CHAPTER 1
Modelling of Gas Separation in Thermally Rearranged Polymeric Membranes
ELENA TOCCI, ALESSIO CARAVELLA, CARMEN RIZZUTO, GIUSEPPE BARBIERI, YOUNG MOO LEE AND ENRICO DRIOLI
1.1 Introduction
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.
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.
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. 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).
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.
Thermally rearranged (TR) polymers are an example of microporous polymers with high permeability and selectivity for the separation of gas mixtures. In particular, they have shown outstanding molecular and ionic transport, as well as separation performance, beyond the limits of more conventional polymers.
An example of process design using TR polymer membranes can be found in the work of Dong et al. from 2015.
1.1.1 Thermally Rearranged (TR) Polymers
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 hydroxylpolyimide (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. 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.
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. Furthermore, TR copolymers have also been investigated in terms of the concerted effects of different TR polymers with several glassy polymers. 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.
Moreover, this causes an increase of the free volume distribution, which improves their general mass transport performance. 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.
The physical properties of TR polymer membranes depend on the polymer backbone structure, as well as on the imidisation method. A great advantage of TR polymers lies in the possibility of determining their cavity size by designing appropriate polymer structures and thermal reaction mechanisms.
1.1.2 Computational Approach to Polymeric Membranes: From Macro- to Atomistic Scale
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, as it allows for dealing with different types of polymers as well as, for example, with polymer colloids such as cement slurries, the thermal conductivity of composites, and advanced batteries.
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.
1.1.3 Micro- and Macroscopic Simulation Methods
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, the Ideal Adsorption Solution Theory (IAST), the corresponding non-ideal one – i.e., the Real Adsorption Solution Theory (RAST), which makes use of activity coefficients in the adsorbed phase –, and the Vacancy Solution Theory (VST).
As for diffusion, the most complete macroscopic approach is the Maxwell–Stefan model, which can be applied to both bulk diffusion and surface diffusion. 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.
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.
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.
1.2 Thermodynamics and Transport in Polymeric Membranes
1.2.1 Solubility
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 following (eqn (1.1)):
[MATHEMATICAL EXPRESSION OMITTED] (1.1)
where Ci [mol kg] is the loading of the i-th species and ITLpITL [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.
[MATHEMATICAL EXPRESSION OMITTED] (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. The interaction energy of a gas particle inserted within the accessible free volume of a polymer matrix is calculated and the excess thermodynamic potential µ can be estimated from eqn (1.3):
µexcess = RTln(exp(-Eint/kT) (1.3)
The solubility S is then obtained from eqn (1.4):
s = exp(-µexcess/RT (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.
1.2.2 IAST
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. Although the details of such a theory can be found in the paper by Myers and Prausnitz, 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:
PTyi = xip0i n equations (.15)
[MATHEMATICAL EXPRESSION OMITTED] (1.6)
[MATHEMATICAL EXPRESSION OMITTED] (1.7)
where Cµ,i0 [moli kg-1] is the single-gas loading of the species; ? 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 O [J kg-1] is the specific Gibbs free energy of immersion, i.e., the minimum work required for the isothermal "immersion" of the gas.
