Modeling Gas and Vapor Sorption and Swelling in Triptycene-Based

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

pubs.acs.org/Macromolecules

Modeling Gas and Vapor Sorption and Swelling in Triptycene-Based Polybenzoxazole: Evidence for Entropy-Driven Sorption Behavior Valerio Loianno,†,‡ Qinnan Zhang,§ Shuangjiang Luo,∥ Ruilan Guo,§ and Michele Galizia*,†

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†

School of Chemical, Biological and Materials Engineering, University of Oklahoma, 100 E. Boyd Street, Norman 73019, Oklahoma, United States ‡ Department of Chemical, Materials and Industrial Production Engineering, University of Naples Federico II, p.le Tecchio 80, Naples 80125, Italy § Department of Chemical and Biomolecular Engineering, University of Notre Dame, 205 McCurtney Hall, Notre Dame 46556, Indiana, United States ∥ Beijing Key Laboratory of Ionic Liquids Clean Process, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China S Supporting Information *

ABSTRACT: The nonequilibrium lattice fluid model was used to describe and sometimes predict nitrogen, methane, carbon dioxide, ethane, and water vapor sorption at multiple temperatures (5−50 °C) and pressures (up to 32 atm) in novel triptycene-based polybenzoxazole (TPBO) prepared via a thermal rearrangement (TR) process from an ortho-functional polyimide precursor. The polymer lattice fluid parameters were determined using a few sorption data in the limit of infinite dilution and used to predict the solubility of nonswelling gases at several temperatures with no adjustable parameter. To calculate the solubility of swelling gases, the polymer−penetrant interaction parameter was adjusted to experimental sorption data at low pressure at a reference temperature. The second adjustable parameter, that is, the swelling coefficient, was calculated at each temperature using only one experimental sorption datum at high pressure. TPBO exhibits better dimensional stability upon exposure to swelling penetrants relative to previously reported TR polymers. Finally, it was demonstrated that the larger sorption capacity exhibited by TPBO relative to iptycene-free TR polymers has a purely entropic origin.

1. INTRODUCTION The possibility of predicting elementary transport properties of polymers with a minimal number of experimental information would expedite the design and optimization of membrane materials for gas, vapor, and liquid separation, as well as polymers for barrier packaging applications.1,2 The dual-mode model and its subsequent modifications have been used to describe gas sorption and transport in glassy polymers.3−6 Unfortunately, despite its simplicity, the dual-mode model exhibits poor predictive ability, as the three model parameters depend on the polymer−penetrant couple, as well as temperature and pressure range over which the experimental data are fit.7 In contrast, it becomes entirely predictive for mixed gas sorption.8,9 In recent years, molecular modeling emerged as a promising tool to simulate the thermodynamic and transport properties of polymers.10 However, when considering glassy polymers, long equilibration times are needed to optimize the polymer structure.10 Moreover, molecular modeling is computationally intensive. Since the late 80s, macroscopic thermodynamic modeling has been © XXXX American Chemical Society

successfully used to simulate the physical and transport properties of glassy polymers.11−13 For example, the nonequilibrium lattice fluid (NELF) model12 has been used to describe and sometimes predict pure and mixed gas sorption, diffusion, and permeability coefficients in glassy polymers,12,14−16 polymer blends,17 and mixed matrix membranes.18 Compared to the dual-mode model, which is fully empirical, the lattice fluid model is grounded on rigorous fundamental basis and requires a minimal number of adjustable parameters. Compared to molecular simulation, macroscopic thermodynamic modeling is less computationally intensive. In this study, we propose, for the first time, a theoretical analysis of gas and vapor sorption in a novel triptycene-based polybenzoxazole (TPBO). Triptycene-containing polymers exhibit an optimal combination of transport, thermal, and mechanical properties that makes them promising for Received: March 20, 2019 Revised: May 19, 2019

A

DOI: 10.1021/acs.macromol.9b00577 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules Table 1. Relevant Parameters and Equations for the NELF Modela symbol

Property

Mi ρi ρ ωi T*i p*i ρ*i

molar mass of species i density of species i density of the mixture mass fraction of species i characteristic temperature of pure component i characteristic pressure of pure component i characteristic density of pure component i

definition

Φi

volume fraction of species i at close-packed conditions

kij r0i ri

binary parameter number of lattice cells occupied by a molecule of pure component i number of lattice cells occupied by a molecule in mixture

T̃ i

reduced temperature of pure component i

Tĩ =

T Ti*

p̃i

reduced pressure of pure component i

pi ̃ =

p pi*

ρ̃i

reduced density of pure component i

ρi =

T̃

reduced temperature of the mixture

T̃ =

p̃

reduced pressure of the mixture

T T* p p̃ = p*

ρ̃

reduced density of the mixture

ρ =

ρ ρ*

vi*

volume occupied by a mole of lattice site of pure component i

vi* =

RTi* pi*

ωi /ρi*

Φi =

∑i ωi /ρi*

∼

∼

ρi ρi*

Mixing Rules (For Binary Mixtures)

T* =

p*

T*

characteristic temperature

p*

characteristic pressure

p* = Φ1p1* + Φ2p2* − Φ1Φ2[p1* + p2* − 2(1 − k12) p1* p2* ]

ρ*

characteristic density

ω ω 1 = 1 + 2 ρ* ρ1* ρ2*

Φ1p1* T1*

+

Φ2p2* T2*

ÅÄÅ ∼ 2 ÑÉÑ ÅÅ ρ i p̃ ∼ ∼j 1 zyÑÑÑ Å ρ1 = 1 − expÅÅÅÅ− 1 − 1 − ρ1jjjj1 − 0 zzzzÑÑÑÑ ÅÅ T1̃ T1̃ r1 {ÑÑÑ k ÅÅÇ ÑÖ ∼ ÑÉÑ ÅÄÅ ∼ 0 ÅÅ ρ p̃ μ1 1 − ρ1 ∼ ∼ Ñ Ñ 1 = r10ÅÅÅÅ− 1 + 1∼ + 0 ln ρ1 + ln(1 − ρ1)ÑÑÑÑ ∼ ÑÑ ÅÅ T1̃ RT r1 ρ1 T1̃ ρ1 ÑÖ ÅÇ

Relevant NELF Model Equations LF equation of state for pure penetrant

pure penetrant equilibrium chemical potential

μi ne RT nonequilibrium chemical potential of species i in mixture

ij ∼ ∼ r − r 0 yz = ln(ρ ϕi) − jjjjri0 + i ∼ i zzzz ln(1 − ρ ) − ri ρ { k ÑÉÑ ÅÄ Np+ 1 0 *Å ÑÑ Å ∼r v Å Ñ Å − ρ i i ÅÅÅpi* + ∑ ϕj(p*j − Δpij* )ÑÑÑ ÑÑ Å RT ÅÅ ÑÑÖ = j 1 ÅÇ

a

Details are reported in previous studies.32

application in membrane separations.19−22 Specifically, the controlled ultramicroporosity provided by triptycene units enables superior size sieving ability,19−22 which pushes the performance of these materials far above the 2008 upper bound23 in several industrially relevant separations. Despite the attention raised in the last decade, no attempt has been made to model the transport properties of these materials. This study aims at filling this gap.

2. THEORETICAL BACKGROUND The lattice fluid equation of state developed by Sanchez and Lacombe24,25 and its subsequent modifications26,27 provided a good prediction of gas and vapor sorption isotherms in several rubbery polymers, such as polydimethylsiloxane,24 cross-linked poly(ethylene glycol) diacrylate,28 and poly(caprolactone).29 However, the equation-of-state approach cannot be used to predict gas sorption isotherms in polymers below their glassB


ÄÅ É l o ij v1* yz ρ2* ÑÑÑÑ jij TSTP zyz oÅÅÅÅ 0 j z zz + r1 mÅÅ1 + jj ln(Sinf ) = lnjjj − 1zz 0 ÑÑÑ o j v* z ρ ÑÑ ÅÅ o j p T zz o k 2 { 2 ÑÖ k STP { nÅÇ 0 yz ρ zy ij v * ji lnjjjj1 − 2 zzzz + jjj 1 − 1zzz j z j ρ2* z{ k v2* { k | 0 o ρ T* 2 o (1 − k12) p1* p2* o + 2 1 } o o ρ2* T p1* o ~

Macromolecules transition temperature.12 Glassy polymers exhibit an excess free volume which originates from drawing below their glasstransition temperature,12 and, as such, they are nonequilibrium materials. The lattice fluid theory envisages mixtures of polymers with low-molecular-weight species as a three-dimensional lattice.25 Each small molecule occupies a cell into the lattice. Polymer chains are assumed as a sequence of rigid beads, each of which occupies a cell in the lattice. Unlike the primitive lattice fluid model developed by Flory,30 modern lattice fluid theories assume that some cells in the lattice can be empty, which permits to account for volume changes upon polymer− penetrant mixing. In the late 90s, Sarti introduced the nonequilibrium thermodynamic of glassy polymers (NETGP), a new approach to calculate the thermodynamic properties of glassy polymers.12 The NET-GP approach can be applied to virtually any equation of state: when applied to the Sanchez−Lacombe equation of state, it gives rise to the NELF model. The NELF model considers glassy polymers as isotropic and homogeneous materials12 and provides an expression for the nonequilibrium Gibbs free energy, from which the penetrant chemical potential in nonequilibrium conditions can be calculated and used to solve phase equilibrium problems. The polymer density, given by the ratio of the polymer mass to the total volume, including the volume occupied by the polymer chains plus the free volume, is treated as the order parameter because it directly quantifies the out-of-equilibrium degree.31 Noteworthy, the NELF model requires the same characteristic parameters and uses the same mixing rules as the original lattice fluid model developed by Sanchez and Lacombe for rubbery polymers.25 The thermodynamic state of mixtures comprising low-molecular-weight species and glassy polymers is defined by the usual state variables, that is, temperature, pressure, composition, and the polymer density, which quantifies the departure from equilibrium conditions.12,31 The lattice fluid model requires three characteristic parameters to describe the pure component thermodynamic properties: T*, p*, and ρ*. The characteristic temperature of component i, T*i , defines the interaction energy between two molecules (or two molecular segments) occupying adjacent positions in the lattice sites.25 The characteristic pressure, pi*, is related to the cohesive energy density of the species i at close-packed conditions (i.e., at 0 K).25 Finally, the characteristic density, ρ*i , is the density of the component i at close-packed conditions.25 The lattice fluid parameters of polymers are usually estimated by fitting pVT data above the glass-transition temperature (i.e., in the rubbery region) to the Sanchez−Lacombe equation of state.12 However, several glassy polymers of interest for membrane applications exhibit a very high glass-transition temperature (Tg), such that volumetric data in the rubbery region are not experimentally accessible. In this situation, the polymer lattice fluid parameters can be estimated using mixture data instead of pure component data.32−34 Specifically, the NELF model provides the following expression for the solubility coefficient in the limit of infinite dilution (i.e., at vanishing pressure)32

Article

(1)

where Sinf is the infinite dilution solubility coefficient, the subscript STP indicates standard temperature and pressure conditions (i.e., 273 K and 1 atm), and subscripts 1 and 2 stand for the penetrant and polymer species, respectively. The meaning of variables in eq 1 is reported in Table 1. For polymers with very high Tg, the lattice fluid parameters can be estimated by fitting eq 1 to experimental infinite dilution sorption data for several penetrants. Once the polymer and penetrant lattice fluid parameters are available, gas and vapor sorption isotherms in glassy polymers can be calculated by satisfying the following two conditions:12 (i) the equality of penetrant chemical potential in the external fluid phase and in the glassy mixture and (ii) the validity of the Sanchez−Lacombe equation of state for the pure penetrant in the external gas phase. Finally, the polymer density has to be known experimentally because it cannot be calculated using an equation-of-state approach. When considering the sorption of light, nonswelling gases, the polymer density can be assumed to be constant and equal to that of the unpenetrated polymer. Therefore, the polymer− penetrant interaction parameter k12 is the sole adjustable parameter. From the physical point of view, k12 quantifies the departure of polymer−penetrant interaction from the Hildebrand mixing rule.35 If the Hildebrand rule applies, k12 = 0, so the procedure is completely predictive at any temperature. To calculate the sorption isotherms of light gases with the NELF model, the polymer−penetrant interaction parameter k12 has to be fit to a few sorption data at a reference temperature. Because k12 does not depend on temperature, pressure, and composition, its optimal value estimated at the reference temperature can be used to predict sorption isotherms at any other temperature with no adjustable parameter.36 In contrast, when considering the sorption of swelling penetrants, such as carbon dioxide, hydrocarbons, and condensable organic vapors, significant polymer dilation may occur. On the basis of experimental observations, the polymer density decreases fairly linearly with penetrant partial pressure in the external gas phase, p37 ρ2 (p) = ρ20 (1 − kswp)

(2)

where ρ02 is the density of the dry, unpenetrated polymer and ksw is the swelling coefficient, which can be evaluated experimentally when dilation data are available. However, dilation data are rarely reported in the literature. In the latter condition, ksw can be estimated by fitting the NELF model to one sorption datum at high pressure, where polymer dilation is likely to occur.37

3. EXPERIMENTAL SECTION 3.1. Polymer Synthesis and Characterization. Details about TPBO synthesis and film casting can be found in our previous study.20 For the sake of clarity, they are briefly summarized in the C


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Macromolecules Table 2. Structure and Relevant Chemical−Physical Properties of TPBO20

glass transition Ta

densityb (g/cm3)

PBO conversionc (%)

d-spacingd (Å)

>400 °C

1.393 ± 0.002

100

6.8

Measured via DSC.20 bMeasured at ambient temperature using the buoyancy method.38 cCalculated as 100 × (experimental mass loss/theoretical mass loss). Experimental mass loss was estimated via thermogravimetric analysis. Theoretical mass loss is the mass loss expected for 100% conversion. Full conversion was confirmed by Fourier transform infrared.20 dMeasured via wide-angle X-ray diffraction.20 a

Supporting Information. Relevant polymer properties are summarized in Table 2. 3.2. Pure Gas and Vapor Sorption Measurements. Pure gas and water vapor sorption isotherms in TPBO up to 32 atm and in the temperature range 5−50 °C were reported in our recent publication.38

4. RESULTS AND DISCUSSION 4.1. Lattice Fluid Parameters and Free Volume of TPBO. The polymer lattice fluid parameters were estimated using the procedure originally reported by Galizia et al.32 Equation 1 was fit to the experimental infinite dilution solubility coefficients available for N2, CH4, CO2, and C2H6 in TPBO at multiple temperatures, ranging from 5 to 50 °C.38 In the fitting procedure, the polymer−penetrant interaction parameter, k12, was set equal to zero, and the three polymer lattice fluid parameters, T2*, p2*, and ρ2*, were treated as adjustable parameters. This procedure was previously used to estimate the lattice fluid parameters of several high Tg polymers, such as polynorbornenes,32 polyimides,34 and thermal rearrangement (TR) polymers obtained from the HAB-6FDA precursor.33 Equation 1 requires the polymer density, ρ2, as an input parameter. The polymer thermal expansion coefficient, αv, is needed to calculate the polymer density at any temperature. Unfortunately, αv is not available for TPBO, so, in first approximation, it was assumed equal to that of other partially fluorinated, high-performance polymers, such as 6FDA-ODA and 6FDA-6FpDA (i.e., αv = 2 × 10−4 K−1).13 Other commercial high Tg polymers, such as Ultem 1000 and Matrimid, exhibit similar values of thermal expansion coefficient (i.e., αv = 9 × 10−5 K−1 and 8.4 × 10−5 K−1, respectively).33,39 Specifically, if αv changes between 2 × 10−4 and 9 × 10−5 K−1 (i.e., by 122%), the corresponding changes in the polymer density and in the calculated gas solubility are less than 0.4 and 3%, respectively. In Figure 1, the experimental infinite dilution solubility coefficients are shown in a parity plot, along with the values calculated from eq 1 using T2*, p2*, and ρ2* as adjustable parameters. A Matlab routine, based on the trust region reflective algorithm, was used to fit eq 1 to the experimental infinite dilution solubility coefficients. In Table 3, the lattice fluid parameters of TPBO are compared with those reported previously for HAB-6FDA-TR450-30min, a thermally rearranged polymer obtained from the HAB-6FDA polyimide (cf. Figure S1, Supporting Information).33 Both TPBO and HAB6FDA-TR450-30min are polybenzoxazoles obtained upon thermal conversion of a corresponding polyimide precursor at 450 °C for 30 min.20,33,38 However, TPBO contains triptycene units and ether linkages on its backbone. Moreover, while TPBO is fully converted to the polybenzoxazole form, HAB-6FDA-TR450-30min exhibits a 76% conversion, which

Figure 1. Calculated vs experimental infinite dilution solubility coefficient in TPBO at multiple temperatures. Green open triangles: N2; green filled triangles: CH4; blue open circles: CO2; red filled circles: C2H6.

means that it is actually a copolymer made up to 76% by polybenzoxazole and 24% by polyimide. As such, differences in lattice fluid parameters of the two materials reflect differences in their structure. TPBO and HAB-6FDA-TR450-30min exhibit fairly similar T*2 values. This fact is not surprising because T*2 is related to the polymer rigidity. Indeed, although ether linkages in TPBO enhance the polymer chain flexibility relative to HAB-6FDATR450-30min, the presence of triptycene units enhances polymer rigidity,20 so the two effects offset each other. TPBO exhibits a higher ρ*2 value relative to HAB-6FDA-TR45030min. This result is consistent with the higher density exhibited by TPBO relative to HAB-6FDA-TR450-30min (1.393 vs 1.340 g/cm3). Interestingly, the ratio

* ρTPBO * ρHAB ‐ 6FDA ‐ TR

is

1.08, which is fairly similar to the density ratio of the two polymers

(

ρTPBO ρHAB ‐ 6FDA ‐ TR

)

= 1.04 . Finally, the two materials

exhibit very similar characteristic pressures, p2*. The slightly larger p*2 value exhibited by TPBO is consistent with the higher density exhibited by this polymer. The uncertainty of the lattice fluid parameters of TPBO was calculated using a likelihood-based data analysis.40 The lattice fluid theory provides the following expression for the polymer fractional free volume42 f=

ρ2* − ρ2 ρ* 2

(3)

Using the ρ*2 value determined earlier and the density determined experimentally at room temperature (cf. Table 2) gives, for TPBO, a fractional free volume of 19.4%. This value is in excellent agreement with that estimated previously from D


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Macromolecules Table 3. Lattice Fluid Parameters of TPBO, Nitrogen, Methane, Carbon Dioxide, Ethane, and Watera polymer

T* (K)

p* (MPa)

TPBO HAB-6FDA-TR450-30min penetrants

900 ± 22 930 ± 23

474 ± 31 446.9 ± 7.3

N2 CH4 CO2 C2H6 H2O

ρ* (kg/m3)

k12

T* (K)

p* (MPa)

0 0 −0.05 ± 0.0025 0.05 ± 0.01 −0.09 ± 0.003

145 215 300 320 670

160 250 630 330 2400

source

1.662 ± 0.030 1.528 ± 0.037 ρ* (kg/m3) 0.943 0.500 1.515 0.640 1.050

this study 33 source 12 12 12 41 42

a

Parameters of HAB-6FDA-TR450-30min are reported for comparison. Uncertainties were calculated using the propagation of errors method.40

Figure 2. Gas sorption isotherms at multiple temperatures in TPBO. (A) Nitrogen; (B) methane; (C) carbon dioxide; (D) ethane. Symbols are experimental data reported previously by Loianno et al.38 Green open triangles: 5 °C; black filled triangles: 20 °C; blue open circles: 35 °C; red filled circles: 50 °C. Continuous lines are NELF model calculations.

the dual-mode analysis of sorption data (i.e., 19%38). On the basis of the NELF and dual-mode calculations, TPBO exhibits much higher fractional free volume relative to thermally rearranged polymers obtained from HAB-6FDA. For example, eq 3 gives, for HAB-6FDA-TR450-30min, a fractional free volume equal to 12.5%. Differences between the two materials may be ascribed to the presence of triptycene units on the TPBO backbone. Triptycene units enhance the free volume available for penetrant transport in two ways: (i) by providing an internal, configurational free volume (i.e., the clefts between the benzene “blades”) and (ii) by preventing efficient chain packing. The larger free volume estimated for TPBO relative to HAB-6FDA-TR450-30min is also consistent with gas diffusivity data. As reported by Loianno et al.,38 gas and water vapor diffusion coefficients in TPBO are larger than in HAB-6FDATR450-30min, which is compatible with the larger free volume exhibited by the former material. 4.2. Predicting Gas Sorption Isotherms in TPBO. Light gases (e.g., N2 and CH4) sorption isotherms in TPBO can be

predicted assuming negligible polymer swelling. Because low condensable gases exhibit quite low solubility in polymers, they are not expected to produce any detectable matrix swelling. This conclusion is supported by the literature data.33,43 To improve the calculation accuracy, the polymer density was estimated at each temperature using the thermal expansion coefficient, as discussed in Section 4.1. Calculation of light gas solubility requires only one adjustable parameter, that is, the polymer−penetrant interaction parameter. k12 was adjusted, for each penetrant, to the experimental sorption isotherm at 35 °C, and the resulting value was used to predict sorption isotherms at any other temperature with no additional adjustable parameters. Remarkably, nitrogen sorption isotherms at any temperature are well described with k12 = 0 (cf. Figure 2A). The maximum deviation of model calculation from the experimental data, observed at 50 °C and pressure higher than 15 atm, is about 20%. Such deviations could be due to the larger uncertainty affecting nitrogen sorption data at high pressures.38 However, the overall quality of the prediction E


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Figure 3. (A) Swelling coefficient for CO2 and C2H6 in TPBO as a function of temperature. Uncertainties were calculated using the propagation of errors method.40 (B) TPBO volume swelling induced by CO2 and C2H6 sorption at 35 °C as a function of concentration, predicted by the NELF model. HAB-6FDA-TR450-30min dilation induced by CO2 sorption at 35 °C is also reported for comparison. It was predicted using the NELF model.

range is required to predict the entire isotherm. Because gas sorption decreases with increasing temperature, less polymer swelling is required to accommodate penetrant molecules in the polymer matrix at higher temperature. According to this picture, the swelling coefficient decreases with increasing temperature (cf. Figure 3A). CO2 sorption in TPBO is well described with k12 = −0.05 (cf. Figure 2C). This slightly negative value indicates that polymer−penetrant interactions are more favorable than those predicted by the Hildebrand rule. This result is not surprising because polar ether linkages on the TPBO backbone are expected to interact favorably with polar CO2 molecules.45 In summary, the model provides a very accurate description of temperature and pressure dependence of carbon dioxide sorption in TPBO. The maximum deviation of the model calculations from the experimental data is less than 4%. Similar conclusions can be drawn for ethane sorption (cf. Figure 2D). Experimental sorption isotherms are well described with k12 = 0.05 and a swelling coefficient that decreases with increasing temperature (cf. Figure 3A). As shown in Figure 3A, the swelling coefficient in the presence of C2H6 is larger relative to that of CO2. Although the two penetrants have the same critical temperature and they are sorbed to a fairly similar extent by TPBO, ethane is a much larger penetrant than carbon dioxide; therefore, a larger polymer swelling is required to accommodate ethane molecules in TPBO. 4.3. Estimation of Polymer Dilation. Once the swelling coefficient is known from the NELF analysis of sorption isotherms, polymer dilation as a function of pressure or penetrant concentration can be predicted by rearranging eq 2 as follows

can be considered satisfactory, as the model is used in a predictive fashion. Methane sorption at any temperature is well predicted with k12 = 0 (cf. Figure 2B). The maximum deviation of model calculations from the experimental values is less than 5%. The possibility of predicting the whole sorption isotherms in a broad temperature and pressure range with k12 = 0 indicates that the interaction of light gases with TPBO does not depart from the Hildebrand model. Remarkably, once the polymer lattice fluid parameters are known, calculation of the solubility of light gases is fully predictive. In contrast, the dual-mode or the Guggenheim−Anderson−de Boer models would require three adjustable parameters for each polymer−penetrant pair at each temperature. Modeling the solubility of swelling penetrants, such as carbon dioxide and ethane, requires an additional adjustable parameter, that is, the swelling coefficient. Because of their larger condensability (i.e., higher critical temperature), carbon dioxide and ethane are sorbed by TPBO to a larger extent relative to light gases, which may cause a non-negligible polymer swelling. To predict CO2 solubility, the interaction parameter k12 was adjusted to the experimental sorption data at 35 °C and in the low-pressure range (below 2 atm), where swelling is negligible. The resulting value of k12 was used to calculate sorption isotherms at any other temperature, analogous to the case of light gases. To properly account for matrix dilation, the swelling coefficient was optimized, at each temperature, to the sorption datum available at the highest pressure, where swelling is likely to occur. To this aim, the polymer density was assumed to decrease linearly with penetrant partial pressure (cf. eq 2). Such assumption is supported by experimental dilation data for several glassy polymers.43 Therefore, the swelling coefficient at any temperature was calculated as follows44 ksw =

kswp ΔV = V0 1 − kswp

ρ20 − ρ2 (p) ρ20 p

(5)

Equation 5 assumes isotropic swelling. This assumption is generally verified for amorphous unoriented polymers, such as TPBO. In Figure 3B, the volume dilation induced by CO2 and C2H6 sorption at 35 °C in TPBO is reported as a function of penetrant concentration. Even though ethane and carbon dioxide exhibit fairly similar solubility in TPBO, ethane sorption swells the polymer matrix much more than carbon dioxide. As discussed previously, ethane (kinetic diameter = 4.4 Å) is a larger molecule relative to carbon dioxide (kinetic diameter = 3.3 Å),46 so it cannot fit into the internal volume of

(4)

where ρ2(p) is the polymer density calculated by fitting the NELF model to the solubility datum available at pressure p (≈30 atm in the case of CO2). In summary, CO2 sorption at the reference temperature (Tref, i.e., 35 °C) is calculated with two adjustable parameters, k12 and ksw. At any other temperature, just one adjustable parameter, ksw, is needed. Thus, at T ≠ Tref, just one sorption datum in the high-pressure F


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Figure 4. Enthalpic (A) and entropic (B) contributions to gas sorption coefficient in TPBO (filled circles) and HAB-6FDA-TR450-30min (open circles) as a function of TC/T. Data for HAB-6FDA-TR450-30min were from ref 33.

triptycene units, whose average internal dimension is 3−4 Å.20 This means that the probability of accommodating ethane molecules in these internal free volume sites is lower than the probability of accommodating carbon dioxide molecules. Therefore, larger matrix dilation is required to open gaps between polymer chains for accommodating ethane molecules. For the sake of comparison, CO2 sorption-induced dilation in HAB-6FDA-TR450-30min is also reported in Figure 3B.33 Interestingly, at fixed CO2 concentration, TPBO exhibits much higher dimensional stability relative to HAB-6FDA-TR45030min, as it can accommodate the same amount of CO2 molecules with lower matrix dilation. This result reflects the different structures of the two polymers. Unlike HAB-6FDATR450-30min, TPBO contains triptycene units, which can accommodate CO2 molecules without inducing matrix swelling. TPBO dilation induced by CO2 and C2H6 sorption at multiple temperatures is shown in Figure S2 (Supporting Information). 4.4. Entropic and Enthalpic Contributions to Sorption Coefficient. Gas sorption isotherms are predicted by equating the equilibrium penetrant chemical potential in the external gas phase to the nonequilibrium penetrant chemical potential in the polymer−penetrant mixture. Because chemical potential exhibits, by its definition, an entropic and an enthalpic contribution (owing to the definition of the partial molar Gibbs free energy, i.e., G̅ i = H̅ i − TS̅ i), the sorption coefficient can be expressed as the sum of an enthalpic part (ϕH) and an entropic counterpart (ϕS). ϕH and ϕS can be calculated by rearranging eq 1 as follows32,42 ij T yz ln(Sinf ) = lnjjjj STP zzzz + ϕS + ϕH jp Tz k STP { where

reflects polymer−penetrant interactions (it depends explicitly on the interaction parameter, k12), as well as penetrant condensability, and it increases with increasing penetrant critical temperature.32,42 The entropic contribution reflects the free volume available to accommodate penetrant molecules in the polymer matrix. Because it is less likely to accommodate bulky, condensable penetrants in the polymer relative to smallsized, less condensable penetrants, the entropic contribution to the sorption coefficient decreases with increasing penetrant critical temperature.32,42 As noted in previous studies, the enthalpic contribution has the major influence on the overall solubility coefficient, which increases with increasing penetrant critical temperature.32,42 The enthalpic and entropic contribution to the infinite dilution sorption coefficients in TPBO are reported in Figure 4 as a function of TC/T, where TC is the penetrant critical temperature and T is the experimental temperature. The behavior of the enthalpic and entropic contributions to the sorption coefficient in TPBO is qualitatively similar to that reported for other glassy polymers.32,42 For the sake of comparison, data for another thermally rearranged polymer, HAB-6FDA-450-30min, are also shown in Figure 4.33 Differences between the two materials reflect their different structures. As reported by Loianno et al.,38 gas sorption in TPBO slightly exceeds that in HAB6FDA-450-30min. The origin of this difference is fully entropic. Indeed, the enthalpic contribution to gas sorption in HAB-6FDA-450-30min slightly exceeds (by 4% or less) that in TPBO. This result reflects the much larger free volume exhibited by TPBO relative to HAB-6FDA. Indeed, eq 7 can be rewritten in terms of free volume, f, as follows42 ϕH = r10(1 − f )

(6)

0 | l o o o o ρ2 T1* 2 * * ϕH = r10m (1 − k ) p p } 12 o o 1 2 * * o o T ρ p o o (7) 1 n 2 ~ Ä É l o oÅÅÅ ij v1* yz ρ2* ÑÑÑÑ ijj yz| ρ20 yzz ij v1* o 0oÅ j z j zzo Å Ñ j z j ϕS = r1 o − 1zz 0 ÑÑ lnjj1 − + − 1 mÅÅ1 + jj } z j z o z j z j z Å Ñ * * * o o j z ρ2 { k v2 Å o k v2 { ρ2 ÑÑÖ k {o nÅÇ ~

T1* 2 (1 − k12) p1* p2* T p1*

(9)

Equation 9 indicates that the enthalpic contribution to sorption coefficient decreases, all the other conditions being equal, with increasing free volume. Because TPBO exhibits larger free volume relative to HAB-6FDA-TR450-30min (cf. Section 4.1), ϕHAB‑6FDA‑TR > ϕTRPBO . In contrast, the entropic H H contribution to gas sorption in TPBO exceeds by 9% that in HAB-6FDA-450-30min. Indeed, ϕS, which provides a negative contribution to gas sorption, is, in absolute value, smaller in TPBO relative to HAB-6FDA-450-30min. Therefore, the larger overall gas sorption capacity exhibited by TPBO has an entropic origin. Unlike HAB-6FDA-450-30min, TPBO contains triptycene groups that effectively disrupt polymer chain

(8)

The lattice fluid model provides a simple way to deconvolute the overall solubility coefficient into its elementary contributions. The enthalpic contribution to the sorption coefficient G


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Figure 5. (A) CO2 partial molar volume in TPBO as a function of CO2 concentration and temperature. Red open triangles: 50 °C; blue open circles: 35 °C; black open squares: 20 °C; green open diamonds: 5 °C. Continuous lines are a guide for the eye. CO2 partial molar volume in organic liquids is from ref 50. (B) CO2 partial molar volume at infinite dilution as a function of temperature. Continuous line is a linear interpolation.

are available, the penetrant partial molar volume can be calculated, at any temperature, as follows43,44,49 ÄÅ ÉÑ ÅÅ ∂ i ΔV y ÑÑ dp j z Å Ñ V1̅ = 22 414ÅÅÅ jjj zzz + β ÑÑÑ ÅÅ ∂p j V0 z ÑÑ dC (12) { ÅÇ k ÑÖ

packing and introduce a significant amount of internal, configurational free volume. Therefore, the probability of accommodating penetrant molecules in the TPBO matrix is larger than that of accommodating them in HAB-6FDA-45030min. Equation 8 can be re-written in terms of free volume as follows42 Ä ÉÑ | l Ñ oÅÅÅ o jij v1* zyz 1 ÑÑÑ jij v1* zyzo 0oÅ Å j z j z 1 1 ln f 1 ϕS = r1 m + − + − Å Ñ jj * zz jj * zz} ÅÅ Ñ o o Ñ o o o k v2 { 1 − f ÑÑÖ k v2 {o (10) nÅÅÇ ~

where β is the polymer isothermal compressibility and C is the penetrant concentration expressed in units of cm3 (STP)/cm3 polymer. In eq 12, the term

which indicates that the entropic contribution to sorption coefficient becomes less negative with increasing fractional free volume, f. This conclusion is consistent with the observed behavior. A fundamental assumption of the NELF model is that glassy polymer−penetrant mixtures can be arranged in a threedimensional lattice. However, from a molecular perspective, the complex polymer structure being considered here is far different from the linear structures typically considered in the literature (such as polyimides, e.g.). On the basis of the results presented above, one could speculate that bulky triptycene groups cause a lattice distortion that, in turn, affects the TPBO transport properties. For example, let us consider a lattice that contains triptycene units. Unlike an ideal fluid, where there are six connected lattice sites, all of identical size, forcing triptycene into one of these sites could distort the lattice. To accurately predict the pressure and temperature of formation of natural gas hydrates, Martiń and Peters developed a modified lattice fluid theory which takes into account the lattice distortion.47 However, although the modified model exhibited a slightly enhanced agreement with experimental data, it introduced additional adjustable parameters, which makes it difficult to understand if the improvement has theoretical roots or it is due only to a mathematical artifact. 4.5. Partial Molar Volumes. For a binary mixture of components 1 and 2, the partial molar volume of species 1 is defined by48 ij ∂V yz z V1̅ = jjj j ∂n zzz k 1 {T , p , n2

∂ ∂p

( ) ΔV V0

dp dC

accounts for the

volume change as a function of penetrant concentration dp because of polymer swelling, and the term β dC accounts for the volume change caused by the mechanical effect of pressure. As noted by several researchers,43,44,49 when considering highly soluble penetrants, the isothermal compressibility provides a negligible numerical contribution to eq 12. Because CO2 solubility in TPBO is very high, in this study we neglected the contribution of β to CO2 partial molar volumes. Moreover, owing to its highly rigid structure, TPBO isothermal compressibility is expected to be extremely small. CO2 partial molar volumes in TPBO at several temperatures are reported in Figure 5A as a function of CO2 concentration. At low concentrations, partial molar volumes are very small, which indicates that TPBO takes up a significant amount of carbon dioxide molecules without exhibiting relevant volume changes. This result is consistent with the picture that at low pressure, the vast majority of penetrant molecules are accommodated into preexisting free volume elements, that is, the conformational free volume associated with the nonequilibrium state of TPBO and the configurational free volume provided by the triptycene units.20,38 Interestingly, the infinite dilution CO2 partial molar volume in TPBO at 35 °C, that is, the CO2 partial molar volume in the limit of vanishing concentration, is 2 cm3/mol, which is smaller than that measured in conventional high free volume glassy polymers, such as PTMSP and Teflon AF. Specifically, the CO2 infinite dilution partial molar volumes at 35 °C in PTMSP, Teflon AF2400, and Teflon AF1600 are 5, 5.5, and 6 cm3/mol, respectively.43 This comparison indicates that the volume change experienced by TPBO at low CO2 concentration is smaller than that experienced by PTMSP and Teflon AF. This result is consistent with the configuration-based ultramicroporosity provided by triptycene units, which can

(11)

where V is the total mixture volume and n1 and n2 are the number of moles of species 1 and 2, respectively. For a polymer−penetrant mixture, when sorption and dilation data H


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In the case of CO2, the model overpredicts ΔHS,inf by 9% relative to the experimental values. For N2 and C2H6, the model predictions deviate from the experimental data by about 17%. Finally, the heat of sorption of methane is overpredicted by 12% relative to the experimental value. In previous studies, deviations up to 50% were reported.43 The model predictions indicate that in the limit of infinite dilution, the sorption process becomes more exothermic with increasing penetrant condensability. For example, carbon dioxide and ethane exhibit fairly similar critical temperatures (i.e., condensability), and the corresponding enthalpies of sorption are also comparable. This trend is confirmed by the experimental data. 4.7. Predicting Water Vapor Sorption in TPBO. Because the NELF model does not account for the contribution of penetrant self-interaction (e.g., clustering) to the total Gibbs free energy of the polymer−penetrant mixture,13 it cannot be used to predict the solubility of associating penetrants, such as water and lower alcohols, in polymers. Nevertheless, the model has been successfully used to predict water sorption in Matrimid polyimide at activities far from one,34 where penetrant clustering is expected to be negligible. Likewise, the model provided a reasonable description of alcohol sorption in polynorbornenes at low activities.32 As discussed in our previous study, clustering of water molecules in TPBO is negligible in the activity range 0−0.5.38 Such conclusion was supported by the shape of sorption and diffusion isotherms and was further confirmed by the Zimm− Lundberg analysis. 38,53 The NELF model provides a reasonable description of water vapor sorption isotherm in TPBO at 35 °C with k12 = −0.09 and ksw = 0 (cf., Figure 6).

accommodate penetrant molecules without causing polymer swelling. The infinite dilution CO2 partial molar volume in TPBO increases linearly with temperature (cf. Figure 5B). As reported by Ribeiro and Freeman, CO2 partial molar volumes in rubbery PEO follow a similar qualitative trend as a function of temperature.51,52 However, because rubbery polymers do not contain preexisting free volume holes, the total volume change with CO2 concentration is expected to be much more pronounced than in glassy polymers. This hypothesis is confirmed by the fact that the CO2 partial molar volumes in PEO are 1 order of magnitude larger than in TPBO.51,52 At higher penetrant concentrations, the TPBO free volume sites reach saturation; therefore, matrix swelling is needed to accommodate additional penetrant molecules. As expected, CO2 partial molar volumes markedly increase with increasing concentration. The maximum value reached at 35 °C, 34 cm3/ mol, is comparable to that previously observed in PTMSP and Teflon AF.43 Within the pressure range investigated, CO2 partial molar volume in TPBO is far below that measured in organic liquids. Indeed, while a significant amount of CO2 molecules sorbed in TPBO are accommodated into preexisting voids, any CO 2 molecules sorbed by a liquid are accommodated into transient gaps between the liquid molecules. Moreover, at any temperatures, CO2 partial molar volume in TPBO is smaller relative to the pure saturated liquid CO2 molar volume (48.5 cm3/mol at 5 °C). Again, this result reflects the ability of CO2 molecules to fit into the conformational and configurational free volume of TPBO rather than between previously sorbed CO2 molecules. Similar conclusions can be drawn in the case of ethane sorption in TPBO. For the sake of brevity, relevant results for ethane are reported in Figure S3 (Supporting Information). As expected, ethane partial molar volumes in TPBO largely exceed those of carbon dioxide because of the larger size of the former penetrant. 4.6. Enthalpy of Sorption at Infinite Dilution. Penetrant infinite dilution enthalpy of sorption in glassy polymers can be predicted as follows36 | l 0 o p2* o o o o o 0 ρ2 * 2 (1 ) ΔHS,inf = −R m T + r T − k } 1 1 12 o o * * o o p ρ o o 2 1 n ~

(13)

The quantities appearing in eq 13 have the usual meaning. Because eq 13 is valid in the limit of infinite dilution, where both sorption and swelling are vanishing, ρ02 represents the density of the unpenetrated polymer. While the experimental determination of sorption enthalpies is time-consuming, eq 13 provides a rapid estimate of ΔHS,inf with a minimum number of inputs. As shown in Table 4, the agreement between the experimentally determined infinite dilution sorption enthalpies and the model prediction is reasonable.

Figure 6. Water vapor sorption isotherm in TPBO at 35 °C. Blue filled circles represent experimental data and continuous red line represents NELF calculation.

Interestingly, the model capability of describing water vapor sorption with ksw = 0 indicates that water vapor does not swell TPBO. This important conclusion is supported by the experimental data in two ways. As discussed in a previous study,38 if full vacuum is pulled for a few hours after the first water vapor sorption experiment and a second sorption experiment is run, the two sorption isotherms are practically superimposed. This fact rules out the occurrence of matrix swelling and any long lasting conditioning. Moreover, water vapor diffusion coefficients in TPBO are fairly constant with concentration.38 In the absence of clustering, this behavior is consistent with having ksw = 0. Indeed, in the presence of

Table 4. Calculated vs Experimentally Determined Infinite Dilution Enthalpy of Sorption in TPBO calculated ΔHS,inf (kJ/mol) N2 CH4 CO2 C2H6

−16.30 −21.0 −30.4 −32.1

experimental ΔHS,inf (kJ/mol) −19.82 −18.55 −27.72 −27.30

± ± ± ±

1.58 1.48 2.21 2.20 I


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polymer swelling, water vapor diffusion coefficients would increase with increasing water vapor concentration in the polymer. Because diffusion coefficients exhibit a fairly constant trend with concentration or activity, the occurrence of polymer swelling during water vapor sorption can be ruled out. The lack of swelling suggests that a significant amount of water is likely sorbed into the configurational free volume provided by triptycene units. This picture seems physically sound because water molecule size (kinetic diameter = 2.65 Å) is smaller than the internal size of the iptycene units (3−4 Å). The negative value of the water−TPBO binary parameter, k12, is indicative of favorable interactions between water molecules and polar ether linkages on the polymer backbone. Molecular modeling is underway to shed fundamental light on this aspect.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1(405) 325 5807. ORCID

Ruilan Guo: 0000-0002-3373-2588 Michele Galizia: 0000-0001-5430-4964 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was partially supported by the University of Oklahoma, VPR Office. The University of Naples Federico II is acknowledged for supporting the V.L. stay at the University of Oklahoma. R.G. acknowledges the financial support from the Division of Chemical Sciences, Biosciences, and Geosciences, Office of Basic Energy Sciences of the U.S. Department of Energy (DOE), under award no. DE-SC0019024.

5. CONCLUSIONS A theoretical interpretation of gas and vapor sorption in a novel triptycene-based thermally rearranged polymer (TPBO), based on the lattice fluid theory, has been presented. Because of the inaccessibility of pVT data in the rubbery region, the polymer lattice fluid parameters were estimated using a collection of mixture data in the limit of infinite dilution. Nitrogen and methane sorption isotherms were predicted in a broad range of temperatures and pressures with no adjustable parameters. The solubility of carbon dioxide and ethane in TPBO was calculated with two adjustable parameters, that is, the polymer−penetrant interaction parameter and the swelling coefficient. The model parameters have physical meaning and provide information about polymer−penetrant binary interactions, as well as the polymer dimensional stability upon carbon dioxide and ethane sorption. TPBO exhibits improved dimensional stability in the presence of CO2 at high pressure relative to previously reported TR polymers. This result reflects the presence of triptycene units on the TPBO backbone, which (i) enhance the polymer rigidity and (ii) accommodate CO2 molecules in their internal, configurational free volume, without causing polymer swelling. The analysis of carbon dioxide and ethane partial molar volumes confirms the low TPBO tendency of swell during gas sorption. The larger sorption capacity exhibited by TPBO relative to previously reported TR polymers without triptycene groups has an entropic origin, that is, it is ascribable to the configurational free volume and to the chain packing disruption introduced by triptycene groups, instead of polymer−penetrant interactions. Finally, water vapor sorption at 35 °C was calculated with one adjustable parameter. The modeling outcomes suggest that polymer dilation upon water vapor sorption is negligible. This result is consistent with previously reported diffusivity data. Equally important, the modeling outcomes indicate the NELF model’s ability to correctly describe the gas transport properties of microporous polymers, other than those of dense polymers.

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REFERENCES

(1) Galizia, M.; Chi, W. S.; Smith, Z. P.; Merkel, T. C.; Baker, R. W.; Freeman, B. D. 50th anniversary perspective: Polymers and mixed matrix membranes for gas and vapor separation: A review and prospective opportunities. Macromolecules 2017, 50, 7809−7843. (2) Sanders, D. F.; Smith, Z. P.; Guo, R.; Robeson, L. M.; McGrath, J. E.; Paul, D. R.; Freeman, B. D. Energy-efficient polymeric gas separation membranes for a sustainable future: a review. Polymer 2013, 54, 4729−4761. (3) Feng, H. Modeling of vapor sorption in glassy polymers using a new dual mode sorption model based on multilayer sorption theory. Polymer 2007, 48, 2988−3002. (4) Koros, W. J.; Chan, A. H.; Paul, D. R. Sorption and transport of various gases in polycarbonate. J. Membr. Sci. 1977, 2, 165−190. (5) Park, G. S. Transport principlessolution, diffusion and permeation in polymer membranes. Synthetic Membranes: Science, Engineering and Applications; Springer, 1986; pp 57−107. (6) Paul, D. R.; Koros, W. J. Effect of partially immobilizing sorption on permeability and the diffusion time lag. J. Polym. Sci., Polym. Phys. Ed. 1976, 14, 675−685. (7) Bondar, V. I.; Kamiya, Y.; Yampol’Skii, Y. P. On pressure dependence of the parameters of the dual-mode sorption model. J. Polym. Sci., Part B: Polym. Phys. 1996, 34, 369−378. (8) Koros, W. J. Model for sorption of mixed gases in glassy polymers. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 981−992. (9) Koros, W. J.; Chern, R. T.; Stannett, V.; Hopfenberg, H. B. A model for permeation of mixed gases and vapors in glassy polymers. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 1513−1530. (10) Larsen, G. S.; Lin, P.; Siperstein, F. R.; Colina, C. M. Methane adsorption in PIM-1. Adsorption 2011, 17, 21−26. (11) Conforti, R. M.; Barbari, T. A.; Vimalchand, P.; Donohue, M. D. A lattice-based activity coefficient model for gas sorption in glassy polymers. Macromolecules 1991, 24, 3388−3394. (12) Doghieri, F.; Sarti, G. C. Nonequilibrium lattice fluids: a predictive model for the solubility in glassy polymers. Macromolecules 1996, 29, 7885−7896. (13) Scherillo, G.; Sanguigno, L.; Galizia, M.; Lavorgna, M.; Musto, P.; Mensitieri, G. Non-equilibrium compressible lattice theories accounting for hydrogen bonding interactions: modelling water sorption thermodynamics in fluorinated polyimides. Fluid Phase Equilib. 2012, 334, 166−188. (14) Galizia, M.; Stevens, K. A.; Paul, D. R.; Freeman, B. D. Modeling gas permeability and diffusivity in HAB-6FDA polyimide and its thermally rearranged analogs. J. Membr. Sci. 2017, 537, 83−92. (15) Minelli, M.; Campagnoli, S.; De Angelis, M. G.; Doghieri, F.; Sarti, G. C. Predictive model for the solubility of fluid mixtures in glassy polymers. Macromolecules 2011, 44, 4852−4862.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.9b00577. Experimental details and partial molar volumes of ethane in TPBO (PDF) J


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Macromolecules (16) Toni, E.; Minelli, M.; Sarti, G. C. A predictive model for the permeability of gas mixtures in glassy polymers. Fluid Phase Equilib. 2018, 455, 54−62. (17) Baschetti, M. G.; De Angelis, M.; Doghieri, F.; Sarti, G. Solubility of gases in polymeric membranes. Chemical Engineering: Trends and Developments; Wiley: Chichester (U.K.), 2005; pp 41−61. (18) Ferrari, M. C.; Galizia, M.; De Angelis, M. G.; Sarti, G. C. Gas and vapor transport in mixed matrix membranes based on amorphous Teflon AF1600 and AF2400 and fumed silica. Ind. Eng. Chem. Res. 2010, 49, 11920−11935. (19) Luo, S.; Wiegand, J. R.; Kazanowska, B.; Doherty, C. M.; Konstas, K.; Hill, A. J.; Guo, R. Finely tuning the free volume architecture in iptycene-containing polyimides for highly selective and fast hydrogen transport. Macromolecules 2016, 49, 3395−3405. (20) Luo, S.; Zhang, Q.; Zhu, L.; Lin, H.; Kazanowska, B. A.; Doherty, C. M.; Hill, A. J.; Gao, P.; Guo, R. Highly selective and permeable microporous polymer membranes for hydrogen purification and CO2 removal from natural gas. Chem. Mater. 2018, 30, 5322−5332. (21) Weidman, J. R.; Guo, R. The use of iptycenes in rational macromolecular design for gas separation membrane applications. Ind. Eng. Chem. Res. 2017, 56, 4220−4236. (22) Weidman, J. R.; Luo, S.; Zhang, Q.; Guo, R. Structure manipulation in triptycene-based polyimides through main chain geometry variation and its effect on gas transport properties. Ind. Eng. Chem. Res. 2017, 56, 1868−1879. (23) Robeson, L. M. The upper bound revisited. J. Membr. Sci. 2008, 320, 390−400. (24) Hariharan, R.; Freeman, B. D.; Carbonell, R. G.; Sarti, G. C. Equation of state predictions of sorption isotherms in polymeric materials. J. Appl. Polym. Sci. 1993, 50, 1781−1795. (25) Sanchez, I. C.; Lacombe, R. H. Statistical thermodynamics of polymer solutions. Macromolecules 1978, 11, 1145−1156. (26) Panayiotou, C.; Sanchez, I. C. Hydrogen bonding in fluids: an equation-of-state approach. J. Phys. Chem. 1991, 95, 10090−10097. (27) Panayiotou, C.; Vera, J. H. Statistical thermodynamics of r-mer fluids and their mixtures. Polym. J. 1982, 14, 681. (28) Minelli, M.; De Angelis, M. G.; Doghieri, F.; Sarti, G. C.; Ribeiro, C. P.; Freeman, B. D., Jr; Freeman, B. D. Equation of state modeling of the solubility of CO2/C2H6 mixtures in cross-linked poly (ethylene oxide). Ind. Eng. Chem. Res. 2015, 54, 1142−1152. (29) Scherillo, G.; Sanguigno, L.; Sansone, L.; Di Maio, E.; Galizia, M.; Mensitieri, G. Thermodynamics of water sorption in poly (εcaprolactone): A comparative analysis of lattice fluid models including hydrogen bond contributions. Fluid Phase Equilib. 2012, 313, 127− 139. (30) Flory, P. J. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1942, 10, 51−61. (31) Sarti, G. C.; Doghieri, F. Predictions of the solubility of gases in glassy polymers based on the NELF model. Chem. Eng. Sci. 1998, 53, 3435−3447. (32) Galizia, M.; De Angelis, M. G.; Sarti, G. C. Sorption of hydrocarbons and alcohols in addition-type poly (trimethyl silyl norbornene) and other high free volume glassy polymers. II: NELF model predictions. J. Membr. Sci. 2012, 405−406, 201−211. (33) Galizia, M.; Stevens, K. A.; Smith, Z. P.; Paul, D. R.; Freeman, B. D. Nonequilibrium lattice fluid modeling of gas solubility in HAB6FDA polyimide and its thermally rearranged analogues. Macromolecules 2016, 49, 8768−8779. (34) Minelli, M.; Cocchi, G.; Ansaloni, L.; Baschetti, M. G.; De Angelis, M. G.; Doghieri, F. Vapor and liquid sorption in matrimid polyimide: experimental characterization and modeling. Ind. Eng. Chem. Res. 2013, 52, 8936−8945. (35) Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions: The Solubility of Gases, Liquids, and Solids; Van Nostrand Reinhold Co., 1970. (36) Galizia, M.; Smith, Z. P.; Sarti, G. C.; Freeman, B. D.; Paul, D. R. Predictive calculation of hydrogen and helium solubility in glassy and rubbery polymers. J. Membr. Sci. 2015, 475, 110−121.

(37) Baschetti, M. G.; Doghieri, F.; Sarti, G. C. Solubility in glassy polymers: Correlations through the nonequilibrium lattice fluid model. Ind. Eng. Chem. Res. 2001, 40, 3027−3037. (38) Loianno, V.; Luo, S.; Zhang, Q.; Guo, R.; Galizia, M. Gas and water vapor sorption and diffusion in a triptycene-based polybenzoxazole: effect of temperature and pressure and predicting of mixed gas sorption. J. Membr. Sci. 2019, 574, 100−111. (39) Walsh, D.; Zoller, P. Standard Pressure Volume Temperature Data for Polymers; CRC Press, 1995. (40) Bevington, P. R.; Robinson, D. K.; Blair, J. M.; Mallinckrodt, A. J.; McKay, S. Data reduction and error analysis for the physical sciences. Comput. Phys. 1993, 7, 415−416. (41) De Angelis, M. G.; Merkel, T. C.; Bondar, V. I.; Freeman, B. D.; Doghieri, F.; Sarti, G. C. Hydrocarbon and fluorocarbon solubility and dilation in poly (dimethylsiloxane): comparison of experimental data with predictions of the Sanchez−Lacombe equation of state. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 3011−3026. (42) De Angelis, M. G.; Sarti, G. C.; Doghieri, F. NELF model prediction of the infinite dilution gas solubility in glassy polymers. J. Membr. Sci. 2007, 289, 106−122. (43) De Angelis, M. G.; Merkel, T. C.; Bondar, V. I.; Freeman, B. D.; Doghieri, F.; Sarti, G. C. Gas sorption and dilation in poly (2, 2bistrifluoromethyl-4, 5-difluoro-1, 3-dioxole-co-tetrafluoroethylene): comparison of experimental data with predictions of the nonequilibrium lattice fluid model. Macromolecules 2002, 35, 1276−1288. (44) Galizia, M.; De Angelis, M. G.; Yampolskii, Y. P.; Sarti, G. C.; Sarti, G. C. Sorption and transport of hydrocarbons and alcohols in addition-type poly (trimethyl silyl norbornene). I: Experimental data. J. Membr. Sci. 2011, 385−386, 141−153. (45) Bondar, V. I.; Freeman, B. D.; Pinnau, I. Gas sorption and characterization of poly (ether-b-amide) segmented block copolymers. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 2463−2475. (46) Li, P.; Chung, T. S.; Paul, D. R. Gas sorption and permeation in PIM-1. J. Membr. Sci. 2013, 432, 50−57. (47) Martín, A.; Peters, C. J. New thermodynamic model of equilibrium states of gas hydrates considering lattice distortion. J. Phys. Chem. C 2009, 113, 422−430. (48) Sandler, S. I. Chemical, Biochemical, and Engineering Thermodynamics; John Wiley & Sons, 2017. (49) Moon, J. D.; Galizia, M.; Borjigin, H.; Liu, R.; Riffle, J. S.; Freeman, B. D.; Paul, D. R. Water Vapor Sorption, Diffusion, and Dilation in Polybenzimidazoles. Macromolecules 2018, 51, 7197− 7208. (50) Kamiya, Y.; Terada, K.; Mizoguchi, K.; Naito, Y. Sorption and partial molar volumes of organic gases in rubbery polymers. Macromolecules 1992, 25, 4321−4324. (51) Ribeiro, C. P., Jr.; Freeman, B. D. Sorption, dilation, and partial molar volumes of carbon dioxide and ethane in cross-linked poly (ethylene oxide). Macromolecules 2008, 41, 9458−9468. (52) Ribeiro, C. P., Jr.; Freeman, B. D. Solubility and partial molar volume of carbon dioxide and ethane in crosslinked poly (ethylene oxide) copolymer. J. Polym. Sci., Part B: Polym. Phys. 2010, 48, 456− 468. (53) Zimm, B. H.; Lundberg, J. L. Sorption of vapors by high polymers. J. Phys. Chem. 1956, 60, 425−428.

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Modeling Gas and Vapor Sorption and Swelling in Triptycene-Based

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