules on polymer surfaces, has been used to describe sorption isotherms in both rubbery and glassy polymers ((133)). This model is more flexible than the DMS model and can represent different types of isotherms, including S-shaped ones ((136)).

Another approach is the Non-Equilibrium Thermodynamics for Glassy Polymers (NET-GP), which extends equation-of-state theories to the non-equilibrium glassy state ((54)). This framework provides a more rigorous method for calculating gas solubility in glassy polymers by introducing an internal state variable, the polymer density, to describe the out-of-equilibrium degree of the system ((54)).

For ultra-high free volume glassy polymers, such as polymers of intrinsic microporosity (PIMs), the dual-mode sorption model has been found to provide reliable estimations of mixed-gas sorption data despite its limitations ((130)). This is because these materials typically exhibit limited swelling and are dominated by competitive sorption effects rather than synergistic ones ((130)).

### Practical Considerations for Laboratory Implementation

When implementing mixed-gas sorption capabilities in a laboratory setting, several practical considerations should be taken into account ((327)). The design of the sorption apparatus is crucial, with a focus on minimizing dead volumes to maximize sensitivity ((329)). This optimization is particularly important for measuring the sorption of gases with low solubility, such as methane at high partial pressures ((329)).

Calibration of the sorption apparatus is essential for accurate measurements ((329)). This typically involves a gas expansion procedure using a known reservoir volume, which is calibrated via water filling at room temperature ((329)). Regular recalibration after any maintenance operations is recommended to ensure data accuracy ((329)).

The choice of experimental method also affects the quality of the data obtained ((330)). For pure-gas sorption, both gravimetric and barometric techniques can be used ((333)). However, for mixed-gas sorption, a barometric system equipped with a gas chromatograph is necessary to determine the composition of the gas phase at equilibrium ((337)).

Sample preparation and conditioning are important factors that can influence sorption results ((325)). Before measurements, polymer samples should be thoroughly degassed to remove any pre-sorbed gases or solvents ((331)). The thermal and mechanical history of the polymer sample should also be carefully controlled, as these factors can affect the free volume distribution and, consequently, the sorption behavior ((337)).

### Conclusion

The study of mixed-gas sorption in glassy polymers is essential for understanding and predicting the performance of membrane materials in real-world separation applications ((12)). The dual-mode sorption model provides a useful framework for describing both pure-gas and mixed-gas sorption behaviors, but it has limitations that must be considered when implementing it in a laboratory setting ((203)).

The main differences between pure-gas and mixed-gas sorption lie in the competitive interactions that occur when multiple gas species vie for the same sorption sites within the polymer matrix ((223)). These interactions typically lead to lower sorption capacities for each gas compared to pure-gas conditions, but they can also result in enhanced solubility-selectivity under certain conditions ((175)).

The challenges in accurately predicting sorption behaviors using the DMS model include parameter sensitivity, limitations in representing all types of isotherms, and assumptions regarding the independence of parameters in the presence of other penetrants ((307)). Despite these challenges, the model remains a valuable tool for qualitative predictions and for understanding the fundamental mechanisms of gas sorption in glassy polymers ((266)).

For researchers implementing mixed-gas sorption capabilities in their laboratories, careful attention to apparatus design, calibration procedures, and experimental methods is essential for obtaining reliable data ((329)). By combining well-designed experiments with appropriate modeling approaches, a deeper understanding of mixed-gas sorption in glassy polymers can be achieved, leading to improved membrane materials for gas separation applications ((37)).

I am a researcher interested in bringing mixed-gas sorption capabilities to my lab. Please discuss the differences between pure- and mixed-gas sorption for glassy polymers, how the dual-mode sorption model can be used to predict mixed-gas sorption behavior in glassy polymers (include equations where applicable), and what challenges there are in accurately predicting pure- and mixed-gas sorption using the dual-mode sorption model. 