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During the last few decades, important progress has been achieved in both materials research and processing of device technologies. Individually, these efforts have already had an impact on process design and the modelling and process performance. In particular, membrane processes are change agents that promote a more sustainable and higher global standard of living. The development of advanced membrane technologies and the industrial application of polymeric membranes require good understanding of the materials properties and their transport mechanisms, as well as the realisation of innovative functional materials with enhanced properties. Thermally rearranged polymers show impressive performance for important applications such as gas separation. The reason lies in their microporous structure with enhanced rigidity. In addition, owing to their good scalability, these materials will enable the next step in molecularly selective membranes. The rapid improvement in numerical simulations and modelling methods and algorithms, together with the continuous increase in computer speed and increasing improvements in supercomputer architectures, accelerate research and development efforts by contributing to the improvement of materials research and process design and optimisation.

The sustainable development of chemical and related process-oriented industries critically depends on the development of new innovative processes that use materials and energy more efficiently.1,2 

The development of advanced membrane technologies and the industrial application of polymeric membranes require good understanding of the materials properties and their transport mechanisms, as well as the realisation of innovative functional materials with enhanced properties. Those different aspects have been discussed in detail in previous lectures and have been recently reviewed in the literature.3–5 

Membrane separation technologies have profited from the progress in materials research and processing for device technologies. In particular, membranes for gas separation made mainly of polymer materials, owing to their easy processability and good mechanical properties, compete with other separation processes such as cryogenic distillation and adsorption due to their easy operational handling, relatively small size, low energy consumption, and space efficiency.

Breakthroughs in the development of highly permeable materials for membranes are essential. In this context, microporous polymers are considered efficient membrane materials and good candidates to overcome the well-known Robeson’s upper bound.6  Specific tailoring of the molecular structure can be regarded as a viable approach to obtain improvements on membrane permselectivity due to (i) the loss of inter-segmental packing with a simultaneous inhibition of the intra-segmental (backbone) mobility, and (ii) the weakening of inter-chain interactions (reduction of charge transfer complexes).7 

Recent progress in the development of microporous polymers as gas separation membranes has been achieved by improving the rigidity of the entire polymer structure to improve the separation performance, since a rigid polymer structure enhances the separation properties and durability of the membranes used for gas separation and storage materials.8 

Thermally rearranged (TR) polymers are an example of microporous polymers with high permeability and selectivity for the separation of gas mixtures.9–12  In particular, they have shown outstanding molecular and ionic transport, as well as separation performance, beyond the limits of more conventional polymers.13–16 

An example of process design using TR polymer membranes can be found in the work of Dong et al. from 2015.17 

TR-polybenzoxazole (PBO) polymers are examples of novel membrane materials with high free volume elements and narrow cavity size distribution based on rigid microporous structures. TR-PBO polymers are glassy aromatic polymers with heterocyclic rings prepared by an in situ thermal treatment (350–450 °C) of hydroxyl-polyimide (HPI) precursors with functional groups at the ortho-positions. Since 2007, Lee’s group has been studying the thermal conversion mechanism of TR polymers for their application as membrane materials.16–24  A hydroxyl-polyimide is prepared by a conventional polycondensation reaction of dianhydrides and diamines with hydroxyl functional groups, obtaining a hydroxyl-containing poly(amic acid) (HPAAc) (Figure 1.1). Then, the HPAAc is converted to hydroxyl polyimide by various imidisation methods, such as thermal, chemical, and azeotropic imidisation, based on the dehydration of the poly(amic acid) structure. The final thermal rearrangement of the hydroxyl polyimide into TR-PBO is carried out at a temperature of 350–450 °C under an inert atmosphere after membrane formation.

Figure 1.1

(a) Atomistic model and (b) chemical structure of hydroxyl-containing polyimide (HPI) and thermally rearranged polybenzoxazole (TR-PBO). Reprinted with permission from Chi Hoon Park, Elena Tocci, Young Moo Lee, Enrico Drioli Thermal Treatment Effect on the Structure and Property Change between Hydroxy-Containing Polyimides (HPIs) and Thermally Rearranged Polybenzoxazole (TR-PBO) The Journal of Physical Chemistry B 2012, 116(42), pp 12864–12877. Copyright © 2012 American Chemical Society.

Figure 1.1

(a) Atomistic model and (b) chemical structure of hydroxyl-containing polyimide (HPI) and thermally rearranged polybenzoxazole (TR-PBO). Reprinted with permission from Chi Hoon Park, Elena Tocci, Young Moo Lee, Enrico Drioli Thermal Treatment Effect on the Structure and Property Change between Hydroxy-Containing Polyimides (HPIs) and Thermally Rearranged Polybenzoxazole (TR-PBO) The Journal of Physical Chemistry B 2012, 116(42), pp 12864–12877. Copyright © 2012 American Chemical Society.

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Another strategy used in the thermal process is the introduction of thermally labile molecules in a cross-linkable polyimide to prepare highly permeable polyimide membranes by thermal decomposition of the labile units in the solid state.26–28  Furthermore, TR copolymers have also been investigated in terms of the concerted effects of different TR polymers with several glassy polymers.29–34  These polymers show outstanding physical properties and high permeability, exceeding the limits of more conventional polymers due to their unusual microstructure, a phenomenon that has been explained as the result of the modifications in the polyimide chain during rearrangement into the solid state structure. Such a process leads to the formation of rigid rods with a concomitant conformation randomisation resulting from the formation of meta- and para-linked chains.16 

Moreover, this causes an increase of the free volume distribution, which improves their general mass transport performance.16,24,29–37  In fact, during thermal rearrangement into the solid state, a microporous structure with interconnected microcavities is obtained with a distribution of narrow cavities accessible to small gas molecules.

However, it is still challenging to demonstrate how and to what extent the thermal treatment affects the polymer structure at the atomistic and molecular levels, specifically in terms of its configuration, conformation, glassy transition temperature, and/or free volume. In particular, if a chemical reaction occurs during the thermal treatment, the structure–property relationships of the starting and final structures become more and more complex.38,39 

The physical properties of TR polymer membranes depend on the polymer backbone structure, as well as on the imidisation method.20  A great advantage of TR polymers lies in the possibility of determining their cavity size by designing appropriate polymer structures and thermal reaction mechanisms.16 

The design and optimisation of polymeric membranes for gas separation by numerical simulation would be possible if reliable predictions of material and transport properties could be made significantly more rapidly than the corresponding syntheses and experiments. During the last decade, computational chemistry has had a favourable impact in almost all branches of materials research, ranging from phase determination to structural characterisation and property prediction,40–45  as it allows for dealing with different types of polymers as well as, for example, with polymer colloids such as cement slurries,46  the thermal conductivity of composites,47  and advanced batteries.48,49 

New materials are often developed not so much based on rational considerations, but rather by trial-and-error decisional processes, in part due to the challenging time and length scales involved in modelling transport phenomena in polymeric membranes. However, the rapid progress in computational methodologies and the development of new simulation tools have been gradually improving the understanding of different facets of gas transport in polymeric membranes for their effective use in materials design.50–53 

When describing computational methodologies to study certain types of materials, the main question that a scientist has to answer is “Which properties do I need to get from my material?”. In fact, for separation purposes, the two main phenomena that end-users like engineers and technicians need to quantify are generally adsorption and diffusion. The former is more related to the different affinities of a material towards the species involved in the separation, whereas the latter is more related to the resistance offered to the motion of the species, although diffusion is also strongly dependent on adsorption. In this regard, such transport resistance is generally offered not only by the material itself, but also by the presence of other species in the mixture, which can also affect the adsorption of target species on the material surface.

These general considerations reflect the importance of studying the separation performance of a material by adopting a multicomponent approach, taking into account not only the material–species interactions but also the species–species ones.

As for adsorption, models accounting for the influence of species–species interactions are, just to cite the most complete and used ones, Dual Mode Sorption,54,55  the Ideal Adsorption Solution Theory (IAST),56  the corresponding non-ideal one – i.e., the Real Adsorption Solution Theory (RAST), which makes use of activity coefficients in the adsorbed phase57–64  –, and the Vacancy Solution Theory (VST).65 

As for diffusion, the most complete macroscopic approach is the Maxwell–Stefan model, which can be applied to both bulk diffusion and surface diffusion.66,67  Moreover, it can also be coupled to non-selective bulk transport mechanisms, such as Knudsen diffusion and viscous flow, obtaining in this way the Dusty-Gas Model by Mason and Malinauskas.68 

In the next section, we will present some examples of how some of these models, the IAST and Maxwell–Stefan models, are used synergistically to characterise the separation properties of TR-PBO polymeric membranes.

Regarding the modelling and simulation methods at a molecular level, these usually involve atoms, molecules, or their clusters as basic units. Atoms or molecules interact with each other through a force field (or intermolecular potential energy), and the accuracy of this force field directly determines the accuracy of the resulting calculations.

The common simulation methods dealing with many-body systems can be divided into stochastic and deterministic ones. The first class is represented by the Monte Carlo method, whilst the second one concerns molecular dynamics. The computer-aided molecular design of polymeric membrane models at detailed atomistic level has been reported in the literature for the investigation of the sorption and diffusion of small gas molecules.69–85 

In this context, this contribution focuses on the simulation of TR-PBO polymers at both the atomistic and macroscopic levels, providing examples illustrating the use of existing numerical simulation and modelling approaches that complement the experimental work.

More specifically, after a brief description of the main methodologies used to characterise gas transport through polymeric membranes, the computational approaches used to cover different aspects of TR polymeric membrane simulations are detailed. It must be noted that the successful application of modelling approaches to gas separation by membrane technologies requires the development of models dealing with multicomponent gas mixture transport through model membranes. Moreover, for a given polymeric membrane, both the gas diffusivity and gas solubility depend strongly on process parameters such as the pressure difference, feed composition, and temperature. The effects of process parameters on the selectivity should be thoroughly considered in order to identify membrane materials suitable for specific applications.

An important aspect to be pointed out is the definition of sorption, which is composed of adsorption and absorption phenomena, with the former being related to the interaction of a species in the bulk phase with the surface of the material, and the latter being related to the interaction of a species with the material internal structure (i.e., it is related to the material volume or mass).

However, while it is relatively easy to distinguish the two processes in the case of dense membranes (like metal or perovskite membranes), such a distinction becomes thin and even questionable for microporous materials, for which there is no clear difference between the internal and external surfaces in terms of their potential field.

The situation becomes confusing especially for polymers, as there is a conceptual problem in defining rather than distinguishing the dense zones from the microporous ones at a scale of the order of a few nanometres (1–5 nm). A distinction between the two cases can be made, for example, by considering whether or not the potential field range of the surface occupies all the available internal volume: in the former case, one could say that the material is dense, i.e., no bulk phase can be recognised inside the structure, whereas, in the latter case, the material can be considered to exhibit a certain degree of microporosity.

As mentioned above, solubility is a direct measure of the efficiency of sorption, which is usually considered an equilibrium process, even though it actually is a dynamic one. The definition of Si is the following86,87  (eqn (1.1)):

Equation 1.1

where Ci [moli kg−1] is the loading of the i-th species and pi [Pa] is its partial pressure in the bulk phase considering the whole system is at equilibrium. Based on this definition, it is straightforward to conclude that the solubility values can be directly calculated from sorption isotherms. Under pure-gas conditions and fixed temperature, the solubility is only a function of the partial pressure, whereas, in a mixture, it is a function of the content of all species, as all of them generally affect the sorption of each single species.

In order to acquire non-exhaustive information on the solubility power of a material towards a particular species, one can evaluate the so-called Henry’s constant for the i-th species (eqn (1.2)), which physically represents the reverse solubility value in conditions of infinite dilution.

Equation 1.2

This parameter is actually useful for several reasons and its definition is conceptually coherent since, under conditions of infinite dilution, the presence of other species does not affect the adsorption of a single species. Therefore, Henry’s constant depends on the temperature and each particular material–species pair and its value can be found in the form of tables for several compounds of interest.

The solubility coefficient can be calculated via simulations in a canonical ensemble, in which the chemical potential is calculated using the Widom particle insertion method.88  The interaction energy of a gas particle inserted within the accessible free volume of a polymer matrix is calculated and the excess thermodynamic potential µexcess can be estimated from eqn (1.3):

Equation 1.3

The solubility S is then obtained from eqn (1.4):

Equation 1.4

It is also interesting to note that this approach can be and, in fact, is used for liquids and adsorbents, highlighting the analogy between the thermodynamics of sorption in liquids and the thermodynamics of adsorption in/on solids.

In computer simulations, Henry’s constant is usually calculated via Monte Carlo statistical mechanics methods. Two equivalent modalities are used to perform such calculations. The first requires the evaluation of the simulation-cell loading at several fixed pressures (Grand Canonical Ensemble (GCE)).

Interested readers are referred to the relevant books, reviews, and research articles for more details.89 

Given the relative difficulty in the experimental evaluation of activity coefficients in adsorption systems, the state of the art of complex adsorption studies is based on the IAST, which uses the same formalisms of mixture thermodynamics to deal with the equilibrium of a species on adsorbent surfaces.90–92  Although the details of such a theory can be found in the paper by Myers and Prausnitz,56  we provide here its basic concepts to clarify its application for TR polymers. In particular, Raoult’s law is applied to the adsorbed phase, which the theory defines as “ideal”.

The basic equations characterising the equilibrium and the equations of consistency (mass balance) are as follows:50,86 

Equation 1.5
Equation 1.6
Equation 1.7

where Cµ,i0 [moli kg−1] is the single-gas loading of the species; Π is the so-called spreading pressure [J m−2 = Pa m], which is a sort of bi-dimensional pressure exerting its influence on the surface, analogous to that exerted by the total pressure in the bulk phase; xi is the molar fraction of the adsorbed species; Pi0 is the virtual single-gas pressure that the i-th adsorbed species would exert as a pure species at the same pressure, temperature, and spreading pressure as those of the mixture; A [m2 kg−1] is the adsorbent specific area; and Ω [J kg−1] is the specific Gibbs free energy of immersion, i.e., the minimum work required for the isothermal “immersion” of the gas.

Typical application of the IAST affords the value of the adsorbate composition, total loading, and spreading pressure once the external conditions of temperature and pressure are fixed.

The convenience of the IAST consists in the fact that its implementation can be made using whatever type of isotherm to perform the calculations. For example, in the recent work by Rizzuto et al.,93  the Langmuir model was used to fit Grand Canonical Monte Carlo data and to forecast the adsorption of binary mixtures (Figure 1.2, see next sections for details).

Figure 1.2

Grand Canonical Monte Carlo (GCMC) data (open symbols) and regression curves (continuous lines) under single-gas conditions using the Langmuir model for CO2 and N2 at 25–75 °C. Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

Figure 1.2

Grand Canonical Monte Carlo (GCMC) data (open symbols) and regression curves (continuous lines) under single-gas conditions using the Langmuir model for CO2 and N2 at 25–75 °C. Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

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The Monte Carlo (MC) technique is a stochastic simulation method designed to generate a long sequence (or ‘Markov chain’) of configurations that asymptotically sample the probability density of an equilibrium ensemble of statistical mechanics.94–96  Since its development, MC has been used to test statistical mechanics theories. Today, several advances have been made towards the design of new statistical mechanical ensembles and MC moves for the efficient sampling of complex configuration spaces. A comprehensive review on the progress and outlook of MC simulations has been documented by Theodorou.97 

For the construction of polymeric membrane models filling a basic cubic volume element under periodic boundary conditions, a rotational isomeric state (RIS) Monte Carlo technique incorporating long–range interactions98  can be used.

The simulation of the sorption properties of gas molecules in the amorphous cells of a glassy polymer can also be achieved using Grand Canonical Monte Carlo (GCMC) calculations. This approach requires the structural model of the amorphous cells and the force fields describing the sorbate–sorbent and sorbate–sorbate interactions as input. For the prediction of gas sorption in the generated amorphous cells, the interaction potential is the most important parameter. A simplified interaction potential including only a dispersive–repulsive short-range potential is used, represented by a Lennard-Jones 6–12 potential combined with electrostatic interactions between partial charges on the adsorbent and guest atoms. The multipole–multipole interactions are calculated according to:

Equation 1.8

where Aij is the repulsion constant, Bij the dispersion constant, and qi the point partial charges located at the atomic positions of the adsorbent and sorbate molecules.

Molecular dynamics (MD) is an atomistic simulation method for studying a wide class of materials, such as polymers, metals, ceramics, and biomolecules under ambient as well as extreme conditions. MD allows one to predict the time evolution of a system of interacting particles (e.g., atoms, molecules, etc.) and estimate the relevant physical properties.99–101  It generates information such as the atomic positions, velocities, and forces from which the macroscopic properties (e.g., pressure, energy, heat capacities) can be derived by means of statistical mechanics. MD simulations usually consist of three elements: (i) a set of initial conditions (e.g., the initial positions and velocities of all the particles in the system), (ii) the interaction potentials to represent the forces between the particles, and (iii) the evolution of the system with the time by solving a set of classical equations of motion for all the particles in the system. MD methods are governed by a Hamiltonian system and the Hamilton equations of motion are integrated to move particles to new positions and to assign new velocities at these new positions.

Given a force field for the potential energy, the Hamiltonian of a system of N atoms can be written as:

Equation 1.9

where it is assumed that the kinetic energy, K, depends only on the momenta (mṙ) and it is separable from the potential energy, ϕ, that depends only on the atomic positions.

Particles in MD move naturally under their own intermolecular forces and follow Newton’s second law:

Equation 1.10

where mi, i, and ri are the mass, acceleration, and position of particle i, respectively. During the simulation, both configuration space and phase space are explored, allowing the extraction of information on the dynamics of the system. In order to simulate gas diffusion in a polymeric membrane, a force field representing the interactions between all the atoms of the system (composed of the polymer amorphous cell and penetrant molecules) is required. The force field has to be validated against experimental results and theoretical constraints.

The gas diffusivity can be estimated either from MD simulations by using Einstein’s formulation (1.11) or by means of the Maxwell–Stefan expression (eqn (1.12)) in a binary mixture:

Equation 1.11

where N is the total number of molecules and ri(t) is the unfolded position of gas molecule i at time t.

Equation 1.12

Here, N is the total number of molecules, χi is the mole fraction, Mi is the molecular mass of the i-th component, and ri(t) is the unfolded position of molecule i at time t.

The diffusivity of small gas molecules in glassy polymeric membranes depends on the concentration and reaches a constant value at relatively high concentrations. In fact, glassy polymers are not in thermodynamic equilibrium. For these polymers, the final “metastable” chain configuration depends on the processing history of the membrane. This detail makes the modelling of glassy polymeric membranes even more difficult due to the lack of experimental structural data to validate the computational approaches.

The predicted self-diffusivity depends principally on the quality of the force fields used to model not only the interactions between the penetrant and polymer matrix, but also the intramolecular interactions between polymer chains. The role of chain relaxation and matrix fluctuations has been clearly demonstrated for the explanation of the diffusion mechanism of small gas penetrants (such as N2) in rubber polymeric membranes through MD calculations, in which the polymer matrix is fixed.70,102,103 

The following sub-sections describe some examples of modelling investigations on TR-PBO polymeric membranes.

Jiang et al. carried out initial simulations on the solubility of CO2 and CH4 using the Widom test-particle insertion method.104  Specifically, they studied six TR-PBO polymers and their respective polyimide precursors with hydroxyl groups, showing gas solubilities higher than the experimental ones, which was attributed to the six bulky fluorine groups in the polymeric structure. The simulation results showed that the CO2 and CH4 solubilities increased with the thermal rearrangement, leading to an increase in the permeability in line with that observed for the experimental data. However, the simulated permeabilities were found to be larger than the experimental data only for polymers with low gas diffusivities. This was ascribed to the partial thermal conversion of TR polymers during the experiments.104 

Park et al. calculated the solubilities of five gases (H2, N2, O2, CO2, and CH4) using GCMC simulations.105  They also compared the solubility of the polyimide precursor with two different polybenzoxazole membranes: (i) aTR-PBO (i.e., polybenzoxazole derived from HPI via azeotropic imidisation) and (ii) tTR-PBO (i.e., polybenzoxazole derived from HPI via thermal imidisation). Their values were not significantly improved after the TR reaction and were in a good agreement with the experimental data.105 

Furthermore, the authors identified the reason for the partial increase in solubility (as already observed elsewhere35 ). More specifically, their simulations confirmed two opposite effects. On the one hand, the solubility increased due to the larger free volume elements of the TR-PBO polymers (Figure 1.3). On the other hand, the dual-mode sorption Langmuir affinity parameters slightly decreased as the thermal rearrangement proceeded. This was caused by a reduction of the interactions between the side-chain/groups (i.e., carbonyl and hydroxyl groups) and the gas molecules in the TR-PBO polymers compared to the HPIs. Therefore, both effects compensate each other in such a way that the overall result leads to a gradual increase in gas sorption as the HPIs are converted to TR-PBO polymers.

Figure 1.3

Simulated and experimental solubility data from the literature of gases in TR-PBO membranes. Reprinted with permission from Chi Hoon Park, Elena Tocci, Seungju Kim, Apurva Kumar, Young Moo Lee, Enrico Drioli, A Simulation Study on OH-Containing Polyimide (HPI) and Thermally Rearranged Polybenzoxazoles (TR-PBO): Relationship between Gas Transport Properties and Free Volume Morphology, J. Phys. Chem. B, 2014, 118(10), pp 2746–2757. Copyright © 2014 American Chemical Society.

Figure 1.3

Simulated and experimental solubility data from the literature of gases in TR-PBO membranes. Reprinted with permission from Chi Hoon Park, Elena Tocci, Seungju Kim, Apurva Kumar, Young Moo Lee, Enrico Drioli, A Simulation Study on OH-Containing Polyimide (HPI) and Thermally Rearranged Polybenzoxazoles (TR-PBO): Relationship between Gas Transport Properties and Free Volume Morphology, J. Phys. Chem. B, 2014, 118(10), pp 2746–2757. Copyright © 2014 American Chemical Society.

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Chang et al. simulated the sorption isotherms of various gas molecules in three types of membranes: PBO, poly(benzoxazole-co-imide) (PBO-PI), and polyimide (PI).106  The aim was to analyse how the rigid benzoxazole segments can affect the membrane structure and gas transport behaviour. CO2 exhibited the highest sorption loading for the three membranes considered, followed by CH4, O2, N2, and H2 (Figure 1.4).

Figure 1.4

Sorption isotherms of various gases with (a) PBO, (b) PBO-PI, and (c) PI membranes at 303 K. Reprinted with permission from Journal of Membrane Science, 454, K. S. Chang, Z. C. Wu, S. Kim, K. L. Tung, Y. M. Lee, Y. F. Lin, J. Y. Lai, Molecular modeling of poly(benzoxazole-co-imide) membranes: A structure characterization and performance investigation, 1–11, Copyright (2014), with permission from Elsevier.

Figure 1.4

Sorption isotherms of various gases with (a) PBO, (b) PBO-PI, and (c) PI membranes at 303 K. Reprinted with permission from Journal of Membrane Science, 454, K. S. Chang, Z. C. Wu, S. Kim, K. L. Tung, Y. M. Lee, Y. F. Lin, J. Y. Lai, Molecular modeling of poly(benzoxazole-co-imide) membranes: A structure characterization and performance investigation, 1–11, Copyright (2014), with permission from Elsevier.

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Furthermore, the PBO membranes exhibited the highest sorption loading for all the gas molecules, followed by the PBO-PI and PI membranes, demonstrating the characteristics of high free volume and loose structure. This was explained by the fact that larger cavities effectively provide more space for gas sorption and diffusion, leading to higher gas permeability values (Figure 1.5). Interestingly, the PI membranes exhibited a solubility higher than that of PBO ones at low pressure, which, however, disappeared gradually with the increasing pressure.106 

Figure 1.5

Sorption sites of various gas species in PBO, PBO-PI, and PI membranes. Reprinted with permission from Journal of Membrane Science, 454, K. S. Chang, Z. C. Wu, S. Kim, K. L. Tung, Y. M. Lee, Y. F. Lin, J. Y. Lai, Molecular modeling of poly(benzoxazole-co-imide) membranes: A structure characterization and performance investigation, 1–11, Copyright (2014), with permission from Elsevier.

Figure 1.5

Sorption sites of various gas species in PBO, PBO-PI, and PI membranes. Reprinted with permission from Journal of Membrane Science, 454, K. S. Chang, Z. C. Wu, S. Kim, K. L. Tung, Y. M. Lee, Y. F. Lin, J. Y. Lai, Molecular modeling of poly(benzoxazole-co-imide) membranes: A structure characterization and performance investigation, 1–11, Copyright (2014), with permission from Elsevier.

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Rizzuto et al. calculated the single gas sorption isotherms of N2 and CO2 considering polymeric boxes with simulated chains built using three types of torsional angles, (i.e., random, 90°, and 180°), and compared them to the experimental isotherms of (i) aTR-PBO (i.e., polybenzoxazole derived from HPI via azeotropic imidisation), (ii) tTR-PBO (i.e., polybenzoxazole derived from HPI via thermal imidisation), and (iii) cTR-PBO (i.e., polybenzoxazole derived from HPI via chemical imidisation) membranes (Figure 1.6).93  In general, the chemical imidisation method resulted in large cavities in the TR-polymers. The authors used single-gas data to predict the thermodynamic and transport properties of CO2 and N2 under mixture conditions. The GCMC isotherms were fitted using the Langmuir and dual-Langmuir adsorption models in order to obtain the adsorption parameters for the successive IAST application to the mixture. Then, due to the unavailability of adsorption experimental data for the mixture, they used the GCMC method to obtain the isotherms of the mixture. The obtained results were compared to those achieved from the IAST approach and with other data calculated from literature experimental values.35  The comparison between predicted and experimental adsorption isotherms was satisfactory, with the solubility being directly proportional to the amount of free volume. In particular, the isotherms corresponding to boxes with large free volumes were found to lie near the tTR-PBO isotherms, whereas other boxes with relatively smaller free volumes fitted the aTR-PBO data.

Figure 1.6

Adsorption isotherms under mixture conditions for CO2 and N2 (filled symbols) with a TR-PBO membrane at 35 °C compared to the theoretical single-gas isotherms (empty symbols). The single-gas isotherms are averaged over different polymeric simulation boxes with three torsional angles (random, 90°, 180°). Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

Figure 1.6

Adsorption isotherms under mixture conditions for CO2 and N2 (filled symbols) with a TR-PBO membrane at 35 °C compared to the theoretical single-gas isotherms (empty symbols). The single-gas isotherms are averaged over different polymeric simulation boxes with three torsional angles (random, 90°, 180°). Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

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They also analysed the sorption behaviour of CO2/N2 binary mixtures in order to give a molecular interpretation of the competitive sorption and diffusion processes of these gas molecules.

More specifically, they first calculated the multicomponent adsorption isotherms of CO2 and N2 by GCMC, and found that the adsorption concentration of both gases increased with the increasing pressure; CO2 was preferentially adsorbed over nitrogen at the pressure range studied because of its higher solubility. The pressure dependence of the mixture adsorption was found to be similar to that observed for single gas adsorption, with CO2 showing higher isotherms than N2. Such behaviour was concluded to be due to the competitive adsorption of one gas over the other, which reduces the ability of the first penetrant to be absorbed in the matrix.

Experimental multicomponent sorption in glassy polymers such as PIM-1 107  afforded the same trend: the solubility of all gases was depressed, although to different extents.

Rizzuto et al. also showed the influence of mutual interactions between two types of molecules in a mixture upon performing GCMC simulations by keeping the fugacity of N2 constant and varying the CO2 one at 35 °C and 5 bar. The obtained trend in Figure 1.7 reveals that the loading of N2 decreases as the fugacity of CO2 increases, thus revealing a mutual influence between the species. As isotherm experimental data for binary systems are not available, the IAST approach was used, obtaining the necessary preliminary adsorption parameters from single-gas GCMC data calculated in the temperature range of 25–75 °C. The authors fitted these GCMC isotherms considering both the Langmuir and dual-Langmuir adsorption models, using a multivariate non-linear regression86,92  (Figure 1.8). By this procedure, all the parameters of the models were evaluated as a function of the temperature.

Figure 1.7

N2 loading as a function of the CO2 fugacity for a structure averaged over different polymeric simulation boxes with three torsional angles (random, 90°, 180°). Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

Figure 1.7

N2 loading as a function of the CO2 fugacity for a structure averaged over different polymeric simulation boxes with three torsional angles (random, 90°, 180°). Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

Close modal
Figure 1.8

Comparison among calculated adsorption isotherms under mixture conditions for CO2 and N2 with TR-PBO membranes using the (a1) Langmuir and (b1) dual-Langmuir model. (a2) and (b2) Difference between IAST and GCMC results. Monte Carlo data at 35 °C (colored squares) are shown for comparison for better readability. Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

Figure 1.8

Comparison among calculated adsorption isotherms under mixture conditions for CO2 and N2 with TR-PBO membranes using the (a1) Langmuir and (b1) dual-Langmuir model. (a2) and (b2) Difference between IAST and GCMC results. Monte Carlo data at 35 °C (colored squares) are shown for comparison for better readability. Reprinted from Journal of Membrane Science, 528, Carmen Rizzuto, Alessio Caravella, Adele Brunetti, Chi Hoon Park, Young Moo Lee, Enrico Drioli, Giuseppe Barbieri and Elena Tocci, Sorption and Diffusion of CO2/N2 in gas mixture in thermally-rearranged polymeric membranes: A molecular investigation, 135–146, Copyright (2017), with permission from Elsevier.

Close modal

In the low-pressure range, the GCMC and IAST results were found to be in satisfactory agreement as the system was close to ideal behaviour. In contrast, at higher pressures, the IAST underestimated the molecule adsorption with respect to the GCMC model, as the assumption of ideality was not valid anymore (Figure 1.8(a1) and (b1)).

The free volume elements in TR-PBO polymers consist of three-dimensional networks of intermolecular microcavities, which are accessible for small gas molecules. This peculiar free volume structure is the reason for both the outstanding permeability of TR-PBO polymers with fast gas diffusion and their high permselectivity for the separation of small molecules.

Several studies have been carried out to understand the peculiar behaviour of TR-PBO polymers, such as the analysis of the rotational energy barrier of specific linkages in TR-PBO polymers to explain their increased rigidity compared to that of HPI precursors (Figure 1.9).9,94,106 

Figure 1.9

Simulated rotational distribution of dihedral angles in (a) an imide-phenylene with an ortho-positioned hydroxyl group and (b) a benzoxazole-phenylene (temperatures, A: 25 °C, B: 300 °C, C: 350 °C, D: 400 °C, E: 450 °C). Reprinted from Journal of Membrane Science, 359, Ho Bum Park, Sang Hoon Han, Chul Ho Jung, Young Moo Lee, Anita J. Hill, Thermally rearranged (TR) polymer membranes for CO2 separation, 11–24, Copyright (2010), with permission from Elsevier.

Figure 1.9

Simulated rotational distribution of dihedral angles in (a) an imide-phenylene with an ortho-positioned hydroxyl group and (b) a benzoxazole-phenylene (temperatures, A: 25 °C, B: 300 °C, C: 350 °C, D: 400 °C, E: 450 °C). Reprinted from Journal of Membrane Science, 359, Ho Bum Park, Sang Hoon Han, Chul Ho Jung, Young Moo Lee, Anita J. Hill, Thermally rearranged (TR) polymer membranes for CO2 separation, 11–24, Copyright (2010), with permission from Elsevier.

Close modal

Generally, MD simulations calculate the changes in properties from the precursor polymers to the TR-PBO polymers during thermal rearrangement in terms of the increased free volume elements10,25,94,104  (Figure 1.10), different angle distribution, and increased internal surface area.10  Furthermore, the cavity size distributions104  show an increase in the number and size of cavities from the precursors to the TR-PBO polymers upon thermal treatment, consistent with the observation of the higher permeability of TR-PBO polymers compared to that of their original precursor materials (Figure 1.11(a) and (b)).

Figure 1.10

Free volume distribution (blue color) of (left) HPI and (right) TR-PBO. Reprinted with permission from Chi Hoon Park, Elena Tocci, Young Moo Lee, Enrico Drioli Thermal Treatment Effect on the Structure and Property Change between Hydroxy-Containing Polyimides (HPIs) and Thermally Rearranged Polybenzoxazole (TR-PBO) The Journal of Physical Chemistry B 2012, 116(42), pp 12864–12877. Copyright © 2012 American Chemical Society.

Figure 1.10

Free volume distribution (blue color) of (left) HPI and (right) TR-PBO. Reprinted with permission from Chi Hoon Park, Elena Tocci, Young Moo Lee, Enrico Drioli Thermal Treatment Effect on the Structure and Property Change between Hydroxy-Containing Polyimides (HPIs) and Thermally Rearranged Polybenzoxazole (TR-PBO) The Journal of Physical Chemistry B 2012, 116(42), pp 12864–12877. Copyright © 2012 American Chemical Society.

Close modal
Figure 1.11

(a) Cavity size distribution in TR1 at T = 308 K, and (b) comparison of the cumulative cavity size distributions in TR1 and PIOFG-1. Reprinted from Polymer, 52(10), Yingying Jiang, Frank T. Willmore, David Sanders, Zachary P. Smith, Claudio P. Ribeiro, Cara M. Doherty, Aaron Thornton, Anita J. Hill, Benny D. Freeman, Isaac C. Sanchez, Cavity size, sorption and transport characteristics of thermally rearranged (TR) polymers, 2244–2254, Copyright (2011), with permission from Elsevier.

Figure 1.11

(a) Cavity size distribution in TR1 at T = 308 K, and (b) comparison of the cumulative cavity size distributions in TR1 and PIOFG-1. Reprinted from Polymer, 52(10), Yingying Jiang, Frank T. Willmore, David Sanders, Zachary P. Smith, Claudio P. Ribeiro, Cara M. Doherty, Aaron Thornton, Anita J. Hill, Benny D. Freeman, Isaac C. Sanchez, Cavity size, sorption and transport characteristics of thermally rearranged (TR) polymers, 2244–2254, Copyright (2011), with permission from Elsevier.

Close modal

Different diffusion mechanisms influence the transport of small gas molecules, as determined by the pore size.108,109  The relationship between the polymer structure and cavity size, as well as the transport properties, has been studied by computer simulation approaches.10,94,106 

Jiang et al. studied the diffusivities of CO2 and CH4 in six TR-PBO polymers and their respective polyimide precursors with hydroxyl groups, and they found that both diffusivities were larger than the experimental ones for most of the studied polymers. The reason for the high solubility results was attributed to the six bulky fluorine groups in the polymer structure.104 

As reported by Chang et al., the trend of the self-diffusivity of four gases (H2 > O2 > N2 > CO2) followed mainly the size (kinetic diameters) of the gases, except in the case of CO2. CO2 has a smaller size than O2 and N2 but still shows a lower diffusivity, which might be caused by unfavourable orientations to enter the cavity and a higher affinity to be more easily captured by the membranes.106 

The same conclusion was reported in the paper by Park et al.105  In general, the diffusivities of TR-PBO models were found to be higher than those of HPI polymeric membranes, as found experimentally.9  In particular, larger-size gas molecules presented a higher increasing ratio of diffusivities, indicating that the higher permeability of TR-PBO is due to its much larger diffusivity and that a size-sieving effect is significant in HPI models. Moreover, the authors focused on the shape effect of the free volume elements: TR-PBO membranes showed a higher fraction of elongated free volume elements than the HPI ones, with bottlenecks, which strongly supports the experimental assumption of the so-called “hourglass-shaped” cavities in TR polymers.9  Nevertheless, 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 reduced bottleneck diameters in free volume elements.

Rizzuto et al. performed MD simulations on TR-PBO membranes with gas mixtures to understand the competitive behaviour of gas molecules approaching real conditions (5 bar).93  As a reference, they also simulated the behaviour of TR-PBO boxes containing only N2 or CO2 under the same conditions. All polymeric boxes contained a relatively high concentration of the mixture (around 30 cm3 STP cm−3 polymer). The number of molecules corresponding to the concentration at 5 bar in the relative GCMC isotherms was reported to be 73 for CO2 and 19 for N2 under single gas conditions, respectively; whilst under mixture conditions, 64 CO2 molecules and 10 N2 ones were present. At higher concentrations, the diffusion coefficients of CO2 under both single-gas and mixture conditions were found to be higher than those of N2. Interestingly, (1) the diffusivity values of both gases in the mixture were higher than the respective ones in the single-gas cases, and (2) the N2 diffusivity in the mixture was around three times larger than that of the pure gas, whereas that of CO2 was just slightly higher.

This indicates that, although the space available for the diffusion of N2 is lower in the case of a mixture than in the case of single gas due to the high number of CO2 molecules in the free volume, N2 diffuses much faster in mixtures than as a pure gas, compensating for the preferential adsorption of CO2 in the polymer. Experimental permeation tests of the mixtures indicated a permeance reduction for all the gas species investigated,9  as also documented for other microporous polymeric membranes in comparison with single gas permeation tests.110–112  Due to the higher condensability of CO2, its permeance is not strongly dependent on the amount of N2. Competitive sorption causes a permeance reduction of the least permeable gases (N2), and just a marginal change in the CO permeance. In contrast to CO2, the mixed-gas permeability of N2 is substantially lower than that obtained from the corresponding pure gas. This reduction is explained by considering the influence of both mutual gas diffusion and the tendency of CO2 to preferentially occupy the available free space with respect to N2 (competitive occupancy).

The field of atomistic modelling of gas transport properties in membranes has experienced a rapid progress due not only to the well-known Moore’s law on computational power growth but also to the development of smart algorithms dealing with the simulation of different time-length phenomena important for the realistic description of gas permeation through membranes.

Thermally rearranged polymers show impressive performance for important applications such as gas separation. In general, thermal rearrangement enhances the free volume, leading to diffusion enhancements with only a small increase in the sorption for small molecule transport.

Molecular simulations provide individual particle motions as a function of time, which often makes it possible to answer detailed questions about the properties of a system more easily than through experimentation. Atomistic modelling techniques have proven to be a very useful tool for the investigation of the structure and transport processes in these materials.

It has been recognised37  that forthcoming developments of microporous TR polymeric membranes for gas separation will require the enhanced sorption of target gas pairs while maintaining their extraordinary gas permeability.

Tuning the high free volume elements will be crucial to target gas pairs and increase the membrane performance. For this purpose, molecular simulations, combined with efficient multiscale approaches, will facilitate the design of tailored materials, reducing the laborious experimental trial-and-error procedures.

The Consiglio Nazionale delle Ricerche of Italy, Istituto per la Tecnologia delle Membrane, is gratefully acknowledged for the financial support of the project ACCORDO CNR-NRF-2016–2017 on “Advanced studies to push the limit of CO2 separation: from molecular modelling to experimental preparation and characterization of advanced copolymer membranes with Ionic liquids”.

A. Caravella gratefully acknowledges the “Programma Per Giovani Ricercatori ″Rita Levi Montalcini” granted by the “Ministero dell’Istruzione, dell’Università e della Ricerca, MIUR”.

1.
Charpentier
J. C.
,
Procedia Eng.
,
2016
, vol.
138
pg.
445
2.
Sholl
D. S.
,
Lively
R. P.
,
Nature
,
2016
, vol.
532
(pg.
435
-
437
)
3.
R. W.
Baker
,
Membrane Technology and Applications
,
John Wiley and Sons, Ltd
,
Chichester, UK
, 3rd edn,
2012
4.
E.
Drioli
, in
Encyclopedia of Membranes
, ed. E. Drioli and L. Giorno,
Springer Berlin Heidelberg
,
2016
, pp. 1231–1232
5.
P.
Pullumbi
, in
Membrane Reactor Engineering: Applications for a Greener Process Industry
, ed. A. Basile, M. De Falco, G. Centi and G. Iaquaniello,
John Wiley & Sons, Ltd
,
Chichester, UK
, 1st edn,
2016
, part 2, ch.13, pp. 256–279
6.
Robeson
L. B.
,
J. Membr. Sci.
,
2008
, vol.
320
pg.
390
7.
Xiao
Y.
,
Low
B. T.
,
Hosseini
S. S.
,
Chung
T. S.
,
Paul
D. R.
,
Prog. Polym. Sci.
,
2009
, vol.
34
pg.
561
8.
Maier
G.
,
Angew. Chem., Int. Ed.
,
2013
, vol.
52
pg.
4982
9.
Guiver
M. D.
,
Lee
Y. M.
,
Science
,
2013
, vol.
339
pg.
284
10.
S. H.
Han
and
Y. M.
Lee
, in
Membrane Engineering for the Treatment of Gases: Gas-separation Problems with Membranes
, ed. E. Drioli and G. Barbieri,
RSC
,
Cambridge UK
,
2010
, pp. 84–124
11.
Du
N.
,
Park
H. B.
,
Dal-Cin
M. M.
,
Guiver
M. D.
,
Energy Environ. Sci.
,
2012
, vol.
5
pg.
7306
12.
Sanders
D. F.
,
Smith
Z. P.
,
Guo
R.
,
Robeson
L. M.
,
McGrath
J. E.
,
Paul
D. R.
,
Freeman
B. D.
,
Polymer
,
2013
, vol.
54
pg.
4729
13.
Robeson
L. M.
,
Dose
M. E.
,
Freeman
B. D.
,
Paul
D. R.
,
J. Membr. Sci.
,
2017
, vol.
525
pg.
18
14.
Cersosimo
M.
,
Brunetti
A.
,
Drioli
E.
,
Fiorino
F.
,
Dong
G.
,
Woo
K. T.
,
Lee
J. M.
,
Lee
Y. M.
,
Barbieri
G.
,
J. Membr. Sci.
,
2015
, vol.
492
pg.
257
15.
Brunetti
A.
,
Cersosimo
M.
,
Dong
G.
,
Woo
K. T.
,
Lee
J.
,
Kim
J. S.
,
Lee
Y. M.
,
Drioli
E.
,
Barbieri
G.
,
J. Membr. Sci.
,
2016
, vol.
520
pg.
671
16.
Park
H. B.
,
Jung
C. H.
,
Lee
Y. M.
,
Hill
A. J.
,
Pas
S. J.
,
Mudie
S. T.
,
Van Wagner
E.
,
Freeman
B. D.
,
Cookson
D. J.
,
Science
,
2007
, vol.
318
pg.
254
17.
Dong
G.
,
Woo
K. T.
,
Kim
J.
,
Kim
J. S.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2015
, vol.
496
pg.
229
18.
Park
H. B.
,
Han
S. H.
,
Jung
C. H.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2010
, vol.
359
pg.
11
19.
Han
S. H.
,
Misdan
N.
,
Kim
S.
,
Doherty
C. M.
,
Hill
A. J.
,
Lee
Y. M.
,
Macromolecules
,
2010
, vol.
43
pg.
7657
20.
Han
S. H.
,
Lee
J. E.
,
Lee
K. J.
,
Park
H. B.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2010
, vol.
357
pg.
143
21.
Choi
J. I.
,
Jung
C. H.
,
Han
S. H.
,
Park
H. B.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2010
, vol.
349
pg.
358
22.
Jung
C. H.
,
Lee
J. E.
,
Han
S. H.
,
Park
H. B.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2010
, vol.
350
pg.
301
23.
Calle
M.
,
Lee
Y. M.
,
Macromolecules
,
2011
, vol.
44
pg.
1156
24.
Han
S. H.
,
Kwon
H. J.
,
Kim
K. Y.
,
Seong
J. G.
,
Park
C. H.
,
Kim
S.
,
Doherty
C. M.
,
Thornton
A. W.
,
Hill
A. J.
,
Lozano
A. E.
,
Berchtold
K. A.
,
Lee
Y. M.
,
Phys. Chem. Chem. Phys.
,
2012
, vol.
14
pg.
4365
25.
Park
C. H.
,
Tocci
E.
,
Lee
Y. M.
,
Drioli
E.
,
J. Phys. Chem. B
,
2012
, vol.
116
pg.
12864
26.
XiaO
Y.
,
Chung
T. S.
,
Energy Environ. Sci.
,
2011
, vol.
4
pg.
201
27.
Askari
M.
,
Xiao
Y.
,
Li
P.
,
Chung
T. S.
,
J. Membr. Sci.
,
2012
, vol.
390
pg.
141
28.
Chua
M. L.
,
Xiao
Y. C.
,
Chung
T. S.
,
J. Membr. Sci.
,
2012
, vol.
415
pg.
375
29.
Soo
C. Y.
,
Jo
H. J.
,
Lee
Y. M.
,
Quay
J. R.
,
Murphy
M. K.
,
J. Membr. Sci.
,
2013
, vol.
444
pg.
365
30.
Scholes
C. A.
,
Ribeiro
C. P.
,
Kentish
S. E.
,
Freeman
B. D.
,
J. Membr. Sci.
,
2014
, vol.
450
pg.
72
31.
Jung
C. H.
,
Lee
J. E.
,
Han
S. H.
,
Park
H. B.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2010
, vol.
350
pg.
301
32.
Choi
J. I.
,
Jung
C. H.
,
Han
S. H.
,
Park
H. B.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2010
, vol.
349
pg.
358
33.
Do
Y. S.
,
Seong
J. G.
,
Kim
S.
,
Lee
J. G.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2013
, vol.
446
pg.
294
34.
Wang
H.
,
Liu
S.
,
Chung
T. S.
,
Chen
H.
,
Jean
Y. C.
,
Pramoda
K.
,
Polymer
,
2011
, vol.
52
pg.
5127
35.
Kim
S.
,
Jin Jo
H.
,
Lee
Y. M.
,
J. Membr. Sci.
,
2013
, vol.
441
pg.
1
36.
Han
S. H.
,
Kwon
H. J.
,
Kim
K. Y.
,
Seong
J. G.
,
Park
C. H.
,
Kim
S.
,
Doherty
C. M.
,
Thornton
A. W.
,
Hill
A. J.
,
Lozano
A. E.
,
Berchtold
K. A.
,
Lee
Y. M.
,
Phys. Chem. Chem. Phys.
,
2012
, vol.
14
pg.
4365
37.
Kim
S.
,
Lee
Y. M.
,
Prog. Polym. Sci.
,
2015
, vol.
43
pg.
1
38.
Hodgkin
J. H.
,
Liu
M. S.
,
Dao
B. N.
,
Mardel
J.
,
Hill
A. J.
,
Eur. Polym. J.
,
2011
, vol.
47
pg.
394
39.
Calle
M.
,
Lozano
A. E.
,
Lee
Y. M.
,
Eur. Polym. J.
,
2012
, vol.
48
pg.
1313
40.
Gubbins
K. E.
,
Moore
J. D.
,
Ind. Eng. Chem. Res.
,
2010
, vol.
49
pg.
3026
41.
J.
Karger
,
D. M.
Ruthven
and
D. N.
Theodorou
,
Diffusion in Nanoporous Materials
,
Wiley-VCH Verlag GmbH & Co
,
Weinheim, Germany
,
2012
, p. 227
42.
J.
Baschnagel
,
K.
Binder
,
P.
Doruker
,
A. A.
Gusev
,
O.
Hahn
,
K.
Kremer
,
W. L.
Mattice
,
F.
Müller-Plathe
,
M.
Murat
,
W.
Paul
,
S.
Santos
,
U. W.
Suter
and
V.
Tries
, in
Advances in Polymer Science
, ed. A. Abe, A. C. Albertsson, A. C. G. W. Coates, J. Genzer, S. Kobayashi, K. S. Lee, L. Leibler, T. E. Long, M. Möller, O. Okay, V. Percec, B. Z. Tang, E. M. Terentjev, P. Theato, M. J. Vicent, B. Voit, U. Wiesner and X. Zhang,
Springer
,
Berlin, Heidelberg
,
2000
,
vol. 152
, pp. 41–156
43.
R. A.
van Santen
and
P.
Sautet
, in
Computational Methods in Catalysis and Materials Science: An Introduction for Scientists and Engineers
, ed. R. A. van Santen and P. Sautet,
Wiley-VCH Verlag GmbH & Co. KGaA
,
Weinheim, Germany
,
2009
, p. 441
44.
C. R. A.
Catlow
, in
Computer Modelling in Inorganic Crystallography
, ed. C. R. A. Catlow,
Academic Press
,
1997
45.
Tafipolsky
M.
,
Amirjalayer
S.
,
Schmid
R.
,
J. Phys. Chem. C
,
2010
, vol.
114
pg.
14402
46.
Papatzani
S.
,
Paine
K.
,
Calabria-Holley
J.
,
Constr. Build. Mater.
,
2015
, vol.
74
pg.
219
47.
Ma
L. K.
,
Srivastava
R.
,
Barpanda
D.
,
Fowler
T.
,
Theophanous
T.
,
Verghese
N.
,
J. Reinf. Plast. Compos.
,
2013
, vol.
32
pg.
1916
48.
Christensen
J.
,
Albertus
P.
,
Sanchez-Carrera
R. S.
,
Lohmann
T.
,
Kozinsky
B.
,
J. Electrochem. Soc.
,
2012
, vol.
159
pg.
1
49.
Albertus
P.
,
Girishkumar
G.
,
McCloskey
B.
,
Sanchez-Carrera
R. S.
,
Kozinsky
B.
,
J. Electrochem. Soc.
,
2011
, vol.
158
pg.
343
50.
D. N.
Theodorou
,
Challenges in Molecular Simulations: Bridging the Time-scale and Length-scale Gap (SIMU)
,
2002
, pp. 19–40
51.
K.
Binder
,
W.
Paul
,
S.
Santos
and
U. W.
Suter
, in
Simulation Methods for Polymers
, ed. M. Kotelyanskii and D. N. Theodorou,
M. Decker Inc
,
2004
, p. 491
52.
R. A.
van Santen
and
P.
Sautet
, in
Computational Methods in Catalysis and Materials Science: An Introduction for Scientists and Engineers
, ed. R. A. van Santen and P. Sautet,
Wiley-VCH Verlag GmbH & Co. KGaA
,
Weinheim, Germany
,
2009
, p. 441
53.
C. R. A.
Catlow
, in
Computer Modelling in Inorganic Crystallography
, ed. C. R. A. Catlow,
Academic Press
,
1997
54.
Koros
W. J.
,
Chan
A. H.
,
Paul
D. R.
,
J. Membr. Sci.
,
1977
, vol.
2
pg.
165
55.
Koros
W. J.
,
Paul
D. R.
,
J. Polym. Sci., Polym. Phys. Ed.
,
1978
, vol.
16
pg.
1947
56.
Myers
A. L.
,
Prausnitz
J. M.
,
AIChE J.
,
1965
, vol.
121
pg.
11
57.
Stoeckli
F.
,
Couderc
G.
,
Sobota
R.
,
Lavanchy
A.
,
Adsorpt. Sci. Technol.
,
2002
, vol.
20
pg.
189
58.
Glessner
A. J.
,
Myers
A. L.
,
Chem. Eng. Prog., Symp. Ser.
,
1969
, vol.
65
pg.
73
59.
Costa
E.
,
Sotelo
J. L.
,
Calleja
G.
,
Matron
C.
,
AIChE J.
,
1981
, vol.
27
pg.
5
60.
Talu
O.
,
Zwiebel
I.
,
AIChE J.
,
1986
, vol.
32
pg.
1263
61.
Chen
Y. D.
,
Ritter
J. A.
,
Yang
R. T.
,
Chem. Eng. Sci.
,
1990
, vol.
45
pg.
2877
62.
Karavias
F.
,
Myers
A. L.
,
Chem. Eng. Sci.
,
1991
, vol.
47
pg.
1441
63.
Dunne
J.
,
Myers
A. L.
,
Chem. Eng. Sci.
,
1994
, vol.
49
pg.
2941
64.
Yun
J.
,
Park
H.
,
Moon
H.
,
Korean J. Chem. Eng.
,
1996
, vol.
13
pg.
246
65.
Suwanayuen
S.
,
Danner
R. P.
,
AIChE J.
,
1980
, vol.
26
pg.
76
66.
Krishna
R.
,
Chem. Eng. Sci.
,
1990
, vol.
45
pg.
1779
67.
R.
Krishna
and
R.
Taylor
,
Multicomponent Mass Transfer
,
Wiley & Sons
,
New York, USA
,
1993
68.
E. A.
Mason
and
A. P.
Malinauskas
,
Gas Transport in Porous Media: The Dusty-gas Model, Chemical Engineering Monographs 17
,
Elsevier Science Ltd
,
Amsterdam, The Netherlands
,
1983
69.
Pant
P. V. K.
,
Boyd
R. H.
,
Macromolecules
,
1992
, vol.
114
pg.
494
70.
Müller-Plathe
F.
,
Acta Polym.
,
1994
, vol.
45
pg.
259
71.
Muller-Plathe
F.
,
Van Gunsteren
W. F.
,
Suter
U. W.
,
Adv. Polym. Sci.
,
1994
, vol.
116
pg.
207
72.
Pant
P. V. K.
,
Theodorou
D. N.
,
Polym. Prepr.
,
1994
, vol.
35
pg.
165
73.
Hofmann
D.
,
Ulbritch
J.
,
Fritsch
D.
,
Paul
D.
,
Polymer
,
1996
, vol.
37
pg.
4773
74.
Tocci
E.
,
Bellacchio
E.
,
Russo
N.
,
Drioli
E.
,
J. Membr. Sci.
,
2002
, vol.
206
pg.
389
75.
Heuchel
M.
,
Hofmann
D.
,
Pullumbi
P.
,
Macromolecules
,
2004
, vol.
37
pg.
201
76.
Karayiannis
N. C.
,
Mavrantzas
V. G.
,
Theodorou
D. N.
,
Macromolecules
,
2004
, vol.
37
pg.
2978
77.
Neyertz
S.
,
Brown
D.
,
Macromolecules
,
2004
, vol.
37
pg.
10109
78.
Raptis
V. E.
,
Economou
I. G.
,
Theodorou
D. N.
,
Petrou
J.
,
Petropoulos
J. H.
,
Macromolecules
,
2004
, vol.
37
pg.
1102
79.
J. R.
Fried
, in
Materials Science of Membranes for Gas and Vapor Separation
, ed. Y. Yampolskii, I. Pinnau and B. D. Freeman,
John Wiley & Sons
,
2006
, ch. 3, p. 95
80.
Neyertz
S.
,
Macromol. Theory Simul.
,
2007
, vol.
16
pg.
513
81.
Holck
O.
,
Heuchel
M.
,
Bohning
M.
,
Hofmann
D.
,
J. Polym. Sci., Part B: Polym. Phys.
,
2008
, vol.
46
pg.
59
82.
Heuchel
M.
,
Fritsch
D.
,
Budd
P. M.
,
McKeown
N. B.
,
Hofmann
D.
,
J. Membr. Sci.
,
2008
, vol.
318
pg.
84
83.
Chunhai
L.
,
Shijun
N.
,
Wenkai
C.
,
Junsheng
L.
,
Chengjiang
Z.
,
Comput. Mater. Sci.
,
2010
, vol.
49
pg.
S65
84.
Larsen
G. S.
,
Lin
P.
,
Hart
K. E.
,
Colina
C. M.
,
Macromolecules
,
2011
, vol.
44
pg.
6944
85.
Tocci
E.
,
De Lorenzo
L.
,
Bernardo
P.
,
Clarizia
G.
,
Bazzarelli
F.
,
Mckeown
N. B.
,
Carta
M.
,
Malpass-Evans
R.
,
Friess
K.
,
Pilnáček
K.
,
Lanč
M.
,
Yampolskii
Y. P.
,
Strarannikova
L.
,
Shantarovich
V.
,
Mauri
M.
,
Jansen
J. C.
,
Macromolecules
,
2014
, vol.
47
pg.
7900
86.
S.
Matteucci
,
Y.
Yampolskii
,
B. D.
Freeman
and
I.
Pinnau
, in
Materials Science of Membranes and Gas Separation
, ed. Y. Yampolskii, I. Pinnau and B. D. Freeman,
John Wiley & Sons, Ltd
,
Chichester
,
2006
87.
Wang
R.
,
Cao
C.
,
Chung
T.
,
J. Membr. Sci.
,
2002
, vol.
198
pg.
259
88.
Widom
B.
,
J. Chem. Phys.
,
1963
, vol.
39
pg.
2808
89.
V. N.
Burganos
, in
Comprehensive Membrane Science and Engineering
, ed. E. Drioli and L. Giorno,
Elsevier
,
Oxford
, 1st edn,
2010
,
vol. 1
, pp. 29–74
90.
D. D.
Do
, in
Adsorption Analysis: Equilibria and Kinetics
, ed. D. D. Do,
Imperial College Press
,
London
,
1998
, ch. 10,
vol. 1
, pp. 634–643
91.
Caravella
A.
,
Zito
P. F.
,
Brunetti
A.
,
Drioli
E.
,
Barbieri
G.
,
J. Chem. Eng. Data
,
2015
, vol.
60
pg.
2343
92.
Zito
P. F.
,
Caravella
A.
,
Brunetti
A.
,
Drioli
E.
,
Barbieri
G.
,
J. Chem. Eng. Data
,
2015
, vol.
60
pg.
2858
93.
Rizzuto
C.
,
Caravella
A.
,
Brunetti
A.
,
Park
C. H.
,
Lee
Y. M.
,
Drioli
E.
,
Barbieri
G.
,
Tocci
E.
,
J. Membr. Sci.
,
2017
, vol.
528
pg.
135
94.
J. M.
Haile
,
Molecular Dynamics Simulations, Elementary Methods
,
Wiley, Interscience
,
New York
,
1992
95.
M. P.
Allen
and
D. J.
Tildesley
,
Computer Simulation of Liquids
,
Clarendon Press
,
Oxford, UK
,
1987
96.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulations
,
Academic Press
,
New York, NY, USA
,
1996
97.
Theodorou
D. N.
,
Ind. Eng. Chem. Res.
,
2010
, vol.
49
pg.
3047
98.
Jagodic
F.
,
Borstnik
B.
,
Azman
A.
,
Makromol. Chem.
,
1973
, vol.
173
pg.
221
99.
Maginn
E. J.
,
Elliott
J. R.
,
Ind. Eng. Chem. Res.
,
2010
, vol.
49
pg.
3059
100.
Karplus
M.
,
McCammon
J. A.
,
Nat. Struct. Biol.
,
2002
, vol.
9
pg.
646
101.
Hummer
G.
,
Rasaiah
J. C.
,
Noworyta
J. P.
,
Nature
,
2001
, vol.
414
pg.
188
102.
Van der Vegt
N. F. A.
,
Macromolecules
,
2000
, vol.
33
pg.
3153
103.
Tocci
E.
,
Gugliuzza
A.
,
De Lorenzo
L.
,
Macchione
M.
,
De Luca
G.
,
Drioli
E.
,
J. Membr. Sci.
,
2008
, vol.
323
pg.
316
104.
Jiang
Y.
,
Willmore
F. T.
,
Sanders
D.
,
Smith
Z. P.
,
Ribeiro
C. P.
,
Doherty
C. M.
,
Thornton
A.
,
Hill
A. J.
,
Freeman
B. D.
,
Sanchez
I. C.
,
Polymer
,
2011
, vol.
52
pg.
2244
105.
Park
C. H.
,
Tocci
E.
,
Kim
S.
,
Kumar
A.
,
Lee
Y. M.
,
Drioli
E.
,
J. Phys. Chem. B
,
2014
, vol.
118
pg.
2746
106.
Chang
K. S.
,
Wu
Z. C.
,
Kim
S.
,
Tung
K. L.
,
Lee
Y. M.
,
Lin
Y. F.
,
Lai
J. Y.
,
J. Membr. Sci.
,
2014
, vol.
454
pg.
1
107.
Vopička
O.
,
De Angelis
M. G.
,
Du
N.
,
Li
N.
,
Guiver
M. D.
,
Sarti
G. C.
,
J. Membr. Sci.
,
2014
, vol.
459
pg.
264
108.
Y.
Yampolskii
,
I.
Pinnau
and
B. D.
Freeman
,
Materials Science of Membranes for Gas and Vapor Separation
,
John Wiley & Sons Ltd
,
West Sussex UK
,
2006
, p. 466
109.
Thornton
A. W.
,
Hilder
T.
,
Hill
A. J.
,
Hill
J. M.
,
J. Membr. Sci.
,
2009
, vol.
336
pg.
101
110.
Thomas
S.
,
Pinnau
I.
,
Du
N. Y.
,
Guiver
M. D.
,
J. Membr. Sci.
,
2009
, vol.
333
pg.
125
111.
Thomas
S.
,
Pinnau
I.
,
Du
N. Y.
,
Guiver
M. D.
,
J. Membr. Sci.
,
2009
, vol.
338
pg.
1
112.
Swaidan
R.
,
Ma
X.
,
Litwiller
E.
,
Pinnau
I.
,
J. Membr. Sci.
,
2013
, vol.
447
pg.
387
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