A Simulation Study on OH-Containing Polyimide (HPI) and Thermally

Article pubs.acs.org/JPCB

A Simulation Study on OH-Containing Polyimide (HPI) and Thermally Rearranged Polybenzoxazoles (TR-PBO): Relationship between Gas Transport Properties and Free Volume Morphology Chi Hoon Park,† Elena Tocci,*,† Seungju Kim,‡ Apurva Kumar,§ Young Moo Lee,‡ and Enrico Drioli†,‡,∥ †

Institute on Membrane Technology, ITM-CNR, Via P. Bucci Cubo 17/C, Rende (CS) 87036, Italy WCU Department of Energy Engineering, College of Engineering, Hanyang University, Seoul 133-791, Korea § Indian Institute of Technology Guwahati, Guwahati 781 039, Assam, India ∥ Department of Chemical Engineering and Materials, University of Calabria, Via P. Bucci Cubo 42/A, Rende (CS) 87036, Italy ‡

ABSTRACT: Recently, high free volume polymer materials have been regarded as high potential candidates for gas transport/separation membranes, since the amount of free volume in polymeric membrane can improve the diffusivity and solubility of gas molecules. In this study, we focused on how local changes in polymer structure can affect the performance of a membrane at the molecular level. The transport behavior was theoretically analyzed, and then the differences in the amount and morphology of free volume were characterized. Finally, we suggested how the “evolution of microcavities” affects the gas transport properties of hydroxylcontaining polyimide (HPI) and thermally rearranged (TR) polymers. In particular, using image analysis, we intuitively demonstrate the morphological difference between HPI and TR polymers that have been indirectly explained by experimental analyses using a wide-angle X-ray diffractometer (WAXD) and positron annihilation laser spectroscopy (PALS). Solubility results using the grand canonical Monte Carlo (GCMC) method showed marginal improvement in thermally rearranged polybenzoxazoles (TR-PBOs) from its precursor HPI, which is in good agreement with the experimental tendency. Moreover, higher diffusivities but lower selectivities of TR-PBO models compared with those of HPI models were observed, as reported experimentally. The difference in gas transport abilities between HPIs and TR-PBOs originates from the difference in their diffusion behavior, and this is strongly related to the free volume amount and morphology of polymeric materials. In addition to the higher amount of total free volume in TR-PBO, our image analysis revealed that TR-PBO has a higher amount of interconnected “hourglass-shaped free volume elements”, which consist of larger and more elongated cavities with bottlenecks than the HPI model. In particular, the bottleneck diameters in the TR-PBO models are wider than those in the HPI models, enabling the larger gas molecules to diffuse through the cavities faster. However, the narrower and smaller bottleneck diameters in the HPI model can induce better selectivity for large gas molecules. polymers.8,15 More interestingly, TR-PBOs are prepared from aromatic polyimide containing ortho-positioned hydroxyl groups (HPIs) using thermal treatment, and the resulting rearrangement in the solid-state structure increases the free volume.8 As a result, the gas permeabilities of almost all TR polymer membranes are enhanced by at least 2 orders of magnitude compared with those of the original polymers and typical glassy polymers, with selectivities for a number of gas pairs that lie in Robeson’s upper bound.3 It should be noted that the permeability selectivity of TR-PBOs, particularly the ratio of permeability between smaller gas molecules and larger ones, is almost constant, in some cases reduced, owing to the lower and higher permeability increasing ratios of the smaller

1. INTRODUCTION The design of new functional polymeric membranes that exhibit permselective properties is an important target for membrane technology.1−4 Polymer materials having high free volume have captured the attention of many researchers for use as gas transport/separation membranes, since the diffusivity and solubility of the gas molecules in polymer membranes have a strong relationship with the amount of free volume.5−8 A wellknown method of improving the diffusivity−selectivity, as well as the diffusion coefficient of gas molecules in polymer membranes, involves an increase in free volume, such as in perfluoropolymers,9−11 poly(trimethylsilylpropyne) (PTMSP) and related polymers,12 and polymers with intrinsic microporosity,7,13,14 combined with the stiffness of the main chains. Among these, thermally rearranged polybenzoxazoles (TR-PBOs) have shown not only high permeability but also outstanding mechanical and chemical stability that surpass the limits of conventional © 2014 American Chemical Society

Received: November 26, 2013 Revised: February 10, 2014 Published: March 3, 2014 2746

dx.doi.org/10.1021/jp411612g | J. Phys. Chem. B 2014, 118, 2746−2757

The Journal of Physical Chemistry B

Article

wide-angle X-ray diffractometer (WAXD) analysis, interchain distances increase after a TR reaction, which can be correlated with the higher free volume of TR-PBO.8,15,17 In addition, PALS results have shown that the structure of the cavities is changed by the thermal reaction: from a unimodal cavity distribution to a bimodal distribution.8,15,17 Also, they have compared the positron annihilation laser spectroscopy (PALS) result with poly[1-(trimethylsilyl)-1-propyne] (PTMSP) and introduced the concept of an “hourglass-shaped cavity” enabling adequate separation (i.e., higher selectivity). However, it is still difficult to elucidate the exact shape of the cavities and how they affect gas transport properties only using PALS and WAXD. As a result, a deeper understanding is needed of how local changes in the membrane structure can affect the performance of the membrane at the molecular level. To address this issue, molecular dynamics (MD) simulation is well-known to be a powerful tool to enhance the understanding of nanoscale systems. In our previous study, the higher amount of free volume was already analyzed at the molecular level, and we also confirmed the conversion from a unimodal cavity distribution to a bimodal distribution through coalescence during the thermal treatment process.18 In addition to these general approaches, in this paper, we introduce a new concept in order to analyze the shape factor of free volume elements (i.e., cavities). In particular, we focus on the most important factor, the so-called “hourglass-shaped cavity”, explaining the extraordinary enhancement including selectivity in gas transport properties, in terms of how the differences in the free volume shape and distribution and the “evolution of microcavities” of HPI and TR-PBO membranes affect gas transport properties. Specifically, the novelty of this paper is a detailed analysis of the free volume considering the overall effects such as size, shape, size distribution and interconnectivity. These properties are related not only to permeability but also to the permselective behavior. Moreover, an in-depth analysis of transport behaviors such as solubility, diffusivity, and permeability coefficients was performed. Finally, the results from our molecular dynamics (MD) simulations were integrated with experimental findings already reported in the literature, yielding insightful indications. This approach has finally been able to establish the role of molecular structure on the permselective behavior of HPI and TR-PBO membranes.

and larger gas molecules, respectively. Nevertheless, the higher permeability enables TR-PBO to cross over the upper-bound limitation of conventional polymeric membranes (Figure 1a,b).6,16

2. SIMULATIONS 2.1. Molecular Dynamics Simulation. The Materials Studio (5.0) software package (Accelrys Software Inc., San Diego, CA)19 was used for this simulation. Solubility was predicted using the Sorption module in MS modeling, and diffusivity coefficients were calculated by the mean-square displacement of each gas molecule in the cells simulated by molecular dynamics. Amorphous polymer packings were constructed using the Theodorou/Suter method20,21 as implemented in the Amorphous-Cell module. The molecular dynamics (MD) simulations were performed with the COMPASS force field22−24 with two adjusted parameters: (1) oxygen atoms in PBO rings modified from o2a (oxygen, SP2, aromatic, in 5-membered rings) to o (oxygen, generic) and (2) nitrogen atoms in PBO rings modified from n2a (nitrogen, SP2, aromatic) to n3a (nitrogen, SP2, aromatic).18 COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) is an ab initio force field that has been parametrized and validated using condensed phase properties in addition to various empirical data for molecules in isolation.

Figure 1. Robeson’s upper bound relationship for (a) O2/N2 and (b) CO2/CH4 separation.6,17 tHPI: hydroxyl-containing polyimide by thermal imidization; aPBO: polybenzoxazole derived from HPI via azeotropic imidization; tPBO: polybenzoxazole derived from HPI via thermal imidization; Br-PC: (tetra)bromopolycarbonate; PI: polyimides; PDMS: polydimethylsiloxane; PET: poly(ethylene terephthalate); PES: poly(ether sulfone); CA: cellulose acetate; PSf: polysulfone; PS: polystyrene; PC: polycarbonate; PIM-1: polymers with intrinsic microporosities; PIM-7: catechol-containing PIM; PMP: poly(4-methyl-2-pentene); PPO: poly(phenylene oxide); PTMSP: poly(1-trimethylsilyl-1-propyne). Adapted with permission from ref 17.

The reason for the different gas transport properties is that TR-PBOs are diffusivity-enhanced polymers, with a small enhancement in solubility in comparison to the precursors HPI, as confirmed by a recent study.16 This is in agreement with previous studies,8,15,17 indicating that the outstanding properties are determined by the excess free volume in the polymer membrane matrix that can increase and improve molecular transport particularly in diffusion behavior. For example, using a 2747



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2.2. Preparation of Polymer Models. We generated single repeat units with assigned charge groups and subsequent energy minimization of hydroxyl-containing polyimides (HPIs) and thermally rearranged polybenzoxazoles (TR-PBOs) (Figure 2)

sphere is rolled through the amorphous periodic structure to generate a solvent-accessible, internal void volume that has a corresponding internal surface area. Owing to the size variation of the probe molecules, the same probe can only “see” a subset of the total free volume, which is termed as the “accessible volume”. The accessible surface area and accessible volume were calculated with the Free Volume utility of the Visualizer module of the Materials Studio (5.0) software package19 using the accessible mode together with the fine grid spacing specifications. This approach has also been used for the analysis of the free volume properties on the final equilibrated models and in ref 18. The calculation was performed by varying the probe radius from 1.0 to 2.0 Å in stepwise increase of 0.1 Å. We expect that a small increase of a probe should not strongly vary the accessible volume (also the accessible surface) and consequently its gradient. In addition, since the ratio of accessible volume (AV) to accessible surface area (ASA) indicates the shape of cavity as well as its size,18 large variation in the AV/ ASA ratio indicate a big “hole” inside the polymer box unlike a real membrane so that these samples should be not consider in the simulation. Accordingly, we finally selected three “good” models showing correspondence to the average tendency of the AV/ASA ratios out of 50 boxes of each configuration of random, 90° and 180° torsion restrictions. 3. The experimental density was reached by increasing the pressure using a set of NPT-MD runs (constant number of particles, pressure, and temperature). Simulated annealing NVT-MD runs at high temperatures followed by NVT dynamics at 303 K were used to further relax the polymer structure. Longer MD runs were performed for the final equilibration. We built in a total 300 3D models. 150 boxes were constructed for each polymer type, HPI and TR-PBO, considering 50 boxes for each torsion restriction. Nine boxes for HPI and nine boxes for TR-PBO were finally chosen for this study after the model validation mentioned. The validity check has been performed stating their stable total energy during the long MD runs and their densities deviation in comparison to the experimental densities (for tHPI density is 1.47 g/cm3, for aHPI is 1.49 g/cm3, for tPBO is 1.27 g/cm3, and for aPBO is 1.38 g/cm3).17 It is worth stating that small deviations in obtaining the experimental density can occur for glassy stiff-chain polymer materials,26−28 particularly if the models are rather large. The general simulation conditions used were: minimum image boundary condition to make the system numerically tractable and to avoid symmetry effects and a cutoff distance of 20 Å with a switching function in the interval 18.5−20 Å. Through the dynamics, the Andersen pressure control29 and the Berendsen temperature control method30 were used. The side length of the bulk models was about 40 Å for all HPIs and TR-PBOs. Details are summarized in Table 1.

Figure 2. Chemical structures of hydroxyl-containing polyimide (HPI) and thermally rearranged polybenzoxazole (TR-PBO).

The isolated initial chain configurations were then constructed with various torsional angles, random, 90° and 180°. HPI chains were constructed using a single atactic homopolymer chain with 33 repeat units (2246 atoms). Meta- and para-linked PBO monomers were prepared for the PBO chains. Then, a single atactic copolymer chain with 33 repeat units (2048 atoms) with the appropriate molar ratio of m-PBO:p-PBO of 5:5 was constructed using conditional statistics. For the packing procedure, a general methodology for generating realistic models of membranes was used:18,25 1. An “initial packing” was prepared, where three polymer chains of each polymer type (total 6738 and 6114 atoms for HPI and TR-PBO, respectively) and spacer molecules were packed in an amorphous cell at a lower density (∼10% of the final density), using the Theodorou/Suter method.20,21 Several spacer molecules were introduced to avoid the artifacts of catenated rings spearings. This allows for a much more homogeneously packed chain configuration as well as for a more uniform free volume distribution within the matrix. The spacers, 200 methanol and 600 argon molecules, were added randomly to the simulation box and were later removed in four steps. Each removal procedure was followed by energy minimization and NVT-MD (constant number of particles, volume and temperature) runs at 303 K combined with “scaling” of conformation energy terms and nonbonded interaction energy terms in the force field.26−28 2. After the removal of all spacers and at an intermediate stage of the “equilibration” procedure, the goodness of each model was analyzed in order to choose the candidate materials to be packed at the (final) experimental density by checking the variation of accessible volume (AV) to accessible surface area (ASA) ratio and its gradient with the increasing size of the probe radius.18,25 The probe Table 1. Simulation Model Details and Final Properties simulation models

simulated properies

polymer

monomer

mole ratio

no. of chains

no. of repeat unit

no. of atoms

density (g/cm3)

cell size (Å)

HPI TR

6FDA-APAF meta-PBO/para-PBO

N/A 5:5

3 3

33 33

6738 6144

1.490 ± 0.003 1.382 ± 0.002

44.050 ± 0.029 43.386 ± 0.020

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2.3. Calculation of Solubility. The solubility S were obtained from GCMC simulations by calculating the sorption isotherm obtained from every simulated box.19 In GCMC, two different algorithms, Metropolis and configurational bias, are applied. Metropolis has been preferred because it considers the adsorption of sorbate molecules in porous frameworks without providing them with any internal degrees of freedom. In this method, each conformation is treated as a rigid body. The sorbate molecules, such as those considered in this study, do not have a high degree of torsional flexibility. The Metropolis31 algorithm uses to accept or reject an insertion and deletion of a sorbate molecule. The probabilities of an addition and deletion of a sorbate molecule are given as

where nB is the number of B atoms at a distance r in a shell of thickness dr from atom A, NB is the total number of B atoms in the system, and V is the total volume of the system. 2.7. Free Volume Analysis. Fractional free volume (FFV) is the ratio of the free volume within a polymer and the specific volume of the polymeric material

FFV = Vf /Vsp

where Vf indicates the amount of free volume and the specific volume, Vsp, is defined as a reciprocal density. First, the theoretical fractional free volume (FFV) values were compared with that of Bondi’s group contribution method.17 According to the Bondi’s method, the amount of free volume can be estimated as

⎡ ⎛ −ΔU ⎞⎤ pV 1 exp⎜ Padd = min⎢1; ⎟⎥ ⎢⎣ Ns + 1 k bT ⎝ k bT ⎠⎥⎦ ⎡ Nk T ⎛ −ΔU ⎞⎤ Pdel = min⎢1; s b exp⎜ ⎟⎥ ⎢⎣ pV ⎝ k bT ⎠⎥⎦

Vf = Vsp − 1.3VvdW

(1)

(2)

2.4. Calculation of Diffusion Coefficients. In order to enhance the sampling efficiency, ten gaseous molecules of the same kind were inserted into each polymer structure and the polymeric boxes were then equilibrated. After equilibration, the NPT MD simulation was performed at 300 K for 5 ns with a time step of 1 fs (1 fs = 10−15 s). Diffusion coefficients were calculated from the slope of the plots of the mean-square displacements of gases versus time using the Einstein relation Dα =

1 d lim ⟨[ri(t) − ri(0)]2 ⟩ 6Nα t →∞ dt

(3)

3. RESULTS AND DISCUSSION 3.1. Gas Transport Parameters. Table 2 contains the calculated values for the diffusion coefficient D, the solubility S, and the permeability coefficient P for five light gases (H2, N2, O2, CO2, and CH4) in comparison with the experimental data16,17,36 and the simulated results for CO2 and CH4 (the authors here use a probabilistic approximation of molecular dynamics algorithm).37 Also, the experimental results refer to aPBO and tPBO, whose polyimide precursors were prepared using different synthetic methods; they were thermally converted from HPIs prepared by thermal imidization and azeotropic imidization, respectively.17 3.1.1. Solubility. The calculated solubilities of HPI and TRPBO were investigated with the grand canonical Monte Carlo

where Nα is the number of diffusing molecules of type α, ri(0) and ri(t) are the initial and final positions, respectively, of molecules (mass centers of particle i) over the time interval t, and ⟨|ri(t) − ri(0)|2⟩ is the mean-square displacement (MSD) averaged over the possible ensemble. The Einstein relationship assumes Brownian dynamics for the diffusing particles.34 2.6. Radial Distribution Function. The radial distribution function (RDF) indicates the local probability density of finding B atoms at a distance r from A atoms averaged over the equilibrium density, as follows: RDF(r ) =

nB /4πr 2 dr NB/V

(6)

where the van der Waals volume VvdW is calculated using a group contribution method, and a universal “packing coefficient”, equal to 1.3, is used to convert the van der Waals volume of the repeat unit into the “occupied” volume. In our simulation, the van der Waals volume VvdW was calculated and averaged directly from the simulated structure. As mentioned above, we carried out more specific free volume analysis, considering the size effect of gas molecules, using the Visualizer module of the MS software package.19 First, the van der Waals surface is defined as the surface that intersects with the vdW radii of the atoms in the given structure, where the volume on the atom side of the surface (occupied volume) is used as the van der Waals volume. Based on the van der Waals surface, the accessible solvent surface is also defined as the surface that is the locus of the probe center as the probe rolls over the scaled vdW surface. This surface describes a space which could, in principle, be occupied by a probe of the given radius and is only defined over externally accessible regions, where the volume on the side of the surface without atoms (the free volume) is used as the accessible free volume. 2.8. Image Analysis. Free volume images for analysis were captured from the simulated 3D models sliced into nine crosssectional images along the x-, y-, and z-axes. The slices were then separated and each slice was investigated using the image processing in MatLab 7.0. program, transforming the models as a matrix with fraction pixel values. The area and length of each accessible solvent free volume element were measured and converted to angstrom scale.35 In particular, the longer dimensions, threshold dimensions, and area of the surfaces were calculated, which will be specifically discussed in section 3.2.

where U is calculated from the sum of nonbonded (i.e., Coulombic and van der Waals interaction) energies and Ns is the number of sorbate molecules. The addition is accepted if the energy change ΔU is negative or if the Boltzmann factor exp(−ΔU/kT) is greater than a random number generated between 0 and 1. The solubility S is calculated from the number of gas molecules loaded into the polymer 3D model under 1 bar and 35 °C conditions to compare with the experimental data obtained under the same conditions. The isotherms were obtained in a grand canonical ensemble with periodic boundary conditions. The isosteric heat, Q, of a component is defined as the partial molar enthalpy of the sorbate component in the reservoir minus that in the framework.19,32,33 At equilibrium ⎡ d(ln p) ⎤ Q SF = RT ⎢ ⎥ ⎣ d(ln T ) ⎦

(5)

(4) 2749



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Table 2. Gas Permeation Properties: Permeabilities, Diffusivity Coefficients, and Solubilities polymer system HPI

TR-PBO

tHPI16

tHPI36 b

tHPI37

tPBO16

aPBO16

tPBO17

aPBO17

tPBO37

transport propertya S D P S D P S D P S D P S D P S D P S D P S D P S D P S D P

H2

CO2

simulation results 0.180 22.1 9076 0.420 2150 12.2 0.174 22.6 31 133 1.98 7115 59.0 experimental results 0.630 6.70 42.3 1.13 35.1 10.0 0.098 30.1 331 0.429 42.8 17.0 NA 28.6 NA 1.2 NA 45 1.08 16.5 2010 165 2856 3575 0.70 14.2 443 21.4 408 398 NA NA 397 28.4 4194 4201 1.00 23.3 311 13.0 408 398 NA 35.9 NA 37 NA 1748

O2

N2

CH4

4.33 15.7 89.3 4.73 204 1268

2.90 3.40 13.0 3.80 101 507

5.40 0.243 1.73 8.97 2.80 33.0

1.07 1.84 2.6 NA NA 3.73 NA NA NA 3.75 158 778 1.53 40.3 81.1 2.93 283 1092 1.37 44.9 81 NA NA NA

1.02 0.297 0.4 NA NA 0.550 NA NA NA 2.22 53.2 155 0.99 14.6 19.0 1.57 137 284 NA 31.3 19 NA NA NA

1.60 0.039 0.082 1.75 0.091 0.210 1.95 0.45 1.1 5.90 10.4 80.8 4.96 1.84 12.01 8.89 12.9 151 4.34 2.1 12 5.59 8.6 63.0

a S, solubility under 1 bar and 35 °C conditions (cm3 (STP)/cm3 atm); D, diffusion coefficient (10−8 cm2/s); P, permeability (barrer); 1 barrer = (10−10 cm3 (STP) cm/cm2 s cmHg). bExperimental data measured at 25 °C.

Figure 3. Simulated and experimental solubility data from the literatures of gases in (a) HPI and (b) TR-PBO.

method (GCMC) using the Metropolis algorithm. Results are plotted against the critical temperatures of the gases (Figure 3a,b), in comparison with the reported solubilities.16,17,36 Both TR-PBO and HPI show a linear relationship with critical temperature Tc, demonstrating similar tendencies as typical polymeric membrane materials4 if they do not undergo a strong specific interaction with penetrants. The simulated trend is in good agreement with that reported experimentally.16,17,36,37 Concerning HPI, N2 and O2 show differences between predicted and experimentally determined solubilities, fluctuating between 3 and 4 times. CH4 is slightly higher whereas H2 and CO2 show similar solubilities that are in the range of experimental data: CO2 is between two experimental data that show also discrepancies (3 times)16,36 and at almost the same value of the simulated one.37 H2 is lower (about 3 times) then the experimental value16 and almost the double of the simulated one.37 PBO demonstrates better agreement, except for H2 which is underestimated compared with the experimental results. One reason for the differences could be explained by considering that the relatively small dimension of the boxes does not permit a correct statistic to be reached. However, considering that the accuracy of the predictions of the S values is not very precise due to their crucial sensitivity to many factors and the

appropriate choice of force field as Yampolskii summarized38 and other authors stated,39 our simulation results can be concluded to be in the acceptable range for further discussion in this paper. The general consideration is that in both simulations and experiments, the S values are not significantly improved after the TR reaction. They remain approximately constant with an enhancement of up to ∼3 times for aPBO and ∼2−4 times for tPBO from the solubilities in HPI. The reason for the partial increase in solubility has been previously ascribed,16 and our simulation confirms two opposite effects. On one hand, the increase is due to the larger free volume elements of the rearranged polymer. On the other hand, the dual-mode coefficients indicate that the Langmuir affinity parameter (i.e., the affinity between polymer and gas molecules) decreases slightly as the thermal rearrangement proceeds, due to the lack of a side chain or group interacting with gas molecules in TR-PBO, compared with the various side groups (carbonyl and hydroxyl groups) in HPI. Accordingly, the latter effect offsets the increasing solubility by the former one, so that the overall consequence is a gradual increase in the amount of gas sorption as HPI is converted to TR-PBO. 2750



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TR-PBO membranes: D values decrease with increasing molecular size of gases, indicating that they have the ability to sieve penetrant molecules based on their size. This reconfirms that HPI and TR-PBO, like other glassy polymers, are categorized into diffusion-selective membranes.6 The TR-PBO models show higher diffusivities than the HPI models, where the larger size gas molecules (here, we use kinetic diameters as the sizes of gas molecules) have a higher increasing ratio in diffusivities than the smaller gas molecules. In summary, TR-PBO can have high permeability mainly due to its much larger diffusivity than that of its parent HPI. In our HPI models (see Figure 4a), the diffusivity values are overestimated for relatively small gases such as H2, O2, and N2, compared with the experimental diffusivity values. The errors in the diffusion coefficients in the HPI range are around 1 order of magnitude within their respective experimental values. The largest variation is shown for H2, which is much faster (over 1 order of magnitude). The reason for the discrepancies in the HPI models might be the additional free volume in the 3D models, which will be discussed further in section 3.2. Concerning TR-PBO (Figure 4b), the simulated diffusivities correspond well with those of experimental ones, apart from H2, which shows the same overestimation, and for CO2, which is lower as expected. The CO2 behavior could be ascribed to the rigidity of the matrix together with the typical behavior of small and less-soluble penetrants in glassy polymeric membranes:39 the penetrant motion is strongly restricted by the immediate environment, and the simulations do not always sample large enough time scales to provide reliable information. The difficulties lie in the extremely broad distribution of gas molecule jump rates. It is known that the diffusion of a penetrant in a glassy polymer involves occasional jumps between cavities through the opening of a channel. Jumps are rare events, and the time between them is much shorter than the times governing matrix relaxation processes.40 In the literature, it has been reported that predictions are much less reliable for diffusion coefficients than for solubility, and it is commonly accepted that if the simulated D and the experimental D values differ by a factor of 2−5, then the prediction is considered to be successful.38 Accordingly, the difference between the predicted and experimental diffusion coefficients becomes significant, particularly in HPI models with lower experimental D values and free volume, compared with TR-PBO models with higher experimental D values and free volume that compensate for the simulation errors mentioned above. However, if we relate log D, both theoretically and experimentally, with the kinetic diameter of the gas molecule for HPI and TR-PBO, useful information about their size-sieving properties can be obtained. Figure 4 shows that HPIs show bigger differences in diffusivities in function of the size of the gases, compared to TR-PBO, and therefore have superior molecular size-sieving properties. Consequently, the diffusivities of larger gas molecules (i.e., N2 and CH4) increase more than those of small gas molecules by the thermal conversion from HPI membranes to TR-PBO membranes. 3.1.3. Permeability. As previously mentioned, it is wellknown that TR-PBO membranes have higher gas permeability than their precursor HPIs. A detailed analysis of both diffusion and sorption behaviors indicates that the permeability changes in different ways according to the size of the gas molecules. In particular, improvement in the diffusivities of larger gas molecules (i.e., N2 and CH4) is more significant than that of small gas molecules in TR-PBO membranes. On the contrary,

This tendency is evident in our simulated results: a measured increase of sorption from the HPI to the TR-PBO models. A detailed analysis of CO2 sorption simulation results indicates that the isosteric heat in the HPI models is ∼7.2 kcal mol−1 and that value is reduced to ∼6.4 kcal mol−1 in the TR-PBO models, indicating the high affinity of carbon dioxide to the PI membrane. Consequently, the solubility results indicate that the difference in gas transport abilities of HPI and TR-PBO originates from the difference in their diffusion behavior, and TR-PBO has high permeability mainly due to much larger diffusivity than the parent HPIs. 3.1.2. Diffusion Coefficient. Figures 4a,b show the diffusion coefficient results for the various gas molecules used in this

Figure 4. Simulated and experimental diffusion coefficient data from the literatures of gases in (a) HPI and (b) TR-PBO.

study through both HPI and TR-PBO membranes compared with experimental data.16,17,36,37 Gas permeation properties of the membranes were characterized using a constant-volume/ variable-pressure method, the so-called high vacuum time lag method, and diffusion coefficient values were obtained from values derived from time lag experiments.17,36 Sorption experiments were also performed; in this case, diffusivity was calculated from the direct ratio between permeability and measured solubility.16 The MD diffusion coefficients followed a trend similar to that of the experimental data for both HPI and 2751



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improvement in the solubility of all gas molecules is smaller than in the diffusivities, but TR-PBO still shows higher values than HPIs, particularly for CH4 and CO2 solubility. Consequently, owing to the synergistic effect of improvements in both diffusivity (dominant) and solubility (adequate; not so great, but enough), TR-PBO polymeric membranes can have much better gas transport performance than those of their parent HPI, particularly for large gas molecules and can have lower size-sieving effect. To confirm this interesting behavior, selectivity, one of the important parameters describing gas transport performance, was calculated in Table 3. HPI has higher selectivity than Table 3. Permeability Selectivity Pαi/Pαj polymer system

O2/N2

H2/O2

H2/N2

HPI tHPI16 tHPI36 TR-PBO tPBO16 aPBO16 tPBO17

6.88 6.50 6.78 2.50 5.02 4.27 3.85

24.06 13.54 11.46 5.61 3.67 5.03 3.84

165.46 87.97 77.73 14.05 18.43 21.47 14.77

Figure 5. Torsional angles between the imide ring and the orthopositioned hydroxylphenylene ring in HPI chains and between the phenyl ring and a benzoxazole ring in TR-PBO chains.

study previously reported,15 owing to the multiplicity of conformations that can be obtained with the statistical results averaged over three polymer chains. The simulations reveal that phenylene rings are offset with oxazole units, since HPIs are more flexible than TR-PBOs. On the other hand, TR-PBO has a sharp bimodal profile of distributions due to the rigidity of the coplanar structure of the TR-PBO ring. 3.2.2. Amount of Free Volume in HPI and TR-PBO Models. Regarding the analysis of free volume, it is well-known that free volume morphology, considering overall effects such as the size, shape, size distribution, and interconnectivity, has a dramatic effect on membrane properties. Additionally, the amount of free volume alone does not provide information regarding the connectivity and tortuosity of the pores, which are important issues for describing size sieving effects.7 The first evidence and the main cause of their higher permeability in the thermally rearranged materials is that TR-PBOs have a higher fractional free volume (FFV) than their precursor HPIs regardless of synthetic route. The fractional free volume was simulated and compared with the experimental data calculated using the conventional Bondi’s method (Table 4). As

TR-PBO in all gas pairs regardless of their size difference. However, when the difference in the size of the chosen gas pair becomes larger and larger, the selectivity in the TR-PBO model increases much less than in HPI models. For example, O2/N2 selectivities having the smallest size difference between the gas pairs show 6.88 in HPI and 2.50 in TR-PBO. However, with the increasing size difference of gas molecules, H2/O2 selectivity increases up to 24.06 in HPI, which is a much higher increasing ratio than TR-PBO which increases only up to 5.61. If we replace O2 with N2, which has a larger size, this tendency becomes more apparent: H2/N2 selectivity is 165.46 in HPI but only 14.05 in TR-PBO. This gas transport behavior (i.e., higher increasing ratio of diffusivities in larger gas molecules) is explained in terms of the difference in the free volume morphologies between HPI and TR-PBO (i.e., hourglass-shaped cavities).8,15 However, it is still insufficient to elucidate clearly the decrease in selectivity for TR-PBO membranes. To address this issue, we analyzed and compared the distribution and morphology of free volume (or cavity) in HPI and TR-PBO models. 3.2. Free Volume Analysis. 3.2.1. Relationship between Structural Properties and Free Volume of the HPI and TR-PBO Models. The most significant consequence of the thermal rearrangement of HPI to TR-PBO polymers is the change in physical properties such as the fractional free volume (FFV). Another important effect is the increase in chain rigidity (Figure 5) because “bad” packing produces an increase in free volume and the shape persistence is a key factor in the enhancement of gas selectivity.1 The most interesting current materials, such as PIM and its derivatives, have been obtained using the same strategy.5,7 Figure 5 shows the torsional angles between the imide ring and the ortho-positioned hydroxyl-phenylene ring in HPI chains, and between the phenyl ring and a benzoxazole ring in TR-PBO chains. The simulations were performed by counting all of the torsional angles in both simulated amorphous polymer cells. The range of torsional angles is broadly distributed and rather complex for HPI, compared with the monomer model

Table 4. Fractional Free Volume (FFV) by Bondi’s Methoda HPI PBO

simulatedb

experimentalc

0.214 0.252

0.19 (tHPI), 0.17 (aHPI) 0.28 (tPBO), 0.22 (aPBO)

a

FFV is calculated using the equation of Bondi’s method: FFV = (V − 1.3Vw)/V, where V is the molar volume of the polymers (cm3/mol) and Vw is the van der Waals molar volume. bVw is the simulated van der Waals molar volume in this study, averaged on different models. c Vw is calculated by group contribution method.17

shown experimentally, differences in FFV are visible between HPI and TR-PBO. Our simulated models show similar values for TR-PBO: the simulated value is in between the two experimental values. The FFV of HPI is higher (approximately 10−20%) than both experimental FFVs. The comparison of these FFV values with those of ultrahigh free volume membranes indicates that TR-PBO has a comparable FFV.41−45 In particular, the values are comparable to that of amorphous fluoropolymers such as the Hyflon AD series,41 AD 60X (FFV of 0.34) and AD80X (FFV of 0.37), of 2752



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Figure 6. Simulated free volume distribution image.

highly branched polyacetylenes with extremely bulky side groups such as poly[1-phenyl-2-[p-(triphenylsilyl)phenyl]acetylene] (PPhSiDPA) (FFV of 0.31) and poly[1-phenyl-2-[p-(triisopropylsilyl)phenyl]acetylene] (PPrSiDPA) (FFV of 0.29), although they are lower than that of the ultrahigh free volume polymers such as poly[1-(trimethylsilyl)-1-propyne] (PTMSP) (FFV of 0.47) and polymers with intrinsic microporosity (PIM-1)46 (FFV of 0.45) and of perfluorinated Teflons such as AF1600 (FFV of 0.45) and AF2400(FFV of 0.47). Figure 6 shows the different amounts of free volume in both membranes: the TR-PBO model shows larger regions of free space. We have previously reported18 that HPIs and TR-PBOs have very different free volume morphologies. In particular, HPIs have a unimodal distribution of free volume areas of relatively narrow size. On the contrary, TR-PBO shows a bimodal cavity distribution; i.e., they have two qualitatively different free volume phases: one composed of relatively small isolated holes and one of interconnected micropores, which provide exceptional diffusion pathways. Since the FFV value only gives the bulk property of free space in the membrane, we went one step further in this study: to investigate how the difference in free volume morphology affects the gas transport behavior. When only using the amount of free volume, it is difficult to find the reason for the typical selectivity of TR-PBO membranes in comparison with the starting HPI membranes (i.e., why larger molecules have much more improved transport performance after a TR reaction than smaller molecules). Accordingly, we expected that their morphological difference could play a key role in the peculiar transport behavior in TR-PBO. It should be noted that, in this paper, we concentrate our attention only on the effect of the free volume morphology on gas transport without considering their rotational/vibrational degree of freedom and their activation energy during the jumps (i.e., entropic selectivity of HPI and TR-PBO models).47,48 3.2.3. Ratio of the Free Volume Amount to Its Surface Area in HPI and TR-PBO Models. First, we analyzed FFV and the ratio of free volume to its surface area. The ratio of the volume to the surface (V/S), indicated as the “shape factor”, gives the variations in the contour of the free volume. Modification of the internal surface area has already been proposed experimentally by Han17,49 and in our previous paper,18 where we analyzed the modification of the internal surface through the effects of temperature. Here, the volume/surface ratio of both HPI and TR-PBO at 300 K was analyzed with various probe sizes according

Figure 7. Fractional free volume (FFV) and ratio of free volume and its surface area determined using the accessible solvent method with probe radius calculated from kinetic diameters of gas molecules.

to the kinetic diameter of gas molecules, as shown in Figure 7, using the accessible solvent surface and volume calculation tools in the Accelrys software.19 The probe represents a solvent molecule (or a gas molecule) with a certain radius, and “accessible” means that isolated free volume elements (FVEs) are excluded in calculation. Both HPI and TR-PBO show very different behavior. The accessible FFV decreases with increasing probe radius, suggesting that the size of the free volumes with larger radii are reduced in number and that larger gas molecules have less space to transport through and lower diffusivity. Compared with the HPI models, the TR-PBO model has higher accessible FFV throughout the entire range of probe radii, which can explain its higher diffusivity. The higher volume/surface ratio (V/S) indicates (1) a higher volume and/or (2) a smaller surface; for example, the coalescence of two small free volume holes into a larger one induces an increase in the V/S ratio by the reduction of the surface area of this hole.18 As indicated in Figure 7, HPI models have lower V/S values than TR-PBO models in the whole range of probe radii, and specifically their difference becomes larger with increasing probe radius. As a result, this indicates that the free volume in TR-PBOs consists of bigger and/or connected 2753



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Figure 8. Schematic diagram of free volume shape analysis.

cavities than those in HPIs, which more significantly affects larger-size gas molecules such as N2 and CH4. 3.2.4. Image Analysis for the Cavity Shapes in the HPI and TR-PBO Models. A second procedure to explore the different free volume morphologies, specifically cavity shapes, is to use an image analysis approach. The information from our previous free volume analysis included that the dimensions of holes were larger and interconnected, but any indication of the shape was not given.18 A peculiar free volume morphology with hourglassshaped cavities of TR-PBOs has already been suggested.8,48 However, based on the concept of free volume in a polymer (i.e., a space in a polymer excludes the space occupied by the polymer chains), since it has very complex shapes such as “neither sphere nor column”, it is almost impossible to demonstrate the cavity shapes experimentally, particularly at the angstrom scale. In reality, the shape of a hole is quite complex, as shown in Figure 8. In order to simplify this matter, we assumed that the free volume shape could range between an ideal spherical profile and an ideal elliptical profile (see Figure 8). The shapes have been captured from the image analysis performed on the 3D models sliced, considering the longer (DL), the shorter (Ds), and smallest threshold diameters (Dth) in each free volume element. Then, we assumed an ellipsoid with the same longer diameter and total volume as the free volume element, and we calculated the eccentricity of the ellipsoid, i.e., length ratio of shorter (Ds) and longer (DL) diameters: Ds/DL, as shown in Figure 9. This ratio can give information about the shape of free volume elements, which are elongated or otherwise more spherical.35 The range of the ratio falls between the value “1”, for Ds = DL, which indicates a sphere, and the value “0”, for Ds almost zero or very small, which indicates an ellipsoid. Eccentricity indicates that the highest peaks of both HPI and TR-PBO are in the range of 0.8−1.0; in particular, HPI is higher than TR-PBO only in this range. This means that both models have a large amount of spherical free volume elements. From the figure, it is also evident that the TR-PBO models have a higher fraction of elongated free volume elements than HPI, and they only possesses the most elongated FVEs which can be found in the range of 0−0.2. Moreover, TR-PBO models have

Figure 9. Eccentricity (Ds/DL) of the ellipsoids calculated from the FVEs in HPI and TR-PBO models, where the value “1” indicates a sphere and the value “0” indicates an ellipsoid.

the higher values in the region between 0.2 and 0.8. Considering our previous conclusion from the V/S ratio analysis that TR-PBO has larger and/or connected FVEs, this eccentricity result demonstrates that those FVEs have a stretched shape rather than a large “big sphere”. In order to relate only specific features of the structure with selectivity, i.e. the ability of polymer materials to be selective for the size and shape of the gas molecules, we defined the smallest bottleneck in FVEs for each polymer as a “least threshold diameter” (Dth), as shown in Figure 10. First, all the Ds diameters in each FVE were counted and the smallest bottleneck in the same FVE was considered as the “least threshold diameter” (Dth). Then, we calculated the ratio Dth/Ds, i.e., the ratio of the least threshold diameter (i.e., narrowest width) to the smaller average diameter (i.e., average width) of the FVE (Figure 10). The Dth/Ds ratio only indicates a unitless difference in two diameters of the same FVE distributed through the membrane material. Figure 10 indicates that the highest peaks of both HPI and TR-PBO are in the range of 0.4−0.6. This means that 40% of the FVEs have an average width dimension almost double that 2754



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Figure 10. Ratio of least threshold diameter (Dth) and smaller diameter (Ds) of ellipsoids calculated from FVEs. Figure 11. Distribution of least threshold diameters in the simulation models.

of the least threshold diameter. A much more interesting point is the analysis of the ratio in the region 0.2−0.4 because the difference between the average and the narrowest widths is the highest. Compared with HPI, TR-PBO has a higher fraction in the range of small Dth/Ds ratios, which means that FVEs in TR-PBO have relatively larger average widths in comparison with their bottlenecks. Consequently, after considering both eccentricity and Dth/Ds ratio results, we can reach the conclusion that TR-PBO has elongated cavities with larger average widths, but narrower bottlenecks, than the average width of the cavity. This exactly demonstrates the experimentally assumed “hourglass-shaped cavities” in TR-PBO. These findings lead to the question of how to relate selectivity to morphology, i.e., the ability of polymer materials to be selective for the size and shape of the gas molecules. In the above discussion, we focused on the shape of FVEs and showed that the TR-PBO model has more FVEs having bottlenecks that are relatively narrower than the average width compared with HPI model. However, it should be noted that this result does not mean that the bottleneck of the cavities in TR-PBO is narrower than those in HPI, because the Dth/Ds ratio is a unitless parameter. The size-sieving effect is based on the assumption that the size of the smallest bottleneck in FVEs can dominantly affect the diffusivity−selectivity of the gas molecules. For this reason, we measured the absolute sizes of bottlenecks in both membranes. The absolute values of the least threshold diameters have been calculated first checking all the threshold diameters in each free volume elements per packing model as indicated by the image analysis and then counting the number of times each threshold appear. The comparison of the absolute values of the least threshold diameter of TR-PBO and HPI (in Figure 11) indicates that the bottleneck diameters of the TR-PBO models are wider than those of the HPI models. In particular, in the case of HPI, Dth values concentrate around 2 Å and show a sharp distribution; TR-PBO has a broader distribution and slightly higher count numbers in the range greater than 3 Å. In Figure 11, the dimensions of gases have also been inserted. Compared with these sizes, wider bottlenecks in the FVEs are advantageous for the diffusion of large gas molecules. This could explain the reasons that (1) the diffusivities of larger gas molecules (i.e., N2 and CH4) increase more than those of smaller gas molecules in TR-PBO membranes, due to

the increased size and the elongated/interconnected free volume, and (2) the permeability selectivity of TR-PBOs for coupled larger and smaller gas molecules is almost constant or reduced because both larger and smaller molecules could jump through the bottlenecks, lessening the size-sieving effect of the precursor HPI. On the other hand, the unimodal distribution of relatively small sizes of FVEs and the smaller bottleneck diameters of the HPI models can explain (1) the lower diffusivity/permeability and (2) the better/higher selectivity of HPI compared with TR-PBO.

4. CONCLUSIONS We investigated how the difference in the free volume morphology of HPI and TR-PBO polymers affects the transport properties of gas molecules of various sizes. Solubility simulation results using the GCMC method indicate that HPI and TR-PBO have a linear relationship with critical temperature (Tc) and that their values are not significantly improved after a TR reaction, which is in good agreement with experimental tendencies. Diffusivities of TR-PBO models are higher than those of HPI models, as reported experimentally. In particular, larger-size gas molecules have a higher increasing ratio in diffusivities, indicating that a size-sieving effect is significant in HPI models. As a consequence, TR-PBO has high permeability, mainly due to its much larger diffusivity than that of its parent HPI. Transport behaviors of HPI and TR-PBO models are strongly related to the amount and morphology of the free volume. The higher diffusivity and permeability of TR-PBO is due to the higher amount of interconnected free volume, which consists of larger and/or connected cavities. In particular, we focus on the shape effect of free volume elements using image analysis; the TR-PBO model has a higher fraction of elongated free volume elements than the HPI model, with bottlenecks, which strongly supports the experimental assumption of socalled “hourglass-shaped cavities” in TR-PBO, explaining its high permeability. However, the bottleneck diameters of the TR-PBO models are wider than those of the HPI models, and this is advantageous for the diffusion of large gas molecules. On the other hand, HPI can have better selectivity for large gas molecules, owing to the narrower and sharply decreased bottleneck diameters in FVEs. 2755



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AUTHOR INFORMATION

Corresponding Author

*Tel +39-0984-49-2038; e-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the framework of the NEMOPUR project (FP7Marie Curie Actions ITN: grant no. 214226-2). The “Ministero degli Affari Esteri, Direzione Generale per la Promozione e la Cooperazione Culturale” is gratefully acknowledged for the partial financial support of the project “New highly innovative membrane operations for CO2 capture at medium and high temperature: Experimental preparation and characterization, theoretical study on elementary transport mechanisms and separation design” (2013K1A3A1A25037074) cofunded in the framework of a bilateral agreement between MAE and MOST. Y.M.L. and E.D. acknowledge the Nano-Material Technology Development Program (2012M3A7B4049745) of the National Research Foundation and Korea CCS R&D Center (KCRC) funded by the Ministry of Education, Science and Technology in Korea for the partial support of this work.

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A Simulation Study on OH-Containing Polyimide (HPI) and Thermally

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