Experimental results and theoretical investigation of adsorption of gases in crystalline hydrophobic dipeptides

Experimental results and theoretical investigation of adsorption of gases in crystalline hydrophobic dipeptides Rui Vieira Afonso Dissertation presented to obtain the degree of Doctor in Chemical and Biological Engineering by the University of Porto Supervisors: Luís Miguel Gales Pereira Pinto, Associate Professor Adélio Miguel Magalhães Mendes, Cathedratic Professor Ana Margarida Moreira Leitão de Barros Martins Damas, Cathedratic Professor University of Porto, Porto 2016 ! i Acknowledgements I would like to acknowledge the Portuguese Foundation for Science and Technology (FCT) for my Ph.D. grant (ref SFRH/BD/43821/2008). My gratitude goes to my supervisors Professors Luís Gales, Adélio Mendes and Ana Margarida Damas. I would like to express my gratitude to all the colleagues with which I was lucky to share this long and arduous path. In particular, thanks to Roberto Magalhães and Daniel Ferreira for all the support. My thankful words are also directed to IBMC, LEPABE, DEQ and FEUP for providing the necessary conditions to carry out my work. Special thanks to my close friends, Emanuele, Öznur, Barney and Nira, for their unconditional support. Finally, special gratitude goes to my mother, Dulcínea, grandmother, Setélia, and girlfriend, Gina, who, loyally by my side, have, more than anyone else, made this thesis possible. ii iii Preface The present work was developed in cooperation between IBMC (Institute for Molecular and Cell Biology) and FEUP (Faculty of Engineering of University of Porto) and was carried out at IBMC and Laboratory for Process Engineering, Environmental, Biotechnology and Energy (LEPABE). All work was accomplished under FCT PhD scholarship SFRH/BD/43821/2008. This work emerged from the recognition that hydrophobic dipeptides form crystalline arrays of unidimensional ultramicropores. The work that followed sought to determine the usefulness of this class of materials in the context of existing and potential applications of microporous materials currently known. This thesis comprises three scientific articles published during the PhD period, and two currently under submission. iv v Abstract This work had the objecting of testing the host-guest properties and potential applications of crystalline hydrophobic dipeptides. This kind of peptide-based microporous solid has unidimensional ultramicropores that show remarkable framework flexibility and adsorption properties. The pores of crystals of hydrophobic dipeptides can be either hydrophilic or hydrophobic, with the latter being naturally more interesting than the former. Permeation and adsorption experiments with some such dipeptides were performed, after which models were developed to describe adsorption. Single-crystal permeation of He, N2, O2 and Ar in hydrophobic dipeptides LS, VI and AA showed these dipeptides can indeed act as guests for small gases. Millimetre-sized crystals showed a strong tendency for pore blockage, influencing different crystals and species in a different manner. Ar permeation in VI crystals seems to have been all but eliminated, while N2, O2 and He permeated easily. As Ar adsorbs in crystals tens of micrometres-long, this effect is construed to be due to pore blockage, with framework flexibility possibly playing a role in letting other species permeate. In the case of AA, framework flexibility led to a formally non-porous solid allowing the penetration of gas molecules into the isolated cavities of the crystalline framework. Adsorption isotherms of atmospheric gases Ar, N2 and O2 were determined in the hydrophobic dipeptides VI, IA, IV and VV. These four dipeptides have pore sizes of, respectively, 0.37 nm, 0.37 nm, 0.39 nm and 0.44 nm. In all dipeptides, the preferential adsorption sequence observed was Ar>O2>N2, a highly unusual result. Similarly, the sequence of preferential adsorption observed for each gas was VI<IA<IV<VV. Total adsorption concentration thus increases monotonically with pore size. High Ar/O2 selectivites were observed, especially for VI, a highly prised property in adsorption-based air separation systems. The 1.30 value determined, for VI, at 5 ºC, is the highest ever reported for Ag-free porous materials. Simple thermodynamic models were developed to better interpret the results obtained and gain insight into guest species behaviour inside the pores. By adapting the Gibbs Adsorption Isotherm to 1D adsorption systems, it was possible to derive simple equations that are traditionally only used to describe monolayer surface (2D) adsorption. A 3D adsorption model was also created for implementation in supermicroporous systems, which is not vi entirely a new approach. Considering localised adsorption leads to the derivation of the Langmuir equation, considering distributed adsorption leads to the Volmer equations. Taking into account adsorbate-adsorbate interactions, the Fowler-Guggenheim (localised adsorption) and Hill-de Boer (distributed adsorption) equations are also easily derived. The derivations obtained through this thermodynamic approach decisively show that equations such as Langmuir need not work only as mathematical correlations when applied to microporous systems. The developed models were tested with Xe and CO2 adsorption in VI, IA, IV and VV. Through fitting of adsorption data, it was possible to determine that Xe adsorbs in a more distributed way, while CO2 adsorbs in a more localised way. In order to better visualise the adsorption process in ultramicropores, kinetic derivations were made of all the equations previously derived thermodynamically. The same approach originally used by Langmuir was applied, so the validity of the analogous 1D and 3D equations can be easily inferred. A previously inexistent kinetic derivation of the Volmer equation for 2D systems is also presented. Isolating the adsorbate-adsorbate contribution of the heat of adsorption allows the derivation of the Fowler-Guggenheim and Hill-de Boer equations from the Langmuir and Volmer equations, respectively. vii Sumário O presente trabalho teve como objectivo testar as propriedades de adsorção e aplicações potenciais de dipéptidos hidrofóbicos cristalinos. Este tipo de sólido microporoso peptídico tem ultramicroporos unidimensionais que mostram assinaláveis flexibilidade da rede e propriedades de adsorção. Os poros de cristais de dipéptidos hidrofóbicos podem ser hidrofílicos ou hidrofóbicos, com os últimos sendo naturalmente mais interessantes do que os primeiros. Foram efectuadas experiências de permeação e adsorção com alguns destes dipéptidos, após as quais modelos foram desenvolvidos para descrever a adsorção. Experiências de permeação por cristal-único de He, N2, O2 e Ar, nos dipéptidos hidrofóbicos LS, VI e AA, mostraram que estes dipéptidos podem, efectivamente, ser usados na adsorção de gases com moléculas pequenas. Os cristais milimétricos mostraram uma forte tendência para bloqueio dos poros, influenciando cristais e espécies diferentes de forma diferente. A permeação de Ar em cristais de VI foi completamente eliminada, enquanto o N2, o O2 e o He permearam facilmente. Uma vez que o Ar adsorve em cristais com dezenas de micrómetros de comprimento, este efeito é interpretado como sendo devido ao bloqueio de poros, com a flexibilidade de rede possivelmente a permitir que as outras espécies permeiem. No caso da AA, a flexibilidade de rede levou a que sólidos formalmente não-porosos permitissem a penetração de moléculas de gás em cavidades isoladas na rede cristalina. Foram determinadas isotérmicas de adsorção dos gases atmosféricos Ar, N2 e O2 nos dipéptidos hidrofóbicos VI, IA, IV e VV. Estes quatro dipéptidos têm tamanhos de poro, respectivamente, de 0,37 nm, 0,37 nm, 0,39 nm e 0,44 nm. Em todos os dipéptidos, a sequência de adsorção preferencial observada para cada gás foi VI < IA < IV < VV. A concentração de adsorção aumenta portanto com o tamanho de poro. Foram observadas altas selectividades Ar/O2, especialmente em VI, uma propriedade altamente valorizada em sistemas de separação de ar por adsorção. O valor 1,30, determinado para adsorção em VI, a 5 ºC, é o mais alto algumas vez reportado em materiais porosos não contendo Ag. Foram desenvolvidos modelos termodinâmicos simples para melhor interpretar os resultados obtidos e visualizar o comportamento das moléculas e átomos adsorvidos dentro dos poros. Adaptando a Isotérmica de Adsorção de Gibbs a sistemas de adsorção 1D, foi possível derivar equações simples que são tradicionalmente usadas apenas para descrever adsorção em monocamada (2D). Foi criado um modelo 3D também para implementação em sistemas xiv I One-letter abbreviation of isoleucine Ile Three-letter abbreviation of isoleucine L One-letter abbreviation of leucine L Levorotatory Leu Three-letter abbreviation of leucine MAP Multiple antigene peptide MOF Metal-organic framework NMR Nuclear magnetic resonance OSMS Organic supramolecular microporous solids Phe Three-letter abbreviation of phenylalanine PSA Pressure-swing adsorption PTA Potential theory of adsorption S One-letter abbreviation of serine Ser Three-letter abbreviation of serine SC-XRD Single-crystal X-ray difraction T One-letter abbreviation of threonine Thr Three-letter abbreviation of threonine V One-letter abbreviation of valine Val Three-letter abbreviation of valine TVFM Theory of volume-filling of micropores Greek letters α Prefix referring to an amino acid (or its residue) with a single carbon linking the amino and carboxylic groups. ß Prefix referring to an amino acid (or its residue) with two carbons linking the amino and carboxylic groups. xv ß3 Prefix referring to an amino acid (or its residue) with two carbons linking the amino and carboxylic groups, with the side-chain linked to the carbon next to the amine group. γ Prefix referring to an amino acid (or its residue) with three carbons linking the amino and carboxylic groups. Variables Latin letters ! Specific surface area, m2·kg-1 !! Area of an adsorbed molecule, m2·molecule-1 !!"#$%&#$ Average area occupied by a molecule, m2·molecule-1 !!"#$! Average pore mouth area, m2·mouth-1 ! Molar surface area, m2·mol-1 !! Minimum molar surface area, m2·mol-1 !!"#$%&#$ Molar area of a single adsorbed molecule, m2·mol-1 !!"#$ Interfacial molar area of a given pore, m2·kg-1 !! Molar area of the circle centred on the centre of the impact site, and touching the nearest adsorbed molecule, whether in a surface of 2D adsorption systems or at the pore mouth interface of 3D adsorption systems, m2·mol-1 !! Average of !!, m2·mol-1 ! Affinity constant, bar-1 c Interaction parameter in the Hill-de Boer and Fowler-Guggenheim equations ! Distance between the pore mouth and the adsorbed molecule closest to it (1D adsorption systems), m·mol-1 xvi !∗ Critical distance that allows adsorption of incoming molecules in surface adsorption, m !!"# Critical distance allowing adsorption (1D adsorption systems), m·mol-1 !!"#$% Intermolecular distance between two adsorbed molecules, m !! Activation energy of desorption, J·mol-1 −∆!!"# Heat of adsorption, J·mol-1 −∆!!"#$% Term of the isosteric heat of adsorption due to lateral interactions, J·mol-1 !! Henry’s constant, mol·kg-1·bar-1 !! ! Pre-exponential factor in van’t Hoff’s equation, mol·kg-1·bar-1 !! Minimum frequency of adsorption, s-1 !! Frequency of desorption, s-1 !!!" Frequency with which the molecule closest to the exit “hits” the pore mouth (1D adsorption systems), s-1 ! Specific micropore length, m·kg-1 ! Molar pore length, m·mol-1 !! Minimum molar pore length, m·mol-1 ! Molar mass, kg·mol-1 !!"#$! Average number of mouths per pore, mouth·pore-1 !!"#$ Total specific number of pores, pore·kg-1 ! Adsorbed concentration, mol·kg-1 !!"# Maximum adsorbed concentration, mol·kg-1 !!"#$%&'($ Amount of adsorbed molecules located at the pore mouth interface, mol·mouth-1 ! Three-dimensional pressure, bar !!"# Term of the isosteric heat of adsorption due to adsorbate-adsorbent interactions, J·mol-1 xvii !!" Isosteric heat of adsorption, J·mol-1 ! Ideal gas constant, J·mol-1·K-1 ! ! Rate of adsorption, mol·kg-1·s-1 ! ! Rate of desorption, mol·kg-1·s-1 ! Absolute temperature, K ! Specific pore volume, m3·kg-1 ! Molar pore volume, m3·mol-1 !! Minimum molar pore volume, m3·mol-1 ! Mean speed of adsorbed molecules (1D adsorption systems), m·s-1 ! Energy of the interaction between two adsorbed molecules, J·mol-1 ! Number of positions, adjacent to each adsorbed molecule, that other molecules can occupy Greek letters ! Fraction of successful impacts on adsorption sites in a surface ! Fraction of impacts with an angle that allows a molecule to enter a pore ! Interaction constant in the two-dimensional van der Waals equation of state, mol2·m-4 !!/! Adsorbent selectivity ! Surface tension, N·m-1 !! Surface tension of the free surface, N·m-1 ! Excess concentration at the imaginary interface of the 2D Gibbs adsorption isotherm, mol·m-2 ! Amount adsorbed relative to the maximum that can be adsorbed ! Distribution parameter of the exponential probability distributions, mol·m-2 (2D) and mol·m-1 (1D) ! Chemical potential, J·mol-1 xviii ! Spreading pressure, N·m-1 ! 3D spreading pressure, N·m-2 ! 1D spreading pressure, N 1 Chapter 1. Introduction* 1.1. Microporous Solids 1.1.1. Classical Materials and Their Applications Microporous solids are one of the most successful examples of materials engineering. The two most widely used porous solids, activated carbons and zeolites, are both microporous [1]. The next two, silica gel and activated alumina, can also be microporous, to a varying extent [1]. Adsorption and catalysis constitute the two main applications of microporous solids [1], with separation and reaction processes based on microporous solids being central to the chemical industry. Recent advances have also made the industrial use of microporous solids as membrane material a real possibility [2]. Activated carbon can be described as a network of cross-linked defective carbon graphitic planes. It has been used as a purification agent since antiquity and was the first porous material to be used in adsorption experiments, in 1773. It was also, the first to be used in a modern industrial process, in 1794, in sugar syrup decolouration [1]. It remains, to this day, the most industrially significant microporous solid [1]. Produced initially from coal and charcoal, it was eventually realized that it could be derived from nearly any carbon-rich raw material, through adequate anaerobic thermal processing [3]. The term “zeolite” was originally used for naturally occurring aluminosilicates, having later been extended to synthetic aluminosilicates. A natural zeolite was first identified in 1756, and zeolites were used in adsorption experiments throughout the 19th century [4]. However, it was only after the determination of their crystal structures in the 1920’s and 1930’s [4] that it was possible to create an accurate picture of adsorption in zeolites, with the development and wide acceptance of the “zeolitic solid solution” theory [5-7]. Their potential as selective adsorbents of gases was soon recognised [6, 7] and studies on their use as adsorbents [4, 8] and catalysts [4, 9] paved the way to industrial use [1, 4, 10]. * Sub-chapters 1.1 and 1.2 are adapted from parts of Afonso et al., J. Mater. Chem. 22 (2012), 1709- 1723. Chapter 1. Introduction 2 Many other microporous materials have been developed in the post-WWII period. Interestingly, most are either purely carbonaceous, as activated carbon, or mineral, as zeolites. The several allotropic forms of carbon resulted in the development of many carbon-based microporous solids, with different structures and morphologies. The simplest case is that of carbon molecular sieves (CMSs), activated carbons with pores small enough to exclude certain species, and a narrow pore size distribution [11]. This is an important class of materials given their use in the separation of nitrogen from air by pressure swing adsorption (PSA) [12]. Other, more recent but less important (as adsorbents or catalysts) carbon-based microporous solids include carbon nanotubes (CNTs), pillared graphite, graphite nanofibers (GNFs) and graphene (see Table 1.1). Numerous families of mineral crystalline microporous solids were discovered, building on the success of zeolites. Aluminophosphates were the first of such materials to be discovered, followed by other metallophosphates, metallosilicates and many others (see Table 1.1). There are essentially two main types of microporous frameworks: random three-dimensional frameworks (such as CMSs), with wider pore size distributions, and crystal structures with well-defined pore networks (such as zeolites). Among the latter, a distinction can be made between those having isolated unidimensional pores and those with more complex pore networks and morphologies. The interconnected “cages” constituting most zeolite structures are a good example of a relatively complex three-dimensional framework. Structures with unidimensional pores are relatively rare, although zeolites possessing unidimensional pores have been known for some time [13]. Unidimensional pores offer some significant advantages over multidimensional pore networks. When entering the pores, non-spherical molecules face steric constraints, which reduces their diffusivity [14, 15]. The smaller the pore and the greater the anisotropy of the diffusing molecule, the more important this effect is [16]. For complex pore networks, either crystalline or amorphous, frequent molecular “hopping” between larger “cages” or pore intersections, through smaller “windows”, multiplies this effect, creating an entropic resistance that should be minimised [17]. Unidimensional, uniform and single-sized pores create a system with only two transitions between cavities of different size, pore entrance and pore exit. Unidimensional pores have the disadvantage of being more easily blocked than two- and three-dimensional frameworks, either by contaminants or crystal framework irregularities. Microporous Solids 3 Table 1.1 – Known classes of microporous solids, classified according to their chemical properties. Adapted from [18]. For references for each class, refer to the original publication. Inorganic Carbon Activated Carbon Carbon Molecular Sieves Carbon Nanotubes Graphene Graphite Nanofibers Pillared Graphite Crystalline Minerals Metallosilicates Alumina Metallophosphates Metallogermanates Others Silica Gel Organic Covalent Organic Frameworks (COFs) Polymers of Intrinsic Microporosity (PIMs) Supramolecular Dianin's Compound, Hidroquinone and Alicyclic Diols Hydrogen and Halogen-bonded “Tectons” Calixarenes Cyclodextrins Arylene-ethynylene Macrocycles (AEMs) Other Macrocycles Others Peptide- -based Macrocycles Dendrimeric Hydrophobic dipeptides Hybrid Metal-Peptide Frameworks (MPFs) Metal-Organic frameworks (MOFs) Zeolitic Imidazole Frameworks (ZIFs) Metal Phosphonates Others Chapter 1. Introduction 4 The crystal framework of mineral and carbonaceous solids, made-up of covalent bonds, creates a robust pore network, capable of withstanding the extreme thermal, pressure and chemical environments of industrial processes, being a fundamental factor behind their success. All the previously enumerated inorganic materials share this characteristic with zeolites, allowing a fast and easy transition to industrial applications. Although this constitutes one of their main strengths, it is also their main limiting factor. Table 1.1 summarizes the known classes of microporous solids and classifies them according to chemical composition. 1.1.2. Present Trends and Potential Developments In 2008, the synthetic zeolite market involved a total of 1.8 Mt [19]. 72 % (per volume) of the synthetic zeolite market was used in the manufacture of detergents [19]. Adsorption and catalysis constituted 10 % and 17 %, respectively. However, per market value, the biggest market was that of catalysis, with 55 % of the total. Over 95 % of the catalyst market was due to fluid catalytic cracking (FCC). Synthetic zeolites typically cost 3-4 $/kg for FCC, and no more than 20 $/kg for specialty catalysis applications. For adsorbents, the normal price range is 5-9 $/kg, with some exceptional cases of tens of dollars per kilogram. For use in detergents, prices of ~2 $/kg are typical [19]. Also in 2008, the total market of natural zeolites was of 3.0 Mt [19]. They are mainly used in cement manufacture, with prices of 0.04-0.25 $/kg. It was not possible to uncover equivalent numbers for activated carbon and silica-gel, the other two materials used on a large-scale. One of the most promising areas of development in zeolite research is in the construction of zeolite membranes. They have the potential to greatly reduce the cost of many separations currently achieved by distillation [20], both through kinetic and equilibrium mechanisms. The cost of zeolite membranes is one order of magnitude higher than that of polymeric membranes, due to their complex fabrication procedure. Thus, they will probably only be useful in separations that involve temperatures and solvents incompatible with polymeric membranes. The main challenge in the production of zeolite membranes is creating thin active layers with no holes, creating good compromises between permeance and selectivity [21]. Microporous solids can still be used as membrane material in mixed-matrix membranes, to improve the performance of polymeric membranes. Mixed-matrix membranes are relatively easy to build, but have far less potential than dense zeolite membranes [20]. Microporous Solids 5 Carbon molecular sieve membranes (CMSMs) are also thermally and chemically inert, and have been the subject of research for similar reasons as zeolite membranes. The porous framework of CMS is much more heterogeneous than that of zeolites, with only a few pores having molecular sieving properties. Thus, unlike with zeolite membranes, the selection performance is due to the combined effect of molecular sieving and surface diffusion [22]. CMSMs can be produced more easily than zeolite membranes, and have a pore size that can be regulated by manipulation of production parameters, such as temperature or processing time. Furthermore, different functional groups can be present at the surface of the pores, depending on the precursor used in their production. However, unlike zeolite membranes, CMSMs are extremely brittle, and must be handled carefully [23]. Microporous materials are one of the oldest examples of nanotechnology, as atomic structure is fundamental to their properties, and its understanding and manipulation is the central aspect of the science and engineering of microporous materials. Thus, any technology that uses microporous materials is, in a certain sense, “nanotechnology”. Contemporarily, micropores appear in nanotechnological applications mainly as discrete unidimensional nanotubes, not bulk materials. Bulk microporous solids can, nonetheless, be used in a range of nanotechnological applications where the typical functions of microporous solids as catalyst, adsorbent and molecular sieve fill specific needs in a larger process. In an example of stereoselectivity, microporous solids have recently been proposed as frameworks for confined polymerisation [24, 25], as matrices for the formation of arrays of noble metal nanowires/nanorods [26] and even as drug delivery agents [21, 27]. Given the many current and potential future uses of microporous materials, it is expected that they will continue to be used in increasing amounts, into the foreseeable future. New microporous materials, both amorphous and crystalline, can find use in traditional as well as in new applications. In traditional applications, it is usually desired that new materials present better performance properties, such as faster catalysis or greater adsorption. For new applications, new materials may be desired to present also different properties than traditional materials. For example, microporous thin-films can be used to functionalise surfaces, and are thus envisioned as tools for micro-scale process intensification. Such applications include micro-reactors, permselective barriers for gas sensors and Lab-on-a-chip devices [21]. Engineering of important characteristics such as pore size and shape, pore wall chemistry, tortuosity and chirality, among others, require control at the molecular level, which is difficult to accomplish with hard and inflexible inorganic solids. Several alternatives have been Chapter 1. Introduction 12 The hydrophobic nature of the pores in VA-class dipeptides makes them more appealing for traditional adsorption applications than FF-class dipeptides, given their easier regeneration. VA-class dipeptides have repeatedly been tested as adsorbents [25, 51, 63, 64, 67-74] and as catalysts [62]. They display interesting equilibrium separation properties [25, 64] and have been proposed as an ideal template for the study of single-file diffusion [63]. LS is a dipeptide forming hydrophobic pores, which does not share the crystal structure of the VA-class [75]. It is not an hydrophobic dipeptide, since serine is charged, and seems to be an exception to the rule that only dipeptides with two hydrophobic side-chains form porous structures [61]. VS crystals also have hydrophobic pores, but removal of co-crystallised solvent is significantly more difficult than with the other peptides [76]. FF-class crystals are, in fact, co-crystals, as the pores formed are completely filled with water [77]. It is not clear if the water can be completely removed, which may explain why they have attracted very little attention as potential gas adsorbents. Being the central focus of the work here reported, a more extensive introduction to these materials is given in the next section, in the broader context of peptide-based microporous solids. 1.3. Modelling Adsorption in Micropores Understanding of guest interaction with the host network of microporous solids naturally requires the adoption of phenomenologically correct and mathematically coherent models. At present, this is almost entirely missing for micropore adsorption. Neither surface models nor the filling models based on the Dubinin equations are able to describe in a phenomenologically meaningful way the process of adsorption in micropores. Thus, the need emerged to develop appropriate, yet simple, models for adsorption in the micropores of dipeptide crystals. In turn, these make excellent templates to test the simple models developed, being chemically and morphologically homogeneous. In micropores (especially ultramicropores, < 0.7 nm), the adsorption potentials start to coalesce, forming an adsorption environment where it is more accurate to talk of filling of the pores than layered deposition of molecules. In supermicropores (0.7 – 2.0 nm) a combined mechanism of layering and filling takes place, to different extents depending on pore size and Modelling Adsorption in Micropores 13 the species under consideration [78]. In both cases, adsorption equilibrium typically follows a Type I isotherm (Figure 1.5). The micropore filling concept first emerged in the context of studying microporous activated carbon [79]. Amorphous materials are hard to study, since observational determination of pore size distribution is quite troublesome and using the pore size distribution in calculations is also rather complicated. Thus, the overwhelming majority of authors opt for one of the several Dubinin equations/models. The Dubinin-Radushkevich equation, first proposed in 1947 [80], has the form, !=exp −!!! (1-1) where !≡−∆!!"#!!"# =!" ln ! ! ! (1-2) and ! is the ratio of adsorbed concentration and the maximum adsorbed concentration, ! is the so-called potential of adsorption, defined as the negative of the variation of Gibbs energy from liquid to the adsorbed state, ∆!!"#!!"#, ! is the characteristic energy of the adsorbent, ! is the universal gas constant, ! is absolute temperature, ! is the absolute pressure of the gas phase and ! ! is the saturation pressure of the vapour in the gas phase at !. Figure 1.5. Type I adsorption isotherm. For adsorption from liquid solutions, pressure is replaced by concentration. Chapter 1. Introduction 14 The Dubinin-Radushkevich equation emerged from an effort to apply the Potential Theory of Adsorption [81] to microporous activated carbon. It started as a fully empirical correlation between the potential of adsorption, !, and the relative amount adsorbed, !, for some microporous activated carbons. The fact that the equation is reminiscent of a Gaussian probability distribution function, led the authors to postulate that this reflected a Gaussian pore size distribution. Adsorption would thus proceed as the potential of adsorption of the gas/vapour equalled that of a given pore size, with step adsorption for each pore size. The authors referred to this phenomenon as “micropore filling”, imagining it as analogous to capillary condensation. They called this the Theory of Volume Filling of Micropores (TVFM). That the step-wise adsorption model is incorrect was soon recognised [82], but many attempts to reformulate it and its simplicity managed to not only maintain its popularity but also make it a reference model to understand micropore adsorption. The term “micropore filling” thus came to mean something slightly different than what it had meant originally, and is used to this day. The Dubinin-Asthakov equation was introduced [82] to extend applicability of the TVFM to zeolites. It replaced the exponent of the Dubinin-Radushkevich equation with a variable “n”, called the heterogeneity parameter. Introducing an extra fitting parameter allowed the equation to fit a greater number of adsorption isotherms. The new equation was interpreted as a form of the cumulative expression of the Weibull distribution (making the Dubinin- Radushkevich equation a form of the cumulative expression of a Rayleigh distribution), now expressing a distribution of adsorption energies inside the pores, instead of a pore size distribution. Working with a global variable such as the “adsorption potential” and simple equations such as the Dubinin-Radushkevich and Dubinin-Asthakov equations is an easy and expedient route to modelling and characterising the adsorption system. Which explains, together with the difficulty of working with pore size distributions, why they have been, and are, so successful. Nonetheless, there is a growing consensus [83, 84], even from authors that have been associated with Dubinin [85], that the Dubinin equations should be considered fully empirical. Micropore adsorption modelling is highly dependent on the type of adsorbent being considered. Crystalline materials, contrarily to amorphous materials, whose pore sizes and types are known, are usually modelled using thermodynamics-derived expressions, such as Langmuir, Volmer, Fowler-Guggenheim or Hill-de Boer [86]. These equations were Motivation and Outline 15 originally derived and used to describe surface adsorption, but were shown by Barrer [87] to be valid also for the porous frameworks of zeolites. The most commonly used equation is, by far, the Langmuir equation [25, 67], != !" 1+!" !" = ! 1−! (1-3) where ! is the so-called “affinity constant”. The Langmuir equation is still sometimes considered (wrongly) to be valid only for surface adsorption [86], due to it having been originally derived using kinetic arguments, better understood in the context of free surfaces. Independently of the configuration and morphology of the porous framework (see Section 5), the Langmuir equation applies for cases of localised adsorption with no significant interactions between adsorbed molecules. For distributed adsorption, the Volmer equation should be applied. The Volmer equation [86] is expressed as, !" = ! 1−!∙exp ! 1−! (1-4) During the work here reported, adsorption in the micropores of dipeptide crystals was understood to be well represented by both the Langmuir and Volmer equations, depending on the dipeptide and adsorbate used. 1.4. Motivation and Outline This thesis is primarily the result of a study of adsorption and mass transport properties of microporous crystals of hydrophobic dipeptides. During this study, the need emerged to systematise some of the theoretical models used to analyse the adsorption data, and the result of this investigation is also reported. Chapter 2 describes hydrophobic dipeptides with other peptide-based microporous solids, namely, cyclic peptides, dendritic peptides and metal-peptide frameworks. The evolution of research on these materials is outlined, with particular emphasis on their properties as molecular hosts. The structure of hydrophobic dipeptide crystals is discussed detailedly. Chapter 1. Introduction 16 Chapter 3 presents the experimental results of mass transport and adsorption with atmospheric gases (N2, O2 and Ar). Mass transport experiments were performed by single-crystal permeation of dipeptide crystals AA(non-porous) and LS (0.49 nm), with high permeabilities being observed. Adsorption experiments were performed in the four dipeptide crystals with the smallest pores, VI, IA, IV and VV. The results are analysed from practical and theoretical perspectives. The unusual selectivity†, αAr/O2 > 1, is very interesting for oxygen production from air by pressure swing adsorption. Although the materials are unlikely to be used in this application, they provide guidelines for the design of other adsorbents. This unusual selectivity is apparently connected to the extremely small pore size, 0.37-0.44 nm, although the way it influences adsorption is extremely complex. Chapter 4 formulates a new way of understanding adsorption in micropores, in agreement with the concept of pore filling and using precise thermodynamic arguments. The approach implemented is basically the same that has been applied for two-dimensional adsorbed phases for many decades, only extrapolated to three- and one-dimensional adsorbed phases, typical of adsorption in micropores. The isotherm equations derived for three- and one-dimensional spaces are the same already derived for two-dimensional space, with the exception of those requiring localised adsorption, not applicable to three-dimensional adsorption spaces. A lot of resistance still exists regarding the approach taken in Chapter 5, although it is based in simple and well-established principles of classical thermodynamics. Thus, to improve the ease of visualisation of the dynamics of adsorption, a simple kinetic derivation of the adsorption isotherm equations previously presented is given in Chapter 5. The kinetic derivation is based on kinetic gas theory and assuming the adsorbed phases behave like fluids, so the phenomenological accuracy of the derivation is not perfect. 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D.!D.!Do,!D.!Nicholson,!and!H.!D.!Do,!Adsorption!in!micropores!(nanopores):!a! computer!appraisal!of!the!Dubinin!equations,!Molec.!Sim.!35!(2009),!122–137.! 85.!F.# Stoeckli,# Dubinin#1s# Theory# and# its# Contribution# to# Adsorption# Science,# Russ.! Chem.!Bulletin!Int.!Ed.!50!(2001),!2265-2272.! 86.!D.!D.!Do,!in!Adsorption#Analysis:#Equilibria#and#Kinetics,!Ch.!2,!pp.!11-48!(Imperial! College!Press,!1998).! 87.!R.!M.!Barrer,!in!Zeolites#and#Clay#Minerals#as#Sorbents#and#Molecular#Sieves,!Ch.!3,! pp.!104-161!(Academic!Press!Inc.,!1978).! Chapter 2. Peptide-Based Microporous Solids 28 This same study showed that framework dynamics is as important as structure in determining CPNs’ mass transport properties. The use of α,γ-CPs creates nanotubes with mixed hydrophilic/ hydrophobic inner cavities, allowing for strong bonding both with polar and apolar molecules. However, the hydrophilic rim of the tube’s exterior has dramatic consequences on the behaviour of the permeating molecules. Small polar molecules, such as water, can easily enter the 5.4 Å nanotube, although hydrogen bonds between the molecules and the nanotube slowdown their diffusion through it. Slightly larger methanol molecules are completely unable to enter or, if “placed” inside, exit the nanotubes. Even more awkward, polar chloroform molecules can enter, but not leave, the tube, probably due to the hydrophilic outer rim of the tubes. These results are partly confirmed by the experimental work of Amorín et al. [23, 32], and open very interesting perspectives on storage and capture of small hydrophobic molecules, such as hydrogen or methane. A series of molecular dynamics studies have also shown that CP nanotubes possess remarkable water transport properties [38-42], in-line with results observed in other nanotubes [43, 44]. Strong flux dependence has been observed with pore size [39] and the chemical nature of inner pore walls [42]. Theoretical considerations support these results [41]. As the aforementioned examples show, most of the work so far done with CPs is related to solution-phase applications, where the nanotubes are used discretely to create nanochannels on specific locations of a vaster ensemble. Several peptides have been shown to incorporate into hydrophobic membranes [18, 34, 36], with some even showing anti-bacterial activity [45]. The mode of membrane permeation depends mainly on amino-acid sequence [45, 46]. Successful transport of monoatomic ions [18, 21], glucose [34] and glutamate ion [36] across CPs has been observed repeatedly in solution-phase trials. These materials also have a proven ability to create microcrystalline powders with an intact tubular structure [47]. Therefore, their utilisation as crystalline molecular sieves and gas adsorbents is only limited by their unknown stability after removal of co-crystallised solvent. There has been one instance of CPs being used as nanotubes in a microporous solid. An eight residue L,D–α–CP was used as the self-assembling nanotube-forming motif within a block copolymer (BCP) matrix, in order to create a microporous composite membrane [48]. The material synthesis starts with a pre-templated block copolymer thin-film with monodisperse hollow cylindrical pores (~3 nm). L,D-α-CPs are dissolved into the thin-film, after being conjugated with polymer strands, for improved solubility. Finally, a heating-cooling cycle (up to 180 ºC) of the material leads to self-assembly of the CP-polymer conjugates inside the Cyclic Peptides 29 block copolymer matrix, forming nanotubes within the original pores, but with the smaller CP inner diameter. This synthesis sequence is represented in Figure 2.3. Pore alignment was confirmed by microscopic and spectroscopic techniques. Pore opening was assessed through successful proton permeation experiments on a supported CP/BCP thin-film. These are very exciting results, that prove that CPs can be used in laboratory scale membrane modules. Nonetheless, further testing is necessary to assess the fraction of open pores and effective tortuosity, and determine single-channel permeability and inter-membrane reproducibility. Figure 2.3. Schematic diagram of the sequence of synthetic steps of a CP/BCP composite membrane, as described by Xu et al. [48]. Chapter 2. Peptide-Based Microporous Solids 30 Other intensely tested applications for CPs are related to their electric properties. In a similar way as natural helical peptides, with well known electron transfer properties [49-51], it has been proposed that CPs could be used as “molecular wires”, i.e. 1-D electron transfer structures, on nanoelectronic devices [52-57]. CPs are more stable than natural helical peptides, and easily assemble in 1-D structures, but are not electronically conductive enough to be of any use [54]. This was overcome by chemically manipulating the side-chains [55], with CPs thus serving as templating agents for the electronically active functional groups [52, 53]. Good optoelectronic properties have also led to the successful application of CPs in artificial photosynthetic devices [58]. Use of dendritic CPs enabled the synthesis of porous liquid crystals [59]. Macrocycles are a relatively common micropore-forming molecular architecture, and a few other macrocyclic microporous solids have been identified [60-63]. Xu et al. have used diacetylene macrocycles, self-assembled into columns via amide hydrogen bonds, as a CO2 adsorbent [63]. The material retained crystallinity after solvent removal and displayed Type I adsorption isotherms, consistent with micropore adsorption. Together with the previously mentioned mass transport properties, this indicates CPs should be able to be used as bulk microporous solids. It would also be interesting to assess the influence the gating effect previously tested [18, 34, 36, 47, 64] in solution, could have on gas desorption hysteresis and separation selectivity. 2.2. Dendritic Peptides Dendrimers, or cascade molecules, are three-dimensional polymeric macromolecules, with tree-like branches extending outwards from a multifunctional core [65-68]. Typically, their structure is interpreted as a sequence of layers (or generations) [69], with three components being distinguished in an n-generations dendrimer: the core (or root, generation 0), branching units (branched branches, generations 1 to n-1) and functional units (the last layer of branches, generation n). In solution, dendrimers usually take a globular structure, with functional units constituting a polyvalent outer surface. Dendrimers can be synthesised by: a) an iterative sequential multi-step reaction, where successive generations of branches are added to the original core molecule [70] (the divergent approach), or b) independent synthesis of each branch, sequentially merged from the nth to 0th generations, until all are linked to the core molecule (the convergent approach). Dendritic Peptides 31 When a core molecule, chemically distinct from the branching units, is absent, the resulting branched molecule is called a “dendron”. A dendrimer can be seen as a collection of dendrons linked through a core molecule. When several types of dendrons are present in the same molecule, the dendrimer is called “cascadane” [68]. The first materials combining peptides and dendrimers were peptide bond-containing dendrimers (“peptide dendrimers”). They have been extensively studied as drug and DNA delivery agents, in a wide range of biomedical applications, work that has been thoroughly reviewed before [70-73]. They were designed seeking to merge the water-solubility, biocompatibility and biodegradability of peptides (peptide bonds, in particular) with the stability and protein-like ligand selectivity and drug-delivery properties of dendrimers [70]. They are often used as “artificial proteins”, due to the globular structure adopted in solution and the polyvalence shown by the peptidyl functional units. In particular, lysine-based dendrimers, since their discovery in 1988, by Tam [74], have been extensively used as multiple antigene peptide (MAPs) systems [75-78]. The molecular structure of two such dendrimers is shown in Figure 2.4. The amino group on the side-chain can form amide bonds, just like the main-chain terminal amino group, allowing the construction of branched structures using well-known peptide-forming synthetic paths [74, 79]. Figure 2.4. Peptide dendrimers. (a) Diagram of the original lysine dendrimer used by Tam (adapted from [74]). (b) Diagram of a lipid functionalised lysine dendrimer (adapted from [80]). Dendryl-substituted dipeptides constitute another, more recent, type of material that synergistically combines the peptide bond and dendrimeric structures. These porous materials, stable in solution and solid phase, were first reported in 2004, by Percec et al. [81], Chapter 2. Peptide-Based Microporous Solids 32 following on their research on column-forming dendrons [82]. A representation of a pore formed by dendritic peptides can be seen in Figure 2.5. Several enantiomers of Boc-Tyr-Ala- OMe were used, with a two-generation, benzyl ether-based dendron linked to the oxygen atom of the Tyr residue. They readily self-assemble in bulk and in solution [83], through thermal treatment. The resulting supramolecular polymer is structured by intermolecular hydrogen bonds and hydrophobic interactions between different dendrons. Heating causes two- and three-dimensional organization of the columnar assemblies, in the solid state, while, reversely, in solution, cooling is necessary to promote self-assembly. Careful annealing of solid samples allows formation of a crystalline (as the one shown in Figure 2.5) or fibrillar arrangement of the columnar supramolecular polymer [83]. X-ray diffraction analyses of these samples allowed the determination of crystal structure and, therefore, internal pore size and external diameter of the columnar assemblies. Internal pore diameters are usually within the 6-16 Å range [84, 85], but extension of the lower limit down to 2-3 Å seems easily viable. It has been observed that residue chirality can significantly influence pore size [86, 87]. Figure 2.5. Pore, 9.6 Å-wide, observed in the crystal structure of the dendritic peptide (4-3,4- 3,5)12G2-CH2-Boc-L-Tyr-L-Phe-OMe (adapted from [88]). As represented in Figure 2.6, pores are formed by dendritic dipeptides as individual molecules pack consecutively in a helical fashion, with hydrogen bonds between dipeptide main-chains holding together the pore. Nonetheless, contributions from hydrophobic forces are also essential in stabilising the structure. The interior wall is constituted by the protective groups stemming from the dipeptide molecules’ termini. The formed pore can be seen as a Dendritic Peptides 33 supramolecular quasi-macrocycle, but displays few of the most interesting properties of covalent macrocycles, such as intrinsic microporosity. On the other hand, pore chirality [85] and, probably, radial framework flexibility are interesting characteristics of the material that are practically absent in covalent macrocycles. Pore-size control can be easily achieved by changing the protective groups of the dipeptide chain supporting the dendrons [84]. Unlike what would be intuitively expected, the bigger the protective group, the bigger the pore. This results from the greater difficulty of packing all the molecules, increasing the average distances between molecules and, thus, increasing pore size. However, this also increases the average distance of inter-chain hydrogen bonds making the structure less stable. Figure 2.6. Schematic representation of pore-forming aggregation mechanism of dendritic peptides (adapted from [86]). Results from solution phase proton permeation tests were presented by Percec et al. in the same article where the material was introduced [81]. The dendritic dipeptide nanotubes were allowed to self-assemble onto phospholipidic vesicles, serving as proton conductors in a pH- monitoring assay. Gramicidin was also used in a parallel assay, yielding comparable results. In similar experiments, Kaucher et al. have used nanotubes built from slightly modified dendritic dipeptides, with the benzyl group replaced by a naphthyl group [89]. Once again, protons were successfully transferred across the nanotubes, but assays with Li+, Na+ and Clyielded negative results. The tube’s internal diameter is (14.5 ± 1.5) Å, more than enough to allow the permeation of the hydrated form of these ions [90], being hard to understand what causes this ion selectivity. Furthermore, water transport across the nanotubes was also measured in independent osmotic pressure-driven experiments, based on measuring the volume of a unilamellar vesicle. Therefore, the rejection mechanism for the ions cannot involve water rejection. Further study on this issue could shed light on the rejection mechanism, which would be very interesting, as it could unfold a new “gate-keeping” strategy. Chapter 2. Peptide-Based Microporous Solids 34 Solid phase adsorption or permeation experiments have not, to the best of our knowledge, been performed with these materials although this seems perfectly viable. Important properties such as pore size and inner wall chemistry can also be controlled through substitution of the non-dendron residue. [88] As with CPs, the issue of stability after cocrystallised solvent removal (when present) could be the main unknown to hinder their use as microporous solids. 2.3. Dipeptides Large peptides and proteins form crystals with complex molecular structures, having several organisational levels. Amino acids, on the other hand, crystallise in simple, head-to-tail hydrogen-bonded chains. Small peptides form crystal structures sharing some aspects of both, being sometimes more akin to amino acids and others to proteins [91]. The discovery that hydrophobic dipeptides form porous crystalline frameworks resulted from the exploration of the crystal structures of small peptides. Amino acids crystallise by the creation of parallel infinite head-to-tail hydrogen-bonded chains [91-93], with each of the three amino hydrogen atoms of a given molecule bonding to a carboxylic oxygen atom from three different other molecules. Amino acids with hydrophobic side-chains form segregated layers of hydrophobic and hydrophilic moieties. When the side-chain is able to form hydrogen-bonds, a somewhat more complex network may be created, although segregation of hydrophobic and hydrophilic moieties still occurs. A representation of the typical crystal structure of an amino acid (valine, in this case) with an hydrophobic side-chain is shown in Figure 2.7. The layered segregation of the hydrophobic and hydrophilic moieties is clearly visible, both in Figure 2.7b and Figure 2.7c. The crystalline structure is stabilised by London forces between the aliphatic side-chains and hydrogen bonds between the amine and carboxylate groups. Although the focus is usually on the latter, the former can be just as important, or even more. However, as it is hard to quantify, it typically receives much less attention. This fact will become clear when discussing dipeptides. The hydrophilic layer is formed by head-to-tail hydrogen-bonded chains of carboxylate and amino groups, while the isopropyl side-chains form the hydrophobic layer. Each molecule bonds via six hydrogen bonds to five other molecules, four lateral and one frontal (as shown Dipeptides 35 in Figure 2.7d through Figure 2.7f). The frontal molecule is bound by two hydrogen bonds, one involving the amino group and another involving the carboxylate group. The four lateral molecules surrounding each molecule are connected each by one hydrogen bond. Two sides are connected by each of the oxygen atoms (left and bottom of Figure 2.7c) and the other two sides are connected via the amine group. Figure 2.7. (a) Crystalline conformation of a valine molecule. (b) Crystal structure of valine viewed along the c-crystallographic axis. (c) Crystal structure of valine viewed along the acrystallographic axis (frontal view of (b) is now the bottom view). (d), (e) and (f) are different views of the same representation of the hydrogen bond network in the crystalline structure of valine. Only nine molecules are represented. Hydrogen atoms not shown for clarity. Hydrogen bonds are represented as pale blue lines connecting nitrogen (blue) and oxygen (red) atoms. Peptides (< 50 residues) and proteins (> 50 residues) are polymers of amino acids, and form far more complex crystal structures than single amino acids [94]. The basic supramolecular structures of both large peptides and proteins (the “secondary” structures) are the alpha-helix and the beta-sheet [94]. Representations of these structures are shown in Figure 2.8. Alphahelices are peptide chains coiled on themselves, stabilised by amide hydrogen bonds, O···HN, with 3.6 residues per turn. Beta-sheets are parallel peptide chains stabilised by amide hydrogen bonds between the two chains. Chapter 2. Peptide-Based Microporous Solids 36 Figure 2.8. Representation of secondary structures of peptides and proteins. (a) Alpha-helix. (b) Beta-sheet. Hydrogen bonds represented as dotted green lines. Adapted from [95]. Unlike large peptides and proteins, in the structure of short peptides, the amino and carboxylate termini are fundamental in determining their crystal structure. The longer the peptide, the smaller the influence of these groups, as the relative importance of the functional groups in the side-chains and the amide groups increases. The structure formed by very small peptides is thus somewhere in between those of larger peptides and those of amino acids [93]. There is no cut-off at which peptides of a given size start forming alpha-helices and betasheets. The more hydrogen bonds involving the amide group there are, the more structures will emerge that are more and more like alpha-helices and beta-sheets. The simplest dipeptide, glycylglycine, can form a crystal structure with a beta-sheet consisting of a single amide hydrogen bond [96]. As described below, and as has been summarised in Chapter 1.2, dipeptides with bulkier side-chains will not form these structures. Dipeptides cannot form alpha-helices, but tripeptides can have alpha-helix-like molecular conformations, and some tetrapeptides form full-fledged alpha-helices. For crystal structures of pentapeptides and above, alpha-helices are relatively common [93]. Tripeptide and tetrapeptide molecules can adopt many different conformations in a crystal structure [93], forming very complex three-dimensional hydrogen-bond networks. These will still exhibit separation between hydrophobic and hydrophilic moieties, in the form of columns or layers [93]. Dipeptides, on the other hand, due to the rigidity of the peptide bond, have basically only two possible molecular conformations. They are either in a parallel (side-chains pointing in the Dipeptides 37 same direction) or an antiparallel (side-chains pointing in opposite directions) conformation. As shown in Figure 2.9, Glycylglycine, lacking side-chains, adopts a crystal structure very much like that of glycine, only (as already mentioned) with an extra amide hydrogen bond connecting parallel molecules. Dipeptides with side-chains cannot pack with this structure, as the main chains are too separated from each other to form hydrogen bonds. An antiparallel conformation, where the third amino hydrogen atom is accepted by one of the side-chains, is the most common arrangement [97]. Alternatively, the third amino hydrogen atom may be accepted by co-crystallised solvent molecules, if they can function as an hydrogen bond acceptor. Structures such as the one shown in Figure 2.9, for valine, are made possible for dipeptides precisely if the dipeptide molecules have a parallel conformation and the cocrystallised solvent functions as a proton acceptor in hydrogen bonds [97]. Nonetheless, as discussed below, there is a class of pore-forming dipeptides displaying such a structure, although with columns, not layers (see Figure 2.12). Figure 2.9. Hydrogen bond network in a sheet of the crystal structure of (a) Glycylglycine and (b) Glycine. In both cases, the crystal structure is made by parallel sheets such as those represented, bound by hydrogen bonds between the amine and carboxylic groups. It was curiosity regarding which crystal structure would be favoured if neither the side-chain or solvent hydrogen bond acceptor was available that led C. H. Görbitz to investigate hydrophobic dipeptides [13]. The first report on microporous dipeptide crystals was presented by Görbitz, in 1996 [13], in an article on the crystal structure of Valylalanine (VA). In the crystal structure of this dipeptide, the packing problem is resolved by the formation of a complex three-dimensional network of hydrogen bonds, that does away with the head-to-tail configuration. Chapter 2. Peptide-Based Microporous Solids 44 Figure 2.15. (a) Representation of zones of preferential adsorption in VA-class dipeptides and LS, as determined from Grand Canonical Monte Carlo simulations (adapted from [112]). (b) Adsorption isotherms of CO2 and CH4 in VA and AV, at 195 K (adapted from [113]). (c) Adsorption isotherms of CO2 and CH4 in IV at 195 K (adapted from [113]). This was not the first time guest substitution was observed in these materials. In 2002, Görbitz reported [98] that an AV crystal, synthesised by acetonitrile diffusion into an aqueous solution of the dipeptide, could undergo successive solvent substitution cycles, with different solvents. With 2-propanol, the pores are greatly deformed, trapping the solvent molecules inside the pores, thus forming a clathrate. In 2004, Soldatov et al. reported successful Xe adsorption on Ala-Val and Val-Ala in an article arguing that dipeptide crystals could be used as microporous materials [114]. These early results were even more significant due to the pore expansion that had to take place to allow adsorption to occur, showing great framework flexibility. This property was further investigated and confirmed in a 2006 article [109], this time for all VA-class dipeptides. By using X-ray crystallography and NMR spectroscopy, it Dipeptides 45 was also possible to determine pore size and helicity and internal pore morphology. Results from this article were used in Table 1.2, from Chapter 1.2. Recent experiments confirmed the capacity these materials have to adsorb and transport gases with small molecules [112, 113, 115] proving they can be used as adsorbents. Comotti et al. have used AV, VA, IV and VI as adsorbents of CO2, CH4 and H2, observing high and fully reversible adsorption for all gases. Some of these results for CO2 and CH4 are shown in Figure 2.15b and Figure 2.15c. The authors obtained sorption selectivities of 3.5-5 for CO2/CH4 using IV, Figure 2.15c. It was also found that IV has a particular affinity for hydrogen, adsorbing 0.5 mol / mol of IV at 77 K and atmospheric pressure. In a follow-up study, in 2013 [112], the same authors showed that CO2 and also N2 adsorb in all crystalline dipeptides of the VA-class (plus LS) at room temperature. As expected, CO2 adsorbs much more than N2. GCMC simulations and NMR experiments showed that, for most dipeptides, CO2 adsorption is essentially localised (Figure 2.15a). Only in the straight and large pores of LS, is adsorption almost perfectly distributed. Taking advantage of the high CO2/CH4 adsorption ratios (selectivities), the authors performed breakthrough experiments with equimolar CH4/CO2 mixtures, using VI, IV and AV as adsorbents. Using VI, an exit stream with up to ~ 3:1 CH4/CO2 ratios could be obtained. Another potential use as adsorbent was put forth by Afonso et al. [115], by showing that some VA-class dipeptides display Ar/O2 selectivity. This is a rare characteristic, sought for the production of O2 from air by adsorption-based separation processes. Testing VI, IA, IV and VV (the four VA-class dipeptides with the smallest pores), the authors discovered that all display Ar/O2 positive selectivity. It was also observed that selectivities decrease with increasing pore size, indicating this factor is central to the observed effect. The properties observed in these studies are consistent with those of classical hydrophobic solids, such as pure silica zeolites and activated carbon, with the exception of the anomalous Ar/O2 selectivity. The low adsorption of N2 when compared to O2 and Ar is also observed in MOFs, but completely different from those of zeolites. N2 probably has much higher affinity to organic chains than to the Si-O-Si chains of zeolites. FF-class microporous crystals have never been tested as adsorbents, probably because the cocrystallised water is difficult to remove, especially the innermost layer. In dipeptides with very small pores, it is questionable if it will even be possible to remove any of the cocrystallised water. The work of Amdursky et al. [116, 117] suggests the FF crystal structure Chapter 2. Peptide-Based Microporous Solids 46 undergoes an irreversible phase transition when heated to 150 ºC, creating a cyclic compound with a completely different crystal structure. Although significant hopes have been placed on peptide-based solids to work as smart materials for a number of applications [114], they have yet to show any of the expected properties, in an analogous way as, for example, MOFs have [118, 119]. It would be interesting to see what could be achieved by bringing together the pore-size-based sieving selectivity of VA-class dipeptides and the “gate-keeper” functionality predicted by García- Fandiño et al. [37] for peptide macrocycles, where the functional groups on the rim of the tube entrance determined which species‘ molecules were able to enter it. This would be a first step in mimicking the successful combination found in biological transmembrane channels, which yield nearly single-species specificity without compromising flux [3]. A 2007 study by Sopher et al. [120] discussed the integration of FF “peptide nanotubes” on micro-fabrication lithographic processes. Some of these structures are probably of crystalline nature [121] and provide good indicators for effective crystal stability during processing. The study confirmed FF crystals do not withstand hydrophobic solvents and are etched using oxygen plasma. They are also unable to retain their structure above 100 ºC, and should not be used above 85 ºC. They do, however, not only resist contact with water, but are also able to permeate aqueous solutions. Although only slightly, FF is hydrosoluble, making this quite a remarkable result. Although their slow water solubility kinetics have been observed before [97], it is possible that partial encapsulation in a hydrophobic polymer such as PDMS stabilises the structure even more. This stabilisation effect should be further investigated, since it opens up the possibility of using microporous peptide crystals with all sorts of solvents and reagents previously thought incompatible with them, whether this happens with full or partial encapsulation. FF is also supposed to be able to form “fibrils”, extensively studied by Gazit et al. [120, 122- 124]. However, there has never been any direct evidence that “fibrils” with pores tens of nanometers-long, as originally claimed [122], are produced by FF. The best evidence available indicates that FF dipeptides assemble into fully crystalline structures, possessing unidimensional micropores, as those originally reported by Görbitz, [105, 125], and also that these crystals may, sometimes, form pores a few micrometers-wide [117], as shown in Figure 2.16. Peptide-Based MOFs 47 Figure 2.16. Electron Microscopy photographs of FF PNTs grown by rapid water evaporation from an aqueous FF solution. The inset shows a zoomed-in view of an individual PNT. The hollow crystals are clearly visible. The white scale bar is 20 µm long for the main image and 2 µm for the inset (adapted from [117]). 2.4. Peptide-Based MOFs Although previous work had been done with amino-acids [126], peptide-based metal-organic frameworks (metal-peptide frameworks, MPFs) constitute a relatively recent field, whose uniqueness is still being explored [14, 127-129]. They are not structurally and functionally different from other MOFs, but peptides’ high structural diversity, conformational flexibility and intrinsic chirality raised hopes of unwinding properties such as gated-adsorption and chiral recognition [128]. Metal cations are coordinated by oxygen and nitrogen atoms on the peptides, present, at least, on the amino, carboxylic and amide functional groups of the peptide [14, 129]. Initial works on the subject intended to use peptide-metal complexes as models for more complex protein-metal complexes, ubiquitous in the natural world and whose structure is hard to determine [127]. It was not long before these complexes were identified and studied as MOFs [14, 128], resulting in a variety of 2D and 3D-porous architectures. In 2010, Matthew Rosseinsky and co-workers reported [129] an MPF where tetrahedral zinc ions coordinate GA linkers in Zn(GA)2, resulting in bilobal pores with Ala side-chains Chapter 2. Peptide-Based Microporous Solids 48 pointing torwards the pore interior, as can be seen in Figure 2.17. Each peptide molecule is linked to two zinc ions, forming a grid-like layer of the coordination polymer. Different layers are connected through hydrogen bonds between the amide groups of each dipeptide. CO2 adsorption experiments were performed, yielding very promising results. In particular, a pronounced gating effect, resulting from a conformational shift on the C-C bond of the dipeptide main-chain, was observed, a phenomenon typical of pore-forming proteins. The MPF was synthesised in a liquid mixture, with 90 % methanol and 10 % water. These solvents were co-crystallised with the solid, occupying the pores in the structure shown in Figure 2.17a. Following desolvation, the pores collapse upon changes in the torsional angles of the two residues, bringing the methyl side-chains closer together, and eliminating the solid’s porosity. This is not the first time guest-sensitive porosity has been observed for MOFs. The well-known MIL-53 displays a framework with a so-called “breathing” behaviour, contracting and expanding according to the molecular size of the adsorbed species [130]. However, the behaviour of MPF Zn(GA)2 is a bit more complex. Its desolvated structure is uncapable of adsorbing H2 and N2, even at temperatures as low as 77 K. With regards to these species, the material is, effectively, non-porous. On the other hand, CO2 adsorption and H2O and CH3OH resolvation are very easy to accomplish, almost as if the structure opens specifically for these species. CO2 adsorption is accompanied by gradual pore opening, increasing proportionally to the amount of adsorbed CO2. This mechanism is controlled by (also gradual) changes in the torsion angles of the dipeptide linkers. Although there is a pore-opening minimum pressure for CO2 adsorption, the selection mechanism is thought to be associated with the establishment of polar interactions between CO2 molecules and the amide bonds, thus stabilising the mixture. The inexistence of these bonds in H2 and N2 makes the sorbed state thermodynamically unfavourable. The small energy landscape associated with the structure’s conformational shift makes such a mechanism possible, allowing small differences in pore wall-molecule interactions to determine pore opening or closing. The Rosseinsky group has since continued their work on MPFs. Using peptide linkers only slightly different from GA, the group observed drastic changes in the properties of the porous framework [131, 132]. This effect is reminiscent of the dramatic changes a single-point mutation can create in the structure and functional properties of proteins. It is precisely this kind of structure-function relation that was originally hoped for, when deciding to use peptides as linkers in MOFs. Peptide-Based MOFs 49 Figure 2.17. (a) Structure of the MPF [Zn(GA)2] viewed along the c-axis. (b) Threedimensional perspective view of the pores formed by [Zn(GA)2]. (c) Sorption and desorption CO2 isotherms (closed and open circles, respectively) at 273 K. Adapted from [129]. Using GT instead of GA, the authors were able to synthesise Zn(GT)2, a compound whose crystal structure displays a rigid framework with permanent porosity [131]. The crystal structure of Zn(GT)2 is significantly different from that of Zn(GA)2. The zinc cation now forms six coordination bonds, instead of four. However, because two peptide linkers per node bond twice to Zn, through the amine and the amide groups, the linker to node ratio remains 2:1. The coordination framework formed is now three-dimensional, instead of twodimensional, but the pores are still one-dimensional. The side-chains do not point directly inwards, only the methyl substituent from the threonine side-chain does, with the hydroxyl groups forming hydrogen bonds with other N and O containing groups. These extra hydrogen bonds and the higher coordination number of zinc are probably at the root of the increased stability of the framework. Zn(GT)2 adsorbs CO2 and CH4, in a ratio of 14:1 (w/w), at 195 K, a relatively normal result. The authors attribute this to the polar nature of the pores, which potentiates strong interactions with the large quadrupole moment of CO2. CH4, having a null quadrupole moment is not as strongly adsorbed. The authors also claim that N2 is virtually unadsorbed. Although they did not perform an experiment at 195 K, as for the other gases, a 77 K test did, indeed, reveal some N2 adsorption. Chapter 2. Peptide-Based Microporous Solids 50 Figure 2.18. (a) Phase diagram for crystalline solid mixtures of the three Zn-containing MPFs tested by the Rosseinsky group. (b) CO2 adsorption isotherms for solid mixtures with the formula Zn[(GA)x(GS)y]2. Adapted from [132]. The authors then tested Zn(GS)2 [132], which is isostructural with Zn(GA)2. The inward directed side-chains now have an hydroxyl group, changing pore chemistry and adsorption properties. As in Zn(GA)2, the framework of Zn(GS)2 opens and closes depending on the presence or absence of adsorbate molecules. However, unlike Zn(GA)2, it only opens to polar molecules, capable of interacting strongly with the hydroxyl group inside the pore. Thus, not only N2 and H2 are not adsorbed, as with Zn(GA)2, but also CO2 cannot enter the pores. However, resolvation with methanol is achieved easily, showing that the structure is not permanently closed, but opens or closes to specific chemical species. Once again, this behaviour is typical of biological systems, but almost never found in synthetic nanomaterials. In this paper, the authors also reported the study of the properties of solid crystalline mixtures between Zn(GA)2, Zn(GS)2 and Zn(GT)2. Only one tricomponent mixture was tested, with equal parts of the three linkers, having been observed it adopted the Zn(GA)2 conformation, of opening and closing pores. Binary mixtures incorporating threonine may have Zn(GA)2- like or Zn(GT)2-like structure, depending on the relative proportions of the linkers. Zn((GA)x(GT)y)2) mixtures are isotructural with Zn(GA)2 for GT proportions inferior to 25 %, showing this is a much more stable conformation. Zn((GS)x(GT)y)2) mixtures are isotructural with Zn(GA)2 for GT proportions inferior to ~50 %, as shown in Figure 2.18a. The hydroxyl group in the serine side-chain stabilises the open structure for higher GS proportion than with GA. Conclusions 51 For GA- and GS-containing binary mixtures, crystal structure is independent of linker proportion. Conversely, CO2 adsorption is highly dependent on structure, given the very different behaviour of both pure component adsorbents. As GS proportion increases, the porous solid requires a higher pressure to start adsorbing and exhibits progressively lower affinity to CO2. For GS proportions above 60 %, the material is essentially closed to CO2 molecules (Figure 2.18). Also from the Rosseinsky group, another zinc-based MPF was reported [133], containing imidazole and forming a crystal structure akin to that of a zeolitic imidazole framework (ZIF). The resulting three-dimensional structure of the coordination crystal has unidimensional pores 0.5 nm wide. Both the zinc cation and the peptide form four coordination bonds, all with diferent cations/peptides, resulting in a peptide to metal ratio of 1:1. The peptide, carnosine (ß-alanyl-L-histidin), possesses a beta amino acid residue that confers it extra configurational flexibility. The material is stable in water and organic solvents, but its crystal structure is dependent on the guest. Due to the conformational flexibility conferred by ß-alanyl, adsorbed water is able to form a chain of hydrogen bonds that deforms the pores relatively to adsorption of other species. Another recent study [134] has reported two MPFs co-crystallised with water, where it is not possible to remove the water molecules from the cavities they occupy in the crystal structure. Thus, these two MPFs, are effectively non-porous. There have also been many other cases where, contrary to the claims of the authors, the porous MOFs discussed do not have peptide linkers [135-137]. 2.5. Conclusions A wide range of applications based on microporous solids constitute today a stalwart foundation of the chemical industry, ranging from adsorption based separation processes to catalysis, and, more recently, membrane processes. Many of the new applications envisioned for porous solids require a tight control over very specific material properties. A “bottom-up” approach to doing this involves controlling the chemical nature of the building blocks of the material, thus controlling its morphology and structure. Supramolecular porous solids are one of the most intensely studied alternatives to the creation of such materials, due to their wellknown conformational flexibility and controllable composition and structure. Possessing a unique set of properties, and hoping to replicate the incredibly fast and selective molecular Chapter 2. Peptide-Based Microporous Solids 52 transport occurring in cellular protein channels, several peptide-based porous materials have emerged in recent years as one of the most interesting and promising kinds of supramolecular porous solids. Crystals of cyclic peptides, dendritic peptides, hydrophobic dipeptides and metal-dipeptide frameworks possess permanent microporosity, long-term stability and tunable characteristics, such as pore size, inner wall chemistry and porous network geometry. More significantly, framework and conformational flexibility lead to an adaptable porous network, "sensitive" to the chemical species being adsorbed. These characteristics are controllable through residue substitution, similarly to the structure-property relation of proteins in the natural world. Gatekeeping effects have been theoretically predicted on cyclic peptides, with functional groups far simpler than those of naturally occurring transmembrane ion channels. Dipeptides and metal-dipeptide frameworks have successfully been tested as gas adsorbents, showing their guest-free stability and potential as microporous solids. High gas uptakes were observed in both cases, which are partially explained by the adaptable nature of the porous network. Just as with biological systems, “single-point mutations” can have dramatic effects on both structure and function of these materials. References 53 2.6. References 1.! P.! D.! Santis,! S.! Morosetti,! and! R.! Rizzo,! Conformational! Analysis! of! Regular! Enantiomeric!Sequences,!Macromolecules!7!(1974),!52-58.! 2.! I.! L.! Karle,! B.! K.! Handa,! and! C.! H.! Hassall,! The! conformation! of! the! cyclic! tetrapeptide! 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G.!J.!Miller,!Y.!Z.!Khimyak,!G.!R.!Darling,!N.!G.!Berry,!and!M.!J.!Rosseinsky,!Sidechain! control! of! porosity! closure! in! single-! and! multiple-peptide-based! porous! materials!by!cooperative!folding,!Nature!Chem.!6!(2014),!343-351.! 133.!A.#P.#Katsoulidis,#K.#S.#Park,#D.#Antypov,#C.#Martí-Gastaldo,!G.!J.!Miller,!J.!E.!Warren,! C.!M.!Robertson,!F.!Blanc,!G.!R.!Darling,!N.!G.!Berry,!J.!A.!Purton,!D.!J.!Adams,!and!M.! J.!Rosseinsky,!Guest-Adaptable!and!Water-Stable!Peptide-Based!Porous!Materials! by!Imidazolate!Side!Chain!Control,!Angew.!Chem.!Int.!Ed.!53!(2014),!193–198.! 134.! S.!Emami,!F.!A.!A.!Paz,!A.!Mendes,!and!L.!Gales,!Toward!the!Construction!of!3D! Dipeptide−Metal!Frameworks,!Cryst.!Growth!Des.!14!(2014),!4777−4780.! 135.! B.!Lou!and!X.!Huang,!A!Homochiral!Metal-Dipeptide!Supramolecular!Framework! Including!a!Hydrogen-bonded!Guest!Network!of!Water!and!Uncoordinated!4,4- Bipyridine,!Z.!Anorg.!Allg.!Chem.!638!(2012),!1855–1860.! 136.! R.! Miyake,! C.! Kuwata,! and! Y.! Masumoto,! Selective! CO2! gas! adsorption! in! the! narrow!crystalline!cavities!of!flexible!peptide!metallo-macrocycles,!Dalton!Trans.! 44!(2015),!2993-2996.! 137.!W.-x.! Chen,! L.! Tan,! Q.-p.! Liu,! Y.! Zhou,! Y.-x.! Fan,! G.-r.! Qiang,! and! G.-l.! Zhuang,! Lanthanide-based!metal–peptide!frameworks!prepared!by!ionothermal!method:! Anion! direct! effect,! DFT! calculation! and! luminescence! property,! Inorg.! Chem.! Comm.!42!(2014),!29–32.! 61 Chapter 3. Adsorption and Diffusion Inside Porous Crystals* 3.1. Abstract In the first part of this chapter, permeation results of single dipeptide crystals are shown and discussed. Both high permeabilities and ideal selectivities are reported. These results are then put into perspective with new data obtained afterwards concerning the adsorption isotherms in dipeptide powder samples and additional single crystal permeation experiments. In the second part of this chapter, the adsorption isotherms of nitrogen, oxygen and argon in four VA-class hydrophobic dipeptides are presented. Isotherms were determined at 5, 20 and 35 ºC, for a pressure range of 0-6 bar. Under these conditions, adsorption is still in the Henry region. For all materials and temperatures, the sequence of preferential adsorption is Ar>O2>N2, a highly abnormal result. At 5 ºC, the dipeptide with the smallest pores, VI, has Ar/O2 adsorption equilibrium selectivities up to 1.30, the highest ever measured in Ag-free adsorbents. Gas uptakes, at 1 bar and 20 ºC, are ~0.05 mol·kg-1, very low relative values that are partially explained by the low porosity of the solids (< 10 %). The significance of these results for the development of new materials for the process of O2 generation by pressure swing adsorption (PSA) is discussed. The results hint at some of the structural and chemical properties that prospective Ag-free adsorbents might have in order to exhibit Ar/O2 selectivity; hydrophobic pores, less than 0.5 nm-wide, and porosity of, at least, 20 %. 3.2. Introduction Crystalline hydrophobic dipeptides of the VA-class [1, 2] are a type of ultramicroporous solids that has received considerable attention in recent years [3-6]. The structure of these molecular crystals is stabilised by H-bonds and hydrophobic interactions, being essentially the same for all VA-class dipeptides. This crystal structure displays an array of identical, * Adapted from Afonso et al., Angew. Chem. Int. Ed. 49 (2010), 3034–3036 and Afonso et al., Phys. Chem. Chem. Phys. 16 (2014), 19386-19393. Chapter 3. Adsorption and Diffusion Inside Porous Crystals 62 parallel unidimensional micropores [2, 7]. The pore walls are formed by the aliphatic sidechains of the dipeptides, making the pores highly hydrophobic [1]. Different side-chain combinations create slightly different crystal structures, with different pore sizes [1]. The micropores are helical and have an approximately circular cross-section, somewhat uniform throughout the unit cell [4, 7]. Nominal, average pore sizes of VA-class dipeptide crystals range from 0.37 nm to 0.50 nm [2, 7]. Crystal porosity has been proven several times, by determination of adsorption isotherms [4, 8, 9], and through 129Xe NMR [3, 5-7, 10, 11]. Additionally, when embedded in solvents of different colours, the crystals also become uniformly coloured [12], indicating thorough solvent permeation into the pores. Recent NMR experiments have also shown that the pores in VA and AV dipeptide crystals typically span the entire crystal length [6]. Dipeptide crystals thus have incredibly open pores. Moreover, the supramolecular framework of the dipeptide crystals is very flexible, allowing penetration of guests bigger than the guest-free pore [8, 13, 14]. This observation is supported by vibration spectroscopy [15] and 129Xe NMR [3] studies. For guest molecules smaller than the pore, there has yet to be demonstrated any clear effect resulting from framework flexibility [4]. 3.3. Preliminary Studies: Single-Crystal Permeation of Atmospheric Gases Herein, we report the use of dipeptide crystals as permselective materials. Although this looks like an obvious engineering application for the kind of porous topology present in the crystals, there are issues that call for an experimental support, namely i) potential crystal defects, like twinning or fractures may greatly diminish their actual selectivity and ii) potential lack of rigidity of the crystal structure, allowing the pores to adapt to some extent to the size of the guest molecules. We envisage the selective permeation of argon, nitrogen and oxygen (main components of air) through dipeptide crystals. This is a highly relevant industrial separation process and also a very ambitious one given the similarity of the molecular sizes of the individual components. The dipeptide crystals that were tested as single-crystal membranes are L-leucyl-L-serine (LS), L-valyl–L-isoleucine (VI) and L-alanyl-L-alanine (AA) crystals. The peptides were crystallized and their structures determined by X-ray diffraction (Figure 3.1). The structures of all three peptides had been resolved previously [1, 16, 17]. The VI Preliminary Studies: Single-Crystal Permeation of Atmospheric Gases 63 crystal packing has hexagonal symmetry with molecules forming helices with six dipeptides per turn. LS crystals have a unique crystal packing with the inner walls formed by leucine side chains and with right-handed heliticity. AA packs in the tetragonal space group I4 and the crystal arrangement is characterized by the segregation of the hydrophobic methyl groups into columns. The calculated void volumes in the three crystal structures that are accessible to He, the molecule with the smallest kinetic diameter (2.6 Å), are shown in Figure 3.1a. LS and VI contain nanochannels while AA should be considered nonporous. The average channel diameters of LS and VI are displayed in Table 3.1. The LS, VI and AA single crystal permeabilities towards O2, N2, Ar and He were determined at room temperature (Table 3.1). The LS crystals are permeable to all the gas molecules and the respective selectivities are low, probably because the channels size is much bigger than the van der Waals diameter of the guest molecules. Figure 3.1. (a) Void volumes of the crystals structures of AA, VI and LS that can hold a spherical “probe” with a diameter of 2.6 Å. The calculation and visualisation of the void volumes were carried using the software Mercury 2.2 with 0.1 Å of grid spacing [18]. (b) Crystal structures of the dipeptides viewed along the c-crystallographic axis. Hydrogen atoms omitted for clarity. (c) Structural formulae of the three dipeptides. Thus, we have decided to test VI crystals because they display narrower channels. We observed that VI crystals are permeable to O2 and N2 but not to Ar (Table 3.1). However, the selectivity achieved for the O2/N2 (1.2) separation is too low to be of any practical significance, which prompted us to search for dipeptides forming smaller pores. We decided Chapter 3. Adsorption and Diffusion Inside Porous Crystals 64 to study the nominally non-porous AA crystals, given that, despite the fact of the pores being too small, the dynamics of the crystal matrix had never been investigated. Remarkably, it was observed that the AA crystals are permeable to O2 but not to N2 or Ar (Table 3.1). The permeability of the AA crystals towards the smaller He molecules is lower than towards O2, indicating that the host crystal matrix seems to respond individually to each particular guest molecule. The minimum measurable permeabilities of VI and AA determine the maximum selectivities corresponding to the cases of undetected permeation. For VI, the minimum O2/Ar selectivity that could generate such a result is 135. For AA, it is 124 for both O2/N2 and O2/Ar. The unexpected penetration of guest molecules into too narrow pores was already noticed in three other dipeptide crystals and attributed to the flexibility of the crystals framework [7]. Moreover, the experimental determination of the porosity of eight crystal dipeptides (AV, VA, AI, VV, IA IV, VI and LS) showed that two, AV and VA, undergo pore size expansion upon gas sorption [7]. It was suggested that in the VA class, there are backbone vibrational modes that contribute to the pore permeability [15]. Table 3.1. Dipeptide crystals permeabilities and selectivities towards He, O2, N2 and Ar. The minimum permeate flow rate that can be accurately measured in the setup is ~0.0005 mm3/h which corresponds to a permeability of ~0.25 barrer (AA crystals) and of ~20 barrer (VI crystals). Dipeptide Channel Diameter / nm Permeabilities / barrer He O2 N2 Ar LS 0.49 1.7 x 107 9.5 x 106 1.1 x 107 1.2 x 107 VI 0.37 2.8 x 104 2.7 x 103 2.2 x 103 Not detected AA — 19 31 Not detected Not detected Despite the fact that irreversible changes were found with AV, VA and AI crystals [3], the flexibility of the AA packing seems to be reversible. The AA crystals remain non-permeable to N2 after the O2 experiments and there was a full retention of the crystal structure after 2 months of permeation experiments. Interestingly, traces of oxygen molecules are found in the channels at a pressure of 8.5 bar of pure oxygen (Figure 3.2). Preliminary Studies: Single-Crystal Permeation of Atmospheric Gases 65 There are four symmetry equivalent positions for oxygen molecules in each vacancy void volume. The transport may be described in terms of hopping diffusion along the vacancy void volumes that are limited by the four methyl groups (Figure 3.2). The total O2 occupation per void volume (0.018) can be obtained from the adsorption isotherm. Although it is known that the Knudsen model does not apply to micropores [19], it is interesting to observe that the measured gas flow through the LS channels significantly exceeds Knudsen diffusion predictions (see Appendix A). Very fast air flow rates were already observed through 1.6 nm carbon nanotubes and attributed to the smoothness of the carbon pore walls [20]. Apparently, the weak nature of the interactions produced by the methyl groups that decorate the LS channels walls also allows high gas flow rates. A breakdown in the mass transport rate arises from the size-matching between the guest molecules and channel diameters as shown by the drastic decline of the VI and AA permeabilities (Table 3.1). The decrease in the gas sorption equilibrium, in particular for AA, and the reduction in the gas diffusivities certainly combine for such a strong drop in the permeabilitites. Figure 3.2. Crystal structure of AA with O2 viewed along the c-axis (left) and along the baxis (right). Highlighted are the oxygen molecules trapped inside the pores and the Ala side chains that form the pore constrictions. The LS permeability values were surprisingly high when compared to the behaviour of the other dipeptide crystals and to the values for other microporous materials reported in the literature. Though the experiments had been repeated with crystals with different dimensions it was noticed that more recent results by our group using the same method, Durão and Gales [21], showed that the LS permeability is in fact of the same order of magnitude of the VI permeability reported here. Chapter 3. Adsorption and Diffusion Inside Porous Crystals 66 Durão and Gales [21] have also shown that there is a propensity for pore blocking in dipeptide crystals which increases drastically with the crystal size (diffusing path length) and with the molecular dimension of the diffusing species. In agreement with that, in the next section it is shown that Ar is readily adsorbed in 1-50 mm powder polycrystalline solids (see Appendix B) but becomes blocked in 1-2 mm single crystals (Table 3.1). Nevertheless, it was concluded that dipeptide crystals can indeed be used as permselective materials but, as it happen with any unidimensional pore material, care must be taken concerning pore blocking. 3.4. Adsorption Studies The relatively few gas adsorption studies performed with VA-class dipeptides have yielded some notable results, such as high H2 adsorption capacities [9], high CO2/CH4 selectivities [4] and Xe adsorption in pores nominally too small for that to happen [8]. In this article, we report the adsorption isotherms and heats of adsorption of Ar, O2 and N2 in four crystalline dipeptides. The four dipeptides used have the smallest pores of the VA-class; VI, IA, IV and VV, having, respectively, 0.370 nm, 0.374 nm, 0.390 nm and 0.439 nm average crystallographic diameters [2], corresponding, respectively, to porosities of 5.7 %, 6.0 %, 6.3 % and 8.3 % (as determined from the crystal structure) [7]. The adsorbents show inverted Ar/O2 selectivity, a highly abnormal result, extremely interesting in the context of the very important industrial process of O2 production by PSA. Industrial O2 production relies almost exclusively on air-separation processes [22, 23]. One of these, PSA, is a cyclic chromatographic separation process [24], which, in the last four decades, has steadily been displacing cryogenic distillation as the process of choice for O2 production, in an increasing number of conditions [22, 23]. Presently, improvement of the process of air separation by PSA depends mainly on the development of new adsorbent materials, with greater working capacities [25, 26] and selectivities; this is especially the case for O2, which is particularly hard to separate from Ar. PSA production of O2 with up to 95 % purity is done using zeolites with N2/O2 selectivities of 2 to 10 [22, 27, 28]. In these adsorbents, it is very common that the adsorption isotherms of O2 and Ar are close [29, 30], causing the Ar/O2 ratio to remain unchanged throughout the column, approximately 1/20 (the same as in air), thus limiting the O2 purity achievable to 95 %. Separation of O2 and Ar for high-purity (> 95 %) O2 generation is usually done using a kinetically selective carbon molecular sieve (CMS) adsorbent, where O2 has a much higher Adsorption Studies 67 diffusivity than Ar and N2; that is, the adsorbent has O2/Ar and O2/N2 kinetic selectivities. This separation can be performed before [31] or after [32] removing the bulk of N2 with another PSA unit, using a zeolite adsorbent. For technical reasons [28-30, 33], the optimal situation combines initial N2 removal followed by Ar/O2 separation based on an adsorbent with Ar/O2 selectivity (instead of O2/Ar). Enormous efforts have been directed at finding materials displaying this property [28, 30]. Adsorbents that preferentially adsorb Ar over O2 are rare, and the ones that do exist have marginal selectivities. The first Ar-selective adsorbent discovered was silver-exchanged mordenite, Ag-mordenite, with an Ar/O2 selectivity of 1.13 [34]. The potential for PSA separation of Ar/O2 mixtures of Ag-mordenite was immediately recognised [29]. In the next two decades, several articles and patents followed, describing adsorbents with greater selectivities and capacities, as well as more efficient O2-generating PSA processes [26, 28, 30, 33, 35-38]. The highest Ar/O2 adsorption selectivity value ever measured was 1.65, on a Ag-exchanged zeolite ZSM-5, closely followed by that of a Ag-exchanged zeolite A, 1.63, both at atmospheric pressure [36]. Adsorbents used in PSA or gas chromatography experiments are all also Ag-exchanged zeolites [28-30, 33, 34, 38]. These have the additional advantage of having extremely high N2/O2 selectivities (>10) [26]. However, incorporation of silver also makes these materials quite expensive, greatly hindering their usage. Discovery of Ag-free porous materials displaying the desired selectivities is thus of great commercial and technical interest. The overwhelming majority of porous solids adsorb more O2 than Ar [26, 28, 39, 40]. Metalorganic frameworks (MOFs) typically have the preferential adsorption sequence O2>Ar>N2 [41, 42], zeolites have N2>O2>Ar [43, 44] and carbon-based materials display the two situations with similar frequency [45, 46]. We were able to find a single case of an adsorbent having an O2>N2>Ar sequence of preferential adsorption [47]. To date, to the best of our knowledge, only six known Ag-free porous materials show Ar/O2 equilibrium selectivity; MIL-53(Al) (the “breathing” MOF) [48], mordenite (zeolite) [36, 49], a CMS [50], ZSM-5 (zeolite) [36, 51], a polymer of intrinsic microporosity [52] and an activated carbon [53], having equilibrium Ar/O2 selectivities of, respectively, 1.26, 1.12, 1.11, 1.08, 1.09 and 1.06. Out of the six, the first four adsorb in the sequence N2>Ar>O2, the last two Ar>O2>N2 and not a single instance of adsorption in the sequence Ar>N2>O2. The reason for this exceptional behaviour was not addressed in any case. This limited number and inexistence of appropriate Chapter 3. Adsorption and Diffusion Inside Porous Crystals 68 explanations of the selection mechanism prevent the design of strategies for the development/discovery of new Ag-free Ar/O2 selective adsorbents. The dipeptide crystals tested for this work have the order of adsorption Ar>O2>N2, with a maximum Ar/O2 selectivity of 1.30, at 5 ºC, for VI. A discussion of the reasons for these abnormal results and their significance in the context of the development of new materials for air separation by PSA is given. 3.4.1. Experimental Excess adsorption isotherms of N2, O2 and Ar were determined for crystalline samples of four dipeptide crystals, VI, IA, IV and VV, at 5 ºC, 20 ºC and 35 ºC. Samples of dipeptides VI, IA, IV and VV were purchased from Bachem. Samples were received as a crystalline white powder [9], which were regenerated overnight, under vacuum (< 1 mbar), at 70 ºC, before being used in the adsorption experiments. The adsorption isotherms were determined using the volumetric method. This method is based on measuring the variation of pressure of the gas after an expansion between two tanks of known volume, one of which contains the adsorbent sample. Using an appropriate equation of state for the gas phase (the Peng- Robinson equation was used in this work) and performing a mass balance, it is possible to determine the total amount adsorbed. The sorbate concentration is determined by dividing this value by the sample mass. The sample tank and feed tank volumes were, respectively, 13.32 mL and 54.37 mL. The masses of VI, IA, IV and VV used were, respectively, 2.4927 g, 1.0144 g, 2.4131 g and 1.0064 g. The pressure transducer used was a WIKA P-30, 0-6 bar (accuracy of 0.1 % FS). The temperature of the system was controlled through immersion in a Huber CCE-K12 thermostatic bath. Swagelok tubing and valves were used to connect the tanks, the gas feed and the exhaust. Pressures below 1 atm were achieved using a Vacuubrand RZ 2.5 vacuum pump. Alphagaz 1 gases, supplied by Air Liquide, were used, with purities of 99.99965 % for O2 and 99.99945 % for the remaining ones. 3.4.2. Results and Discussion 3.4.2.1. Adsorption Isotherms and Monocomponent Selectivities The adsorption isotherms of Ar, O2 and N2 in the crystalline powder of dipeptides VI, IA, IV and VV, at 20 ºC, are shown, grouped by material, in Figure 3.3. Graphs of isotherms at 5 ºC Adsorption Studies 69 and 35 ºC and grouped by gas species are presented in Appendix C. Adsorption takes place, essentially, in the Henry region, with isotherms being only slightly bent. 80 % of adsorption took place, at least, in the first 30 s, and very often in the first 5-10 s. The dipeptide crystals in the powder are 1-50 µm in length. Desorption isotherms were also determined in every case (not shown) and hysteresis was never detected. N2 adsorption isotherms have been determined before, for 25 ºC [4], and are similar to the ones here reported. Figure 3.3. Excess adsorption isotherms [54] of Ar, O2 and N2 on VI, IA, IV and VV, at 20 ºC, grouped per material. The adsorption isotherms were numerically fitted with the Volmer equation, !" = ! 1−!exp ! 1−! (3-1) where != !! !!"# (3-2) and Chapter 3. Adsorption and Diffusion Inside Porous Crystals 76 likely that the increased rotational freedom the O2 molecules experience allows for a better matching of the pore morphology, overcoming the decreased adsorption potential effect. Between IV and the much bigger VV pore, the decreasing trend observed for Ar is established. For the bigger N2 molecule, there are no sensible differences between VI and IA, and only for the IV framework is the increase in the heat of adsorption clear. Besides the rotational freedom factor mentioned for O2, there may exist a small contribution from the pore size increase beyond the optimal intermolecular distance of N2, 0.38 nm. Finally, for N2 also, the big increase in pore size associated with VV significantly decreases the heat of adsorption. In fact, from IV to VV, the heat of adsorption of all species decreases. Because the Lennard- Jones diameter/widths of Ar, N2 and O2 are smaller than the nominal size of the smallest pore (VI), the influence of framework flexibility is not especially significant, with the possible exception of N2. If compared for the same material and different species, only VI displays a clear trend in heat of adsorption variation, Ar>O2>N2. For IA, the Ar and O2 values are within the error range, although both are still clearly higher than that of N2, Ar ≈ O 2 > N2. For IV, the heats of adsorption of O2 and N2 are much larger than that of Ar, O2 ≈ N 2 > Ar and, for VV, no sensible differences exist between the three, O2 ≈ Ar ≈ N2. For exclusively dispersive interactions, the heat of adsorption should only depend on the polarisabilities of the different species, so it would be expectable that Ar be more strongly adsorbed than O2. The N2 molecule has a nominal polarisability greater than the Ar atom, but since its confinement limitations are greater than those of Ar, both higher and lower values are reasonable. It is possible that due to their quadrupole moments, the O2 and N2 molecules bond, even if weakly, with the slightly polarised H atoms from the aliphatic side-chains forming the pore walls. Since the polarisability of Ar is only marginally higher than that of O2, that effect could be enough to switch the order of the heats of adsorption of the two species. An heat of adsorption of N2 similar to that of O2, in IV, may be only the reflection of the still important confinement effects having a greater impact on the bigger N2 molecule than on that of O2. This would also explain why N2 heats of adsorption are inferior to those of O2 and Ar for VI and IA, despite having greater polarisability and quadrupole moment. These confinement effects are certainly much less important in VV, if at all significant. Therefore, it would be expectable that a N2>O2>Ar order was well established by then, which seems to not be the case. Adsorption Studies 77 Only for N2 does the isosteric heat of adsorption show some kind of trend. For three of the four dipeptides, the isosteric heat of adsorption of N2 forms a “V-shaped” curve for low adsorbed concentrations, only then stabilizing. The variation patterns observed for the !! ! are even harder to explain. In the four dipeptides considered, the capacities should be similar, therefore differences in !! ! must be explained by variations in entropic contributions. The entropy of the mobile adsorbate depends on translational, rotational and vibrational components [64]. For Ar, one could tentatively try to interpret the increases in !! ! by considering that rotational entropy is absent in Ar atoms, thus the great increases may reflect increased vibrational and translational freedom as the frameworks become more open. In the case of O2 and N2, it is possible the highly hindered non-axial rotation in VI potentiates faster translation inside the pores, while the slightly larger IA and IV allow greater rotational freedom, thus limiting translation inside the pore. This factor is the basis for some separations using microporous membranes [65, 66]. The much bigger VV pores allow such rotational freedom that the increasing trend in !! ! observed for Ar is finally established. Overall, it is astonishing how such regular patterns in amount adsorbed (adsorption isotherms), indicating equally regular free energy variation patterns, stem from such wide and irregular swings in entropy and enthalpy as those shown in Figure 3.7. 3.4.2.3. Significance for PSA Separation of Air In spite of the extraordinary Ar/O2 selectivities reported, use of hydrophobic dipeptides as PSA adsorbents is highly unlikely. Besides good selectivities, good working adsorption capacities are needed to implement any PSA separation [24]. The nearly linear isotherms presented allow a working adsorption capacity essentially proportional to the pressure range used in the PSA, an excellent property. However, the overall capacities are low, when compared to other materials. At ambient temperatures and for 1 bar, values of oxygen/argon adsorption between 0.1 mol/kg and 0.2 mol/kg are the most typical in materials that have been tested as PSA adsorbents [28-30, 33-35, 38]. These results are all significantly above the ~0.05 mol/kg obtained for Ar adsorption in VI. It thus seems these materials show very little potential as a PSA adsorbent, as long as Ag-based adsorbents remain commercially viable. Nonetheless, that the pores of crystalline hydrophobic VA-class dipeptides can generate good Ar/O2 selectivities was totally unexpected. This knowledge may be used to search for new (or Chapter 3. Adsorption and Diffusion Inside Porous Crystals 78 otherwise) materials that combine good selectivities with good capacities. The organic materials having the Ar>O2>N2 adsorption sequence, also observed in this work, also have very hydrophobic pores [52, 53], which leads us to think this is a fundamental property, together with pore size, determining the sequence. This probably results from the limited influence the quadrupolar moment of O2 has in hydrophobic pores. The main causes of the low capacities of the materials are their low porosities and micropore concentrations. The four dipeptide frameworks considered have very similar densities, making porosity highly dependent on pore size. The crystallographic porosities of dipeptides VI, IA, IV and VV are 5.7, 6.0, 6.3 and 8.3 % [7] and the micropore concentrations are 0.048, 0.053, 0.053 and 0.067 cm3/g, respectively. These are very small values, especially the porosities, when compared to materials used in O2-generation PSA. Ag-mordenite has been reported as having values of 17 % and 0.062 cm3/g [34], AgLiLSX has 27 % and 0.118 cm3/g [67] and Ag-ETS-10 has 38 % [37, 68] and 0.13 cm3/g [69, 70]. The low porosity of the dipeptides is a contingent property, which does not need to exist in other materials with unidimensional pores, even for such small pores; it stems mostly from the low pore concentration existent in the dipeptides, 0.45 pores/nm2 to 0.48 pores/nm2. A material possessing cylindrical unidimensional pores has a geometrical limiting porosity of 78.5 %, corresponding, for a material with 0.4 nm-wide pores, to 6.25 pores/nm2. Therefore, there is plenty of room for improvement of porosities. A material with pores having similar morphologies and chemical properties as those of the four dipeptides studied in this work, but with significantly higher porosity could, perhaps, combine both the good selectivity observed with the desired adsorption capacities. Such a material should have highly hydrophobic and less than 0.5 nm-wide pores, and a porosity of, at least, 20 %. With the large numbers of organic microporous solids discovered in recent years, there are plenty of candidates to be tested [71, 72]. 3.5. Conclusions The first part of this chapter outlines the preliminary results obtained with dipeptide singlecrystal permeation experiments, and how they were able to indicate highly positive host-guest interaction potentialities with atmospheric gases. Ar permeation in the millimetre-sized VI crystals seems to have been limited by the presence of pore blockage. Conclusions 79 In the second part, crystalline powders of hydrophobic dipeptides VI, IA, IV and VV were tested as adsorbents of N2, O2 and Ar. The three gases adsorb in the sequence Ar>O2>N2, thus displaying Ar/O2 selectivity, an extremely rare result. The best Ar/O2 selectivities were obtained for VI; at 5 ºC and near vacuum conditions, the Ar/O2 selectivity is 1.30, the highest ever measured in Ag-free adsorbents. Despite the good selectivity, the low capacities of the materials strongly hinder their use in PSA for O2 generation from air. At 20 °C and 1 bar, the adsorption of Ar, O2 and N2 on VI is, respectively, 0.048 mol/kg, 0.038 mol/kg and 0.023 mol/kg. The exceptional adsorption sequence observed, Ar>O2>N2, stems from the very small pore size of the materials. The combination of geometrical confinement effects, limitation of rotational degrees of freedom and low polarity limit both the enthalpic and the entropic contributions to the adsorption of O2 and N2, much more than that of Ar. The low capacity of the dipeptides tested is strongly related to their low porosity, below 10 % in all cases. 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A Surface Thermodynamics Approach to Modelling Single-File Adsorption in Ultramicroporous Materials* 4.1. Abstract A new thermodynamic approach is proposed to interpret adsorption equilibrium in ultramicropores with single-file diffusion. By considering the adsorbed phase as a onedimensional fluid, phase equilibria thermodynamics can be used to derive a one-dimensional analogue of the monocomponent Gibbs adsorption isotherm. Equations such as Langmuir, Volmer, Fowler-Guggenheim or Hill-de Boer can thus be used as representations of phenomenological models of the one-dimensional adsorption system, rather than just as mathematical correlations. The bidirectional relation between the equations of state characterising the adsorbed phase and the adsorption isotherm equations allow great insight to be had simply by adsorption isotherm determination. As an example, the adsorption isotherms of Xe and CO2 in four crystalline hydrophobic dipeptides of the VA-class are analysed with this approach, showing its potential and limitations. 4.2. Introduction Most adsorption separation processes involving gases and vapours are based on microporous solids [1, 2]. This is due to the much stronger adsorption potential experienced by adsorbates in micropores relative to mesopores and macropores, allowing operation at lower concentrations. Interestingly, micropore adsorption is still the most poorly understood of the three, due to the complex interactions occurring in the confined environment of a micropore [3]. Micropores (< 2.0 nm) can be divided into supermicropores (> 0.7 nm) and ultramicropores (< 0.7 nm) [4]. Depending on the size of the adsorbate molecule relative to the pore, * Article currently under submission. Chapter 4. A Surface Thermodynamics Approach to Modeling Single-File Adsorption 92 4.2.2.2.3D Gibbs Adsorption Isotherm Adsorption in solids, porous or non-porous, is seldom confined to a monolayer. Multilayer adsorption, capillary condensation and micropore filling make the traditional 2D Gibbs adsorption isotherm inapplicable (with physical significance) to most adsorbents. However, the application of a 3D Gibbs adsorption isotherm allows extension of the thermodynamicsbased approach to micropore filling [1, 11, 12, 30, 35] and multilayer adsorption [31]. The 3D analogue of Eq. (4-5) is [1, 11, 12], d ln ! d! ! = ! !" (4-8) where ! is the three-dimensional spreading pressure (i.e., in-pore three-dimensional pressure) and ! is the molar pore volume. Application of the 3D Gibbs adsorption isotherm implies the assumption of an adsorption potential field uniform both breadthwise and lengthwise. In systems where the 3D Gibbs adsorption isotherm applies, adsorption isotherm equations can be deduced from Eq. (4-8) by application of an equation of state, just as for 2D. However, because the concept of localised adsorption does not apply to three-dimensional space, only isotherm equations corresponding to distributed adsorption can be used. Thus, for micropores where three-dimensional pore filling is assumed to occur, and there is a uniform adsorption potential, equations such as Volmer and Hill-de Boer can be used as physical models of adsorption, instead of just as mathematical correlations. As previously mentioned, only in zeolite research are the thermodynamic models used and, even in such cases, rarely. As illustrated in Figure 4.1, the assumption there is an adsorption potential approximately uniform breadthwise is typically only correct for small supermicropores [16]. It should be noted that, in the context of adsorption in porous materials, the concept of “excess concentration” has a somewhat different meaning than that originally envisioned by Gibbs for generic phase equilibrium [44]. Excess adsorbate concentration in a pore is usually defined as the amount adsorbed in excess of that occupied by the bulk gas on the free equivalent of total pore volume. Thus, Gibbs’ original “excess concentration” is, in the context of 3D adsorption in solids, equivalent to the “absolute adsorbate concentration” of porous materials studies. Absolute adsorbed concentration can be determined from excess concentration, provided total pore volume is known [44]. In fact, for real 2D monolayers, it is also not correct to talk about “excess adsorbate concentration”, but simply “adsorbate concentration”. Introduction 93 4.2.3. 1D and Single-File Diffusion Systems The derivation of a 1D Gibbs adsorption isotherm follows the same thermodynamic reasoning as for 2D and 3D. Assuming a one-dimensional gas behaviour for single-file systems, the 1D Gibbs adsorption isotherm can be written as, d ln ! d! ! = ! !" (4-9) where, !≡ ! ! (4-10) and ! is total micropore length, ! is molar length and ! is 1D spreading pressure (i.e., linear tension). As for 3D, for 1D adsorption, absolute adsorbate concentration should be used, not excess. Depending on the behaviour of the molecules in the pore, different equations of state and adsorption isotherms may apply. Unlike what happens for 3D systems, purely localised adsorption is a reasonable model of adsorption. Langmuir and Fowler-Guggenheim equations can thus be used in 1D systems as phenomenological relations. Table 4.2 summarises the characteristic variables for 1D, 2D and 3D adsorbed phases. It is worth noticing that, for single-file systems, the capacity-determining variable is micropore length, not micropore volume as with supermicropores. A previous instance exists of a proposal of a 1D Gibbs adsorption isotherm, to model the 1D boundary between two 2D monolayers at the surface of a liquid [45]. This has not happened, to the best of our knowledge, for systems with gas adsorption in ultramicroporous solids. There were, however, instances of implicit assumption of the validity of the 1D Gibbs adsorption isotherm, in which the Langmuir [46] and Fowler-Guggenheim [47] isotherm equations were used to model gas adsorption in ultramicroporous solids with clear physical meaning. The concept of “one-dimensional gas” is also frequently used to describe 1D adsorbed phases [5]. Virial equations have also been repeatedly used to model systems where the adsorbed phase is implicitly assumed to be a 1D gas [48]. The main distinction in adsorption systems represented by the equations in Table 4.1 is between mobile and localised adsorption. Any such representation should always be seen as an approximation of an intermediate reality. If none of the two extremes is a good Chapter 4. A Surface Thermodynamics Approach to Modeling Single-File Adsorption 94 approximation, then adsorption equilibrium must be modelled by adding the two terms representing the two contributions. Similar reasoning is behind the popular dual-site Langmuir isotherm equation. The fact that this modelling approach generates many fitting parameters limits the physical significance that may be attributed to them. The problem will not go away, and the only real alternative is to use more complex statistical mechanical models [11, 49, 50], which, nonetheless, may exhibit the same problem. Table 4.2. Characteristic variables of the adsorbent and adsorbed phase, for the models of 1D, 2D and 3D adsorbed phases. Variables characterising the adsorbed phase Phase Variable characterising the adsorbent Inverse of concentration Pressure equivalent 1D Specific pore length, ! Molar length, ! Tension, ! 2D Specific surface area, ! Molar area, ! Spreading pressure, ! 3D Specific pore volume, ! Molar volume, ! In-pore pressure, ! 4.2.3.1. Slab It is frequently the case that ultramicropores have a slab-like, and not cylindrical, shape. In such cases, if its width is smaller than two times that of the adsorbate molecule, a 2D gas model and isotherm must be used, following Eq. (4-5). If it is bigger (but not too big), the 3D gas model must be used, Eq. (4-8). 4.2.3.2. Heterogeneous Systems We find it important to emphasise that the thermodynamic approach to adsorption modelling presented above clearly shows that simple phenomenological equations may, sometimes, be used with physical significance to describe 1D, 2D and 3D adsorption systems. The Langmuir isotherm should therefore stop being referred to as having an exclusively empirical character when applied to micropore adsorption, even when localised adsorption with no intermolecular interactions is a gross approximation of the physical reality. In fact, when compared to the Introduction 95 Dubinin equations, its phenomenological significance will always be equal or greater. The same is valid for the Volmer isotherm. If the adsorbent has a broad pore size distribution, to obtain physically significant parameters it is necessary to integrate the adequate isotherm equation for all pore sizes. The procedure is exactly the same as for the many methods using the Dubinin equations [27]. There is no intrinsic advantage in using them over thermodynamic isotherm equations. For the Dubinin equations, empirical relations have been developed relating “characteristic energy” to pore size, both for slab and for cylindrical pores. If such relations were developed for the parameters of thermodynamic isotherm equations, the integration could be as easily accomplished, while having the advantage of obtaining more meaningful isotherm parameters for each pore size. A mathematical treatment for this approach has, in fact, been proposed [47]. These relations could be developed either experimentally or from molecular simulations of adsorption on pores of different sizes. Figure 4.1. Representation of the adsorption potential and molecular distribution on a a.) free surface or macropore, b.) mesopore, c.) supermicropore and d.) ultramicropore. Although all molecules in a supermicropore are under the influence of the adsorption potential, only in an ultramicropore can it be considered that all molecules are under the influence of a similar adsorption potential. Adapted from [16]. Chapter 4. A Surface Thermodynamics Approach to Modeling Single-File Adsorption 96 Large supermicropores, with breadthwise uneven adsorption potentials such as that shown in Figure 4.1c, cannot be modelled considering a single phase. Instead, the existence of two phases inside the pores must be considered [16], a two-dimensional phase close to the pore walls and a three-dimensional gas phase at the centre of the pore. The latter is still influenced by the adsorption potential, so that its concentration is much higher than that outside the pore. 4.2.3.3. Use in Real 1D Adsorption Systems Xe and CO2 adsorption in hydrophobic dipeptides [51, 52] was used to test the insight that could be gained from using phenomenological equations in 1D adsorption systems. These fully organic microporous materials possess unidimensional pores, which, depending on the dipeptide, are 0.4-1.0 nm wide, with varying degrees of helicity. They have recently been suggested as an ideal template for testing SFD models [53, 54]. The four dipetides used in this work were VI, IA, IV and VV, with pore sizes of, respectively, 0.37 nm, 0.37 nm, 0.39 nm and 0.44 nm. IA is usually considered larger than VI [52, 55]. 4.3. Experimental Adsorption isotherms of Xe and CO2 in VI, IA, IV and VV were determined. Samples of dipeptides VI, IA, IV and VV were purchased from Bachem. Samples were received as a crystalline white powder, which were regenerated overnight, under vacuum (< 1 mbar), at 70 ºC, before being used in the adsorption experiments. The adsorption isotherms were determined using the volumetric method. This method is based on measuring the variation of pressure of the gas after an expansion between two tanks of known volume, one of which contains the adsorbent sample. Using an appropriate equation of state for the gas phase (the Peng-Robinson equation was used in this work) and performing a mass balance, it is possible to determine the total amount adsorbed. The sorbate concentration is determined dividing this value by the sample mass. The sample tank and feed tank volumes were, respectively, 13.32 mL and 54.37 mL. The mass of VI used for the determination of the Xe isotherm was 2.3770 g and for the CO2 isotherm 2.4927 g. The mass of IA, IV and VV used for the determination of both isotherms was 0.9892, 2.4131 and 1.0064 g. The pressure transducer used was a WIKA P-30, 0-6 bar (accuracy of 0.1 % FS). The temperature of the system was controlled through immersion in a Huber CCE-K12 thermostatic bath. Swagelok tubing and valves were Results and Discussion 97 used to connect the tanks, the gas feed and the exhaust. Pressures below 1 atm were achieved using a Vacuubrand RZ 2.5 vacuum pump. Alphagaz 1 gases, supplied by Air Liquide, were used, with purities of >99.999 %. 4.4. Results and Discussion The absolute adsorption isotherms of Xe and CO2 in VI, IA, IV and VV, at 20 ºC, are shown in Figure 4.2. Excess adsorption isotherms are shown in Appendix D. Due to the low porosity of crystalline dipetides, absolute and excess adsorption values are extremely close. CO2 adsorption isotherms at 25 ºC have been published previously [56], displaying similar values and trends to the ones reported here. Xe adsorption isotherms in VA and AV, at 25 ºC, have been published [57] with similar values and trends as those of the four dipeptides shown here. As for other gases [55], adsorbed concentration and Henry constant of Xe generally increase with pore size, i.e., VV>IV>IA>VI. For CO2, this trend is not observed. The lines in the two graphs of Figure 4.2 represent the adsorption isotherms obtained from a non-linear fitting procedure, with the Hill-de Boer equation being used for Xe and Fowler-Guggenheim for CO2. The fitting parameters used are shown in Table 4.3 and Table 4.4. The fitting procedure and choice of the equations to be fitted in each case are discussed below. The four simple models, Langmuir, Volmer, Fowler-Guggenheim (FG) and Hill-de-Boer (HdB), were used to fit the adsorption isotherms. Instead of fitting the final equations, shown in Table 4.1, directly to the adsorption isotherm data, we chose a two-step procedure based on a graphical evaluation method originally proposed by de Boer [58]. Unlike others [11, 50], this graphical representation does not require a priori knowledge of the adsorption capacity parameter. First, the !-! diagrams were determined, from the adsorption isotherm data, using Eq. (4-7). One such diagram, for Xe adsorption on VI, is shown in Figure 4.3, in the !" vs ! format, normalised for !" (the other seven diagrams are shown in Appendix D) The shape of the curve in a !-! diagram is not dependent on Henry’s constant, so curve fitting involves one less parameter than directly using the adsorption isotherms. Henry constants are determined either directly from the !/! ratio at !→0, or by curve fitting of the ! values of the adsorption isotherms, using the previously determined parameters as constants. Thus, the perfect gas (Henry) model always generates the same !-! relation (! !=!"), Langmuir and Volmer have one parameter (adsorption capacity) and Fowler-Guggenheim and Hill-de Boer have two (adsorption capacity and the interactions parameter). Chapter 4. A Surface Thermodynamics Approach to Modeling Single-File Adsorption 98 Figure 4.2. Absolute adsorption isotherms of Xe and CO2 in VI, IA, IV and VV. The solid lines represent the isotherm obtained from the fitting procedure, Hill-de Boer for Xe and Fowler-Guggenheim for CO2, using the parameters shown in Table 4.3 and Table 4.4, respectively. Results and Discussion 99 Table 4.3. Results of non-linear fitting of Xe adsorption data to the FG and HdB equations. SS1 is the sum of squares for the !!"# and ! fitting and SS2 is the sum of squares for the !! fitting. Table 4.4. Results of non-linear fitting of CO2 adsorption data to the FG and HdB equations. SS1 is the sum of squares for the !!"# and ! fitting and SS2 is the sum of squares for the !! fitting. SS1 c nmax / mol kg-1 nmax / atom u.c.-1 l0 / nm atom-1 SS2 / mol2 kg-2 KH / mol kg-1 bar-1 VI FG 1.58 X 10-3 0.847 1.40 1.9 0.556 2.95 X 10-4 0.394 HdB 4.61 X 10-4 1.943 2.08 2.8 0.373 1.45 X 10-4 0.389 IA FG 1.17 X 10-4 -0.101 1.96 2.4 0.487 7.64 X 10-5 0.514 HdB 6.78 X 10-5 0.347 3.07 3.8 0.311 7.00 X 10-5 0.513 IV FG 8.91 X 10-4 0.17 1.67 2.2 0.537 6.85 X 10-5 1.18 HdB 6.25 X 10-5 1.13 2.40 3.2 0.374 1.76 X 10-5 1.15 VV FG 1.46 X 10-2 -0.238 1.74 2.4 0.454 8.20 X 10-4 3.35 HdB 1.03 X 10-3 0.941 2.37 3.3 0.333 9.33 X 10-5 3.06 SS1 c nmax / mol kg-1 nmax / atom u.c.-1 l0 / nm atom-1 SS2 / mol2 kg-2 KH / mol kg-1 bar-1 VI FG 9.00 X 10-4 -0.516 1.30 1.8 0.589 1.03 X 10-4 0.510 HdB 5.57 X 10-4 -0.253 2.02 2.8 0.380 8.20 X 10-5 0.507 IA FG 1.40 X 10-5 -0.331 2.05 2.6 0.466 6.60 X 10-5 0.406 HdB 1.42 X 10-5 -0.089 3.29 4.1 0.291 6.68 X 10-5 0.406 IV FG 8.13 X 10-5 -0.841 1.70 2.2 0.528 1.55 X 10-5 0.776 HdB 1.15 X 10-4 -0.831 2.68 3.5 0.335 1.99 X 10-5 0.773 VV FG 7.28 X 10-4 -0.324 2.14 3.0 0.369 2.81 X 10-4 0.501 HdB 8.29 X 10-4 -0.076 3.43 4.7 0.231 3.07 X 10-4 0.500 Chapter 4. A Surface Thermodynamics Approach to Modeling Single-File Adsorption 100 Figure 4.3. !-! diagram for Xe adsorption in VI. The points represent experimental results and the line Hill-de Boer fitting. Involving only one parameter, the adequacy of the fitting of Langmuir and Volmer models can be easily assessed by analysing the residues of the fitting procedure. One example of these is shown in Figure 4.4, the rest in Appendix D. It is quite clear that the errors, although small, are not random, so these models are inadequate to represent the system. This was to be expected, since CO2 and Xe are highly non-ideal gases, having strong intermolecular interactions. The FG and HdB models were thus fitted to the !-! data, with results being shown in Table 4.3 and Table 4.4. The sums of squares of residues of the first and second fittings are represented, respectively, by SS1 and SS2. For Xe in IV and VV, the HdB fitting SS1s are an order of magnitude below those of FG, being a clearly better fit. This may indicate that Xe is predominantly mobile inside IV and VV, the two dipeptides with larger pores. For the other two, the picture is less clear; HdB fitting SS1s for VI and IA are, respectively, only 1/3 and 1/2 those of FG. It may be the case that VI and IA pores (0.37 nm) are simply too tight for the 0.394 nm-wide Xe atom to diffuse freely, thus having a more localised character. On the other hand, it may happen that the fact that the isotherms are not as curved as those of IV and VV, and thus closer to the Henry region, makes it harder to identify the best equation. The extensive literature on Xe adsorption in VA-type dipeptides [52-54, 57, 59] does not give, in our view, any hint on which equation would be more appropriate. However, there is still some independent information that can be used to assess this; the size of the Xe atom can be compared to the value of !! obtained from Results and Discussion 101 the fitting procedures. The HdB fitting generates !! values clearly closer to the Lennard-Jones diameter of the Xe atom, while the FG values are completely unreasonable. Furthermore, the FG fitting for IA generates a negative !, indicating net repulsion between adsorbed atoms, which, for Xe, is not possible. The HdB equation is, thus, the one that best describes Xe adsorption on all four dipeptides, indicating the existence of a predominantly mobile adsorbed phase. Figure 4.4. Residual errors of non-linear Volmer and Langmuir fittings of Xe and CO2 adsorption in VI. The errors are dimensionless, as they refer to the fitting of !"/!". For CO2, the situation is even less clear. Grand Canonical Monte Carlo (GCMC) simulations performed by Comotti et al. [56] indicate a predominantly localised character of CO2 adsorption in IA and IV, with two adsorption sites per unit cell. VV and VI seem to have a mixed, localised-distributed adsorption. This is not reflected in the results of Table 4.3 and Table 4.4; values of SS1 are very similar for both FG and HdB fittings in the four dipeptides. If the GCMC result was not available, it would be tempting to conclude that the adsorbed Chapter 5. Kinetic Derivation of Common Isotherm Equations 108 and 3D adsorbed phases. In this article, we propose that some of these isotherm equations can also be derived using a simple kinetic approach. Table 5.1. Different adsorption isotherm equations obtained from the Gibbs equation, for gases, and Type of Adsorption it models. Type of Adsorption can be localised or distributed and with lateral interactions or without lateral interactions. For adsorption of solutes from liquid solutions, pressure is replaced by concentration. The variables in the equations are defined in the text. Name of Isotherm Isotherm Equation Type of Adsorption Henry !" =! Any Langmuir [2] !" = ! 1−! Localised, without lateral interactions Fowler-Guggenheim [3] !" = ! 1−!exp −! ! Localised, with lateral interactions Volmer [4] !" = ! 1−!exp ! 1−! Distributed, without lateral interactions Hill-de Boer [5-7] !" = ! 1−!exp ! 1−!exp −! ! Distributed, with lateral interactions The simplest equations derived from the Gibbs adsorption isotherm are represented in Table 5.1. These five equations represent the most basic types of adsorption; localised or distributed, with or without lateral interactions. All can generate Type I isotherms, while the Hill-de Boer and Fowler-Guggenheim equations can also generate Type V isotherms. For 3D adsorbed phases, the concept of localised adsorption does not apply, and, thus, the Langmuir and Fowler-Guggenheim equations are only considered for localised 1D and 2D adsorbed phases. Langmuir pioneered the kinetic approach in 1918 [2], by deriving an expression that considers adsorption of gas-phase molecules on specific sites of the surface, upon hitting on it at a rate given by the Kinetic Gas Theory. To the best of our knowledge, the kinetic approach to adsorption equilibrium used by Langmuir was not followed for the derivation of other adsorption isotherm equations; the more powerful and simple thermodynamic (classic and Introduction 109 statistical) approaches proved much more attractive and productive. In fact, it is only natural that equilibrium is best-studied using thermodynamics, for adsorption as for any other physical phenomenon. In porous solids, even the kinetics of adsorption could not be appropriately described using the kinetic approach of Langmuir, since diffusion through the pores is typically the rate-controlling mechanism. With the discovery of the distributed nature of London forces and the development of the concept of adsorption potential, it was realised that purely localised adsorption very seldom exists for physical sorption [8]. Nonetheless, the elegance, simplicity and ability to produce fitting parameters with clear physical meaning made the Langmuir isotherm one of the most widely used adsorption isotherms. So much so, that Type I adsorption isotherms are often called “Langmuir-like”, or simply “Langmuir” [9-11]. We believe that other adsorption isotherms would greatly benefit from the insight gained from a kinetic derivation. With this purpose in mind, the kinetic derivation of the five isotherm equations of Table 5.1 will be given below, for 1D, 2D and 3D systems. We also hope that by exemplifying a kinetic derivation of the Langmuir equations for 1D systems we may help dispel the idea that it can never represent a physical model of micropore adsorption. To perform these derivations, kinetic arguments as those of Langmuir [2] and de Boer [12, 13] will be used. It will be assumed that adsorbed molecules are hard spheres, behaving according to Newtonian kinematics, and that impacting molecules only interact with the solid upon hitting the surface (2D) or entering the pore (1D and 3D). These crude simplifications, already implicit in the thermodynamic derivations (Chapter 4), will prove to be very powerful in promoting insight into some of the physical realities behind the isotherm equations under consideration, in the same way it occurred with the Langmuir equation. We should emphasize that we are not proposing that adsorption occurs necessarily or exactly according to the mechanisms proposed in the models. This is especially the case in micropores, where adsorption around the pore mouth plays a decisive role in promoting inpore adsorption. We seek only to show that simple models such as that originally used by Langmuir can be used to derive other simple isotherm equations, and that they can be used as easily in micropore adsorption as in monolayer surface adsorption. Chapter 5. Kinetic Derivation of Common Isotherm Equations 110 5.3. Adsorption on a Planar Surface – 2D 5.3.1. Localised Adsorption – The Classic Langmuir Model The classic Langmuir model describes adsorption on a surface with discrete adsorption sites. It has been described many times before [12, 14], but a small recapitulation will be very helpful in setting the conceptual framework used for the derivations that follow. The rate of adsorption is equal to the rate of impact of gas-phase molecules on unoccupied adsorption sites. The rate of impact on a surface is given by the Kinetic Gas Theory, having SI units of mol·m-2·s-1. If the units desired for the adsorption rate are mol·kg-1·s-1, the rate of impact must be multiplied by the specific surface area, with SI units m2·kg-1. The surface is not entirely covered by adsorption sites and not all impacts on these are successful, so an extra term, !, is included, representing the fraction of successful impacts on the adsorption sites. Finally, not all adsorption sites are free, so the rate of adsorption will be proportional to the fraction of those that are. This fraction is given by 1−!, where ! represents the total amount adsorbed relative to the maximum that can be adsorbed (monolayer coverage). Thus, the equation for the rate of adsorption is ! != ! 2!"#$ ·!·1−!·! (5-1) where ! ! represents the rate of adsorption, ! represents the specific surface area of the adsorbent, ! represents pressure, ! represents molar mass, ! represents the ideal gas constant and ! represents absolute temperature. The first fraction in the equation is the rate of impact on a surface, as given by the Kinetic Gas Theory. Desorption is an activated phenomenon, the rate of which is proportional to the amount adsorbed. Therefore, the rate of desorption can be expressed as: ! != !!·! (5-2) where ! ! represents the rate of desorption (SI units of mol·kg-1·s-1), !! represents the frequency of desorption (s-1) and ! represents the adsorbed concentration (mol·kg-1), The frequency of desorption is given by, !!=!!·exp − !! !" (5-3) Adsorption on a Planar Surface – 2D 111 where !! is the activation energy of desorption and !! is the minimum frequency of adsorption, reached at infinite temperature. The activation energy of desorption is basically the heat of adsorption, that is, !!=−∆!!"#. Equilibrium is reached when both rates are the same. Equating the rate of adsorption with the rate of desorption results in the Langmuir equation, ! !=! !!" = ! 1−! (5-4) where ! is the so-called affinity constant. This can be reformulated as != !! !!"# (5-5) where !!"# represents the maximum specific amount that can be adsorbed and !! is Henry’s constant, which can be expressed as !!= !·! !!2!"#$ (5-6) Unlike for the thermodynamic approach [4, 13, 14], where ! stems from an integration constant, the kinetic approach yields a relation between Henry’s constant and physical variables characteristic of the adsorption system. This is frequently quite helpful in interpreting experimentally determined values of ! and !! especially when comparing different materials. 5.3.2. Distributed Adsorption – Volmer Equation for Surfaces The derivation of an equation for distributed monolayer adsorption will now be described, using the same kinetic approach as for localised adsorption. From the thermodynamic derivation [7, 14], we know the Volmer equation is the correct final result. As a starting point, it must be considered, as for the thermodynamic derivation, that adsorption occurs on a uniform adsorption potential on the surface of the adsorbent. This means that, when in the adsorbed-phase, the molecules are free to slide through the surface and bump into each other, as a two-dimensional gas [13]. One of the key variables used for the characterisation of the adsorbed phase is the molar area of the system, defined as: Chapter 5. Kinetic Derivation of Common Isotherm Equations 112 !≡ ! ! (5-7) When a monolayer has been formed, the specific amount adsorbed is at its maximum, !!"#, and the molar area at its minimum, !!. The minimum molar area is approximately the molar area of a single adsorbed molecule, !!=!!"#$%&#$. The desorption rate is modelled in the same manner as for localised (Langmuir) adsorption, ! != !!·! (5-8) where !!=!!·exp − !! !" , !!=−∆!!"# (5-9) However, the adsorption rate cannot be described by Eq. (5-1). The fraction of unoccupied surface, 1−!, does not correspond to the surface available for impact of incoming gas molecules, as it occurs for localised adsorption. The freedom of movement leads to the existence of a random distribution of intermolecular distances, unlike the uniform distribution of localised adsorption. Unoccupied area located between two or more molecules with intermolecular distances smaller than necessary for a successful impact is not free for adsorption. Therefore, an extra term must be multiplied, corresponding to the probability that the intermolecular distance, !!"#$% , between the adsorbed molecules at the point of impact is large enough for the incoming molecule to hit the surface. It thus follows that ! != ! 2!"#$ ·!·1−!·!·p!!"#$% >!∗ (5-10) where !∗ represents the critical distance that allows adsorption of incoming molecules to take place and p!!"#$% >!∗ represents the probability that !!"#$% >!∗. Figure 5.1 shows a representation of an unsuccessful impact on unoccupied area, due to the existence of insufficient space between two adsorbed molecules adjacent to that area. Eq. (5-10) has the implicit assumption that the speed of molecules on the surface is much smaller than that of incoming gas molecules. The problem with the probability introduced in Eq. (5-10) is that it is very hard to model in a 2D surface. Thus, instead of working with distances, it is far more convenient to work with areas. Under this approach, an impact is successful, or not, depending on the presence of molecules within the area of impact, as represented in Figure 5.2. This area is exactly the Adsorption on a Planar Surface – 2D 113 same as the area of an adsorbed molecule, A0, or, in molar terms, !!. Therefore, the probability that the impact will be successful is the same as that of the existence of one or more molecules on the impact area, !!. Figure 5.1. Representation of the lateral and top views of an unsuccessful impact on the surface. The impacting molecule is repulsed by molecules already adsorbed on the surface, although the projection of its mass centre is outside the projected area of the surface molecule. Let us define a variable !!, the area of the circle centred on the centre of the impact site, and touching the nearest adsorbed molecule, where the s stands for “site”. Figure 5.2 shows !! for a successful impact and an unsuccessful impact. It is thus possible to rewrite the probability in Eq. (5-10) as: p!!"#$% >!∗=p!!>!! (5-11) To determine the new probability, it is now only necessary to know the probability distribution function of !!. This non-negative continuous random variable, !!≥0, has a density distribution function, f!!, such that, p!!>!!=1−F!! (5-12) where F!! is the cumulative distribution function of !! for the molecular area, !!. Chapter 5. Kinetic Derivation of Common Isotherm Equations 114 Figure 5.2. Representation of successful and unsuccessful impacts on the surface, depending on the absence, !!>!!, or presence, !!<!!, of an adsorbed molecule on the impact area of the incoming molecule. To determine the distribution function, it must be considered that, under the model of the twodimensional gas [13], the adsorbed molecules are randomly distributed on the surface. Therefore, the probability that there is a molecule at a given site is always the same, equal to the fraction of surface area occupied, !, and independent from the presence of other molecules nearby. The presence of a molecule anywhere on the surface can therefore be seen as a Bernoulli trial, where a given site on the surface either has, or has not, molecules on it, with probability !. The site area !! needed to encounter the closest molecule to the impact site (Figure 5.2) can be seen as a first “success” in successive Bernoulli trials. For continuous random variables, the variable counting the number of Bernoulli trials until the first success has an exponential distribution [15-17]. The corresponding cumulative function of !! is given by: F!!=1−exp −!·!! (5-13) where the distribution parameter ! is given by the relationship != 1 !! (5-14) Adsorption on a Planar Surface – 2D 115 where !! is the average of !!. It is possible to see !! as the free area at the site of impact, which means its average will be !−!!, the total free molar area at the surface. Hence, Eq. (5-14) can be rewritten as, != 1 !−!! (5-15) and, from Eq. (5-13), F!!=1−exp − !! !−!! (5-16) given that, != !! ! !! !−!! = ! 1−! (5-17) substitution of Eq. (5-17) into Eq. (5-16) yields, F!!=1−exp − ! 1−! (5-18) Substituting back into the probability equation, Eq. (5-12), and into the equation for the adsorption rate, Eq. (5-10), and equating to the desorption rate, Eq. (5-8), the equation describing the Volmer Isotherm is obtained, !" = ! 1−! exp ! 1−! (5-19) where b is the same as for the Langmuir isotherm, Eqs. (5-5) and (5-6). The kinetic approach used to describe the distributed monolayer adsorption on 2D surfaces, which was just shown to allow the derivation of the Volmer isotherm equation, offers some interesting insight into the physical features behind the different shapes of the Volmer and Langmuir isotherms. The Volmer isotherm is significantly more concave than the Langmuir isotherm. Mathematically, this is expressed by the additional exponential factor existent in the Volmer equation. Physically, this reflects the need for the existence of space between the molecules already adsorbed for new adsorption of gas-phase molecules to take place. For high coverages, there is so little free area, that the frequency with which a large enough space is formed, so that adsorption can occur, is extremely small, thus dramatically reducing the adsorption rate compared to that of localised adsorption at the same coverage. In fact, from Eqs. (5-11), (5-12) and (5-18), one obtains: Chapter 5. Kinetic Derivation of Common Isotherm Equations 116 p!!"#$% >!∗=exp − ! 1−! (5-20) Thus, for high surface coverage, the probability there will be enough free space concentrated for adsorption to occur tends to zero. As for localised adsorption, for low coverages, the adsorption rate equation, Eq. (5-10), can be considered coverage-independent, with 1−!≈1 and p!!"#$% >!∗≈1, resulting in the Henry isotherm equation. Naturally, applying this condition directly to the Volmer isotherm leads to the same outcome. Finally, it is worth pointing out that this approach does not assume the existence or inexistence of 2D condensation. All the premises of the model presented hold true regardless. 5.3.3.Henry Equation for Surfaces The Henry isotherm can be obtained directly from the rate equations, Eqs. (5-1), (3-4), (5-8), and (5-10) by considering that, for low coverage, 1−!≈1. That is, the adsorption rate is independent of coverage. The Henry isotherm easily follows from equating the two rates, !!!=! (5-21) Applying the 1−!≈1 condition to the Langmuir and Volmer isotherms themselves, the same result is obtained. This result is consistent with the condition of Ideal Gas behaviour of the adsorbed phase, necessary also in the thermodynamic derivation of the Henry isotherm [13, 14]. This condition also implies dimensionless adsorbed molecules, which, thus, would not influence the adsorption rate, independently of superficial density, just as considered. In real systems, the condition of ideality is only met at very low densities. 5.4. 1D and 3D Adsorbed Phases – Adsorption in Micropores Application of the kinetic approach to the derivation of adsorption isotherm equations for micropores has one significant difference regarding 2D adsorption; the interface between the two phases is available only to a small fraction of the adsorbate, present at the pore mouths. This influences fundamentally the modelling of both adsorption and desorption, as the rates of both are now dependent exclusively on concentration at the pore mouths, not the entire 1D and 3D Adsorbed Phases – Adsorption in Micropores 117 adsorbed phase. Since we are only interested in dealing with equilibrium, it will be considered that concentration at pore mouth is the same as anywhere else in the pore. There are two kinds of micropore adsorption that, we proposed in Chapter 4, can sometimes be modelled using the equations in Table 5.1; 1D and 3D adsorbed phases. As for 2D, the models developed for 1D and 3D will be premised on the (likely) supposition that gas molecules move much faster than molecules already adsorbed. As mentioned before, only distributed adsorption will be considered for 3D phases. 5.4.1. 3D Adsorbed Phases - Distributed Adsorption The key variables relevant for describing a 3D system are the specific pore volume of the adsorbent, ! (SI units of m3·kg-1), the molar volume of the adsorbent/adsorbate system, !≡!/! (m3·mol-1), the average pore mouth area, !!"#$! (m2·mouth-1), the total specific number of pores, !!"#$ (pore·kg-1) and the average number of mouths per pore, !!"!"! (mouth·pore-1). As with ! and !!, for !, when !=!!"#, !=!!. The rate of adsorption on a 3D phase is modelled very similarly to that of a distributed 2D phase, but the area available for adsorption is now only that of the pore mouths. That is, ! != ! 2!"#$ ·!!"#$!·!!"#$ ·!!"#$!·p!"#$% (5-22) where p!"#$% is the probability of entry of each gas molecule upon hitting the pore mouth. p!"#!" is determined by conditions at the interface, the “surface” of the adsorbed phase inside the pore. In order to model this probability, a new variable is introduced for the 3D system: the interfacial molar area, !!"#$. It can be defined as the pore mouth area, !!"#$!, divided by the amount of adsorbed molecules located at the interface, !!"#$%&'($. !!"#$ ≡ !!"#$! !!"#$%&'($ (5-23) A visual representation of the concept of 3D adsorbed phase “surface” can be seen in Figure 5.3. At equilibrium, the interfacial molar area and pore molar volume are related by !! !!"#$ = !! !=! (5-24) Chapter 5. Kinetic Derivation of Common Isotherm Equations 124 !!!" =!·f!=0 (5-55) where ! is the mean speed of adsorbed molecules (m·s-1) and f!=0 is the “frequency” (mol·m-1) with which molecules hit each other inside the pores. From Eq. (5-48), f!=0= 1 !−!! (5-56) Therefore, !!!" =!· 1 !−!! (5-57) and, ! !=!!"#$ ·!!"#$!·!· 1 !−!! ·p!"#$%& (5-58) Using Eq. (5-52), this can be rewritten as ! !=!!"#$ ·!!"#$!· !·p!"#$%& !! · ! 1−! (5-59) Thus, the initially hidden frequency of desorption can now be represented explicitly, !!= !·p!"#$%& !! !!=!!·exp − !! !" (5-60) given in mol·s-1·mouth-1. Equating ! ! and ! !, multiplying by !! on both sides, and solving for !", it results that, !" = ! 1−! exp ! 1−! (5-61) with, != !! !!"# (5-62) and, !!= !·!!"#$!·!!"# !!·2!"#$ = !·!!"#$!·1!!·! !!·2!"#$ (5-63) The expression for !! is, as expected, exactly the same as the one determined for the Langmuir equation. 1D Adsorbed Phases 125 The geometric factor present in !! for 3D phases, !!!!, is now 1!!, given that only pore length occupied by the adsorbed molecule influences equilibrium. It is interesting to notice that, unlike what happens for 1D and 3D, in systems with 2D adsorbed phases, !! does not depend directly on the size and shape of the adsorbate, reflecting the fact that, for low coverages, molecular size does not influence the extent of adsorption on the surface. It is worth noticing that !! is widely used in the derivation of the Volmer equation for distributed adsorption, while it does not feature in the localised adsorption derivation. This happens because, for localised adsorption, !! is variable and does not influence the probabilities of neither adsorption or desorption. !! corresponds to the distance between adsorption sites, with molecular length being merely the minimum for this distance. It does, however, influence adsorption, since it represents adsorption site density of the material, and that is why it appears in the expression for !!. p!"#$% for 1D localised and distributed adsorption is as different between the two as p!!"#$% >!∗ is for 2D adsorption. As with the latter, while for localised adsorption the adsorption rate depends linearly on !, for distributed adsorption it depends exponentially. With localised adsorption sites, for equivalent filling, the frequency with which the pore mouth is available for adsorption of a new molecule depends only on the fraction of time the site closest to the pore mouth is free, not a confluence of probabilities, as for distributed adsorption. In this case, for high fillings, there is so little free length inside the pore that the probability there will be enough of it concentrated at the pore mouth, allowing adsorption to take place, is extremely small. 5.5.3. Henry Equation for 1D Adsorbed Phases For 1D adsorbed phases, as in 2D and 3D, the Henry isotherm is obtained, both for localised and distributed adsorption, by considering the effect of low values of ! on the rate of adsorption. In this case, the rate of adsorption is essentially independent of the amount adsorbed, since the probability of a molecule being close to the pore mouth is very small. Chapter 5. Kinetic Derivation of Common Isotherm Equations 126 5.6. Interactions Between Adsorbed Molecules – The Hill-de Boer and Fowler-Guggenheim Equations Adsorbate-adsorbate interactions (so-called “lateral interactions”) may cause changes in the heat of adsorption, which influences the rate of desorption. We will assume the rate of adsorption is unaffected. For 2D, this is not strictly true, since variations in the strength of adsorption influence parameter !, which was previously assumed to be independent of ! and !. The activation energy needed to pull an adsorbed molecule away from the surface (or pore) is now the energy associated with interactions between the adsorbed molecule and the surface (or pore) plus the energy associated with the interactions between adsorbed molecules !!=−∆!!"# = !!"# +−∆!!"#$% (5-64) where !!"# represents the energy associated with adsorbate-adsorbent interactions and −∆!!"#$% represents the energy associated with lateral interactions. −∆!!"#$% depends on the number of molecules next to each adsorbed molecule, and can therefore be approximated as, −∆!!"#$% =!·!·! (5-65) where ! is the number of positions, adjacent to the molecule, that other molecules can occupy and ! is the energy associated with interactions between two adsorbed molecules. ! is equal to the fraction of positions ! that are occupied. From Eq. (5-3), it is possible to write that, !!=!!·exp −!!"# !" ·exp − !"# !" !!=!! !·exp − !"# !" (5-66) Replacing this expression in Eqs. (3-4), (5-8), (5-27), (5-39) and (5-60), the desorption rate considering interactions between adsorbed molecules is obtained. The Fowler-Guggenheim and Hill-de Boer equations follow naturally from there, !" = ! 1−! exp −! ! (5-67) !" = ! 1−! !"# ! 1−!!"# −! ! (5-68) with, Conclusions 127 != !" !" (5-69) where ! is a system-specific constant. The Henry constants obtained remain unchanged; only !! is replaced by !! !, which are actually the same, since both are exclusively dependent on !!"#. This approach to incorporation of lateral interactions assumes that the distribution of the molar area, molar volume and molar length are unaffected by the existence of intermolecular forces. An equivalent assumption was central to the derivation of the van der Waals equation [18], and, thus, is already implicit when applying its two-dimensional form to the Gibbs adsorption isotherm [7, 14]. Naturally, if the same isotherm expression is to be obtained using the kinetic approach, this assumption must be built into this derivation. It should be pointed out that Eq. (5-64) does not imply that !!"# must be constant regarding !. In fact, !!"# typically decreases with ! [14, 19], and may actually hide the influence of lateral interactions on the heat of adsorption. If the variation is linear, or can be approximated as such, this dependence can be incorporated into an experimentally determined linearity constant, which would also incorporate, but not be equal to, !. If the variation is not linear, the Fowler-Guggenheim and Hill-de Boer equations (as well as the associated equations of state) do not apply to the system in question. 5.7. Conclusions The Henry, Langmuir, Volmer, Fowler-Guggenheim and Hill-de Boer equations were derived using a kinetic approach. The Kinetic Gas Theory was used as a starting point to determine the rate of adsorption, while the rate of desorption was modelled by considering desorption to be an activated process with an activated energy equal to the heat of adsorption. The exponential term that distinguishes equations for localised and distributed adsorption emerges considering a random distribution of intermolecular distances. This distance influences (in different ways for 1D, 2D and 3D phases) the probability of adsorption occurring upon impact of a gas-phase molecule. Considering localised adsorption in 1D and 2D systems leads to the Langmuir equation. When interactions between adsorbed molecules are considered, through their influence on the heat of adsorption, the Fowler-Guggenheim equation is obtained. 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General Conclusions and Future Work Crystalline hydrophobic dipetides are microporous materials that have been considered a very promising kind of peptide-based supramolecular microporous material. Like metal-peptide frameworks, they can only be used in the crystalline solid state as membrane material and as adsorbents. Macrocyclic peptides and dendritic peptides, on the other hand, are used mainly in solution, especially the former. These two have thus been those that have better replicated the properties of natural protein and peptide pores, in aqueous environments, while metal-peptide frameworks and hydrophobic dipeptides have shown how typical protein properties such as framework flexibility and guest-specificity can generate unexpected and unusual properties in the crystalline solid state. The VA-class of crystalline hydrophobic dipeptides (seven dipeptides) displays hydrophobic ultramicropores in the 0.37-0.50 nm range, and is therefore extremely interesting in the adsorption of gases and vapours. AA and LS, technically not included of the VA-class, also form hydrophobic cavities and pores. AA displays isolated cages, while LS displays 0.49 nmwide pores. Single-crystal permeation experiments with AA, VI and LS were performed with atmospheric gases. The four dipetides with the smallest pores in the VA-class, VI (0.37 nm), IA (0.37 nm), IV (0.39 nm) and VV (0.44 nm), were tested as adsorbents of N2, O2, Ar, Xe and CO2. The permeation experiments showed that these materials have very high permeabilities. It was also observed that small pore size, framework flexibility and pore blockage can have a dramatic effect on the properties of framework-guest interactions. Millimetre-sized VI singlecrystals were capable of permeating O2, N2 and He, but not Ar. Initially, it was thought this could be due to the open pore sieving the slightly larger Ar atom and not the other species. As VI crystalline powder (crystal length of 1-50 µm) easily adsorbs Ar and crystalline defects increase with pore length, it is likely that Ar impermeability is due to pore blockage. Framework flexibility probably plays an important role in the permeation of the slightly smaller species of O2, N2 and He. AA, being non-porous and forming isolated cavities, was, nonetheless, able to permeate O2, in a remarkable display of framework flexibility of these “soft” materials. If a hydrophobic dipeptide were to be used in an actual membrane separation process, the retained species would have to be sieved outside the pore, having thus to be clearly bigger than the permeating species. Chapter 6. General Conclusions and Future Work 132 Adsorption isotherms of atmospheric gases unveiled an unusual property; Ar is preferentially adsorbed against O2. N2 adsorption was inferior to both Ar and O2. This preferential adsorption sequence, Ar>O2>N2, is highly unusual. All dipeptides tested, VI, IA, IV and VV, showed the same sequence of preferential adsorption. Similarly, all gases used had the same sequence of preferential adsorption for different dipeptides, VI<IA<IV<VV. Adsorption preference thus follows pore size variation sequence, for these four dipeptides. Given pore morphology and chemistry, a scenario of distributed adsorption with no lateral interactions was assumed for atmospheric gases in the pores of hydrophobic dipeptide crystals. This picture is close to that underlying the derivation of the Volmer equation for two-dimensional surfaces. Although it was, at the time, uncertain whether it would also apply to one-dimensional systems, such as those under consideration, it was decided to use it. It was later possible to show this attribution to be correct. Interpretation of the simple adsorption preference sequences proved more complicated than initially expected. The heats of adsorption of N2 and O2 do not follow such a simple relation, and vary in a non-monotonic fashion with pore size. The heat of adsorption of Ar does vary monotonically, but showing precisely the trend opposite to that of adsorption affinity. Equally, the Ar>O2>N2 sequence of preferential adsorption for each dipeptide is also not observed in the heats of adsorption. These results were interpreted as stemming from a complex interaction of entropic degrees of freedom (influencing the pre-exponential factor of the van’t Hoff equation) and matching between pore and guest molecule morphologies (influencing the heat of adsorption). The heats of adsorption do not vary significantly with the adsorbed concentration, for any host-guest pair, with the possible exception of N2. This indicates, on one hand, the existence of a relatively uniform adsorption potential field inside the pores and, on the other, that interactions between guest molecules never becomes significant within the experimental conditions tested, as expected given all isotherms are, for all practical purposes, linear. The values determined are small when compared to those obtained for other materials. In the case of N2, three of the four values determined are below 15 kJ·mol-1, the minimum reported in the literature. The Ar/O2 selectivity values determined are very high. Those determined for VI, at any of the three temperatures, are better than any previously reported for Ag-free adsorbents. The best result obtained was 1.30, for VI, at 5 ºC, under vacuum. Selectivity decreases slightly with pressure, for all four dipeptides, as expected given it is the ratio of two Type I adsorption 133 isotherms barely out of the Henry region. Temperature dependence of selectivity variation is very pressure dependent. At vacuum conditions, it either decreases or increases with increasing temperature depending on whether the heat of adsorption of Ar is bigger or smaller, respectively, than that of O2. Similar behaviours are observed for O2/N2 selectivities. Both Ar/O2 and O2/N2 selectivities decrease with increasing pore size. Despite the exceptional results obtained, it is unlikely crystalline hydrophobic dipeptides could be used as an adsorbent in PSA-based separation processes, mainly due to poor adsorption capacities; these are caused by the low pore densities of this class of materials. However, the selectivity results obtained could point to extreme confinement in hydrophobic ultramicropores of organic materials as a possible path for systematically obtaining similar selectivities. This can only be confirmed by determination of the adsorption isotherms of other materials with similar structures to those of crystalline VA-class dipeptides. If such structure-property relation does exist, then organic adsorbents with hydrophobic micropores could be a viable alternative to Ag-based adsorbents. Given the large number and wide variety of OSMSs discovered in the last 20 years, materials with such features should be in no short supply. It would also be interesting to assess what are the limits of framework flexibility, in these and other supramolecular adsorbents, and to what extent is the sieving threshold influenced by the intensity of the interactions with the framework. This could possibly have great influence in adsorption and diffusion in the porous framework, with obvious practical implications. In order to better interpret some of the results obtained, and be able to gain insight into the physical phenomena occurring at the atomic level, adsorption models were developed for unidimensional pores such as those of hydrophobic dipeptide crystals. By using a wellestablished thermodynamic approach, embodied by the Gibbs Adsorption Isotherm, it was possible to demonstrate that, under certain assumptions, some simple equations used to describe surface adsorption (2D adsorption systems) and micropore-filling (3D adsorption systems) can also be used with equal validity in single-file adsorption (1D adsorption systems). The approach used was exactly the same as that of 2D and 3D adsorption, and its implementation in 1D systems was quite straightforward once appropriate system variables analogue to the spreading pressure and molar area of 2D systems were developed. The Gibbs Adsorption Isotherm is obtained by equalising the Gibbs energy differentials in the adsorbed phase and the bulk gas/liquid phase. It is, thus, universally applicable, although its phenomenological validity is dependent on the physical significance of the characteristic