A New Statistical Theory for Constructing Sorption Isotherms in Mesoporous Structures Represented by Bethe Lattices.

In the present work, a new statistical theory is developed to describe adsorption and desorption in mesoporous materials (pore sizes ranging from 2 to 50 nm) represented by pore networks in the form of Bethe lattices. The new theory is an extension of a previous theory applied for Statistically Disordered Chain Model (SDCM) structures and incorporates the cooperative effects emerging during phase transitions in pore networks. The theory is validated against simulations and algorithmic models that describe sorption of lattice and real fluids in Bethe lattices. It is seen that the pore network coordination number, or pore connectivity, z, has a significant impact on two important processes observed in pore networks: pore assisting condensation during adsorption and evaporation by percolation during desorption. The inclusion of pore connectivity in the earlier developed framework accounting for cooperativity effects is an important step, rendering the existing models to mimic fluid behavior in real materials more accurately. Hence, the new theory inherently contains all essential elements that may offer the extraction of more reliable pore size distributions utilizing both the adsorption and desorption branches of the isotherm.

Impact of Geometrical Disorder on Phase Equilibria of Fluids and Solids Confined in Mesoporous Materials.

Porous solids used in practical applications often possess structural disorder over broad length scales. This disorder strongly affects different properties of the substances confined in their pore spaces. Quantifying structural disorder and correlating it with the physical properties of confined matter is thus a necessary step toward the rational use of porous solids in practical applications and process optimization. The present work focuses on recent advances made in the understanding of correlations between the phase state and geometric disorder in nanoporous solids. We overview the recently developed statistical theory for phase transitions in a minimalistic model of disordered pore networks: linear chains of pores with statistical disorder. By correlating its predictions with various experimental observations, we show that this model gives notable insight into collective phenomena in phase-transition processes in disordered materials and is capable of explaining self-consistently the majority of the experimental results obtained for gas-liquid and solid-liquid equilibria in mesoporous solids. The potentials of the theory for improving the gas sorption and thermoporometry characterization of porous materials are discussed.

Collagen networks determine viscoelastic properties of connective tissues yet do not hinder diffusion of the aqueous solvent.

Collagen accounts for the major extracellular matrix (ECM) component in many tissues and provides mechanical support for cells. Magnetic Resonance (MR) Imaging, MR based diffusion measurements and MR Elastography (MRE) are considered sensitive to the microstructure of tissues including collagen networks of the ECM. However, little is known whether water diffusion interacts with viscoelastic properties of tissues. This study combines highfield MR based diffusion measurements, novel compact tabletop MRE and confocal microscopy in collagen networks of different cross-linking states (untreated collagen gels versus additional treatment with glutaraldehyde). The consistency of bulk rheology and MRE within a wide dynamic range is demonstrated in heparin gels, a viscoelastic standard for MRE. Additional crosslinking of collagen led to an 8-fold increased storage modulus, a 4-fold increased loss modulus and a significantly decreased power law exponent, describing multi-relaxational behavior, corresponding to a pronounced transition from viscous-soft to elastic-rigid properties. Collagen network changes were not detectable by MR based diffusion measurements and microscopy which are sensitive to the micrometer scale. The MRE-measured shear modulus is sensitive to collagen fiber interactions which take place on the intrafiber level such as fiber stiffness. The insensitivity of MR based diffusion measurements to collagen hydrogels of different cross-linking states alludes that congeneric collagen structures in connective tissues do not hinder extracellular diffusive water transport. Furthermore, the glutaraldehyde induced rigorous changes in viscoelastic properties indicate that intrafibrillar dissipation is the dominant mode of viscous dissipation in collagen-dominated connective tissue.

Revealing the Transient Concentration of CO2 in a Mixed-Matrix Membrane by IR Microimaging and Molecular Modeling.

Through IR microimaging the spatially and temporally resolved development of the CO2 concentration in a ZIF-8@6FDA-DAM mixed matrix membrane (MMM) was visualized during transient adsorption. By recording the evolution of the CO2 concentration, it is observed that the CO2 molecules propagate from the ZIF-8 filler, which acts as a transport "highway", towards the surrounding polymer. A high-CO2 -concentration layer is formed at the MOF/polymer interface, which becomes more pronounced at higher CO2 gas pressures. A microscopic explanation of the origins of this phenomenon is suggested by means of molecular modeling. By applying a computational methodology combining quantum and force-field based calculations, the formation of microvoids at the MOF/polymer interface is predicted. Grand canonical Monte Carlo simulations further demonstrate that CO2 tends to preferentially reside in these microvoids, which is expected to facilitate CO2 accumulation at the interface.

Transport properties of hierarchical micro-mesoporous materials.

Adding mesopore networks in microporous materials using the principles of hierarchical structure design is recognized as a promising route for eliminating their transport limitations and, therefore, for improving their value in technological applications. Depending on the routes of physico-chemical procedures or post-synthesis treatments used, very different geometries of the intentionally-added transport mesopores can be obtained. Understanding the structure-dynamics relationships in these complex materials with multiple porosities under different thermodynamical conditions remains a challenging task. In this review, we summarize the results obtained so far on experimental and theoretical studies of diffusion in micro-mesoporous materials. By considering four common classes of bi-porous materials, which are differing by the inter-connectivities of their sup-spaces as one of the most important parameter determining the transport rates, we discuss their generic transport properties and correlate the results delivered by the equilibrium and non-equilibrium techniques of diffusion measurements.

Modeling the influence of side stream and ink bottle structures on adsorption/desorption dynamics of fluids in long pores.

We apply dynamic mean field theory to study relaxation dynamics for lattice models of fluids confined in linear pores with side streams and with ink bottle structures. Our results show several mechanisms for how the pore structure affects the dynamics, and these are amplified in longer pores. An important conclusion of this work is that features such as side streams and ink bottle segments can substantially slow the equilibration of fluids confined in long pore systems where the pore lengths can be more than 100 micrometers, such as in porous silicon. This may make it difficult to properly equilibrate these systems for states close to those where the pores should be completely filled with liquids. The presence of trapped bubbles in the system may change the desorption characteristics of the system and the shape of the hysteresis loops.

Water-mediated proton conduction in a robust triazolyl phosphonate metal-organic framework with hydrophilic nanochannels.

The development of water-mediated proton-conducting materials operating above 100 °C remains challenging because the extended structures of existing materials usually deteriorate at high temperatures. A new triazolyl phosphonate metal-organic framework (MOF) [La3L4(H2O)6]Cl⋅x H2O (1, L(2-) = 4-(4H-1,2,4-triazol-4-yl)phenyl phosphonate) with highly hydrophilic 1D channels was synthesized hydrothermally. Compound 1 is an example of a phosphonate MOF with large regular pores with 1.9 nm in diameter. It forms a water-stable, porous structure that can be reversibly hydrated and dehydrated. The proton-conducting properties of 1 were investigated by impedance spectroscopy. Magic-angle spinning (MAS) and pulse field gradient (PFG) NMR spectroscopies confirm the dynamic nature of the incorporated water molecules. The diffusivities, determined by PFG NMR and IR microscopy, were found to be close to that of liquid water. This porous framework accomplishes the challenges of water stability and proton conduction even at 110 °C. The conductivity in 1 is proposed to occur by the vehicle mechanism.

Probing mass transfer in mesoporous faujasite-type zeolite nanosheet assemblies.

Pulsed field gradient nuclear magnetic resonance (NMR) diffusion studies are performed by using cyclohexane to probe transport properties in a NaX-type zeolite with a hierarchical pore structure (house-of-cards-like assemblies of mesoporous nanosheets), which is compared with a purely microporous sample. With guest loadings chosen to ensure saturation of the micropores, and the meso- and macropores left essentially unoccupied, guest diffusion is shown to be enhanced by almost one order of magnitude, even at room temperature. Diffusivity enhancement is further increased with increasing temperature, which may, therefore, be unambiguously attributed to the contribution of mass transfer in the meso- and macropores.

Filling dynamics of closed end nanocapillaries.

We have studied the filling dynamics of model capillaries using dynamic mean field theory for a confined lattice gas and Kawasaki dynamics simulations. We have found two different scenarios for filling of capped nanocapillaries from the vapor phase. As compared to channels with macroscopic width, in which the filling process occurs by the detachment of the meniscus from the cap, in mesoscopic channels there is an alternative mechanism associated with the spontaneous condensation of the liquid close to the pore opening and its subsequent growth toward the closed pore end. We show that these two scenarios have totally different filling dynamics, providing an additional mechanism for slow capillary condensation kinetics in nanoscopic objects.

Mass transfer in mesoporous materials: the benefit of microscopic diffusion measurement.

We introduce the various options of experimentally observing mass transfer in mesoporous materials. It shall be demonstrated that the exploration of the underlying mechanisms is excessively complicated by the complexity of the phenomena contributing to molecular transport in such systems and their mutual interdependence. Microscopic diffusion measurement by the pulsed field gradient (PFG) technique of NMR offers the unique option to measure both the relative amount of molecules adsorbed and the probability distribution of their displacements over space scales relevant to fundamental adsorption science just as for technological application. These advantages are shown to have cared for a recent breakthrough in our understanding. The examples presented include the measurement of diffusion in purely mesoporous materials and the rationalization of the complex concentration patterns revealed by such studies on the basis of suitably chosen micro-kinetic models. As an interesting feature, transition into the supercritical state is shown to become directly observable by monitoring a jump in the diffusivities during temperature enhancement, occurring at temperatures notably below the bulk critical temperature. PFG NMR studies with hierarchical materials are shown to permit selective diffusion measurement with each of the involved subspaces, in parallel with the measurement of the overall diffusivity as the key parameter for the technological exploitation of such materials. We refer to the occurrence of diffusion hysteresis as a novel phenomenon, found to accompany phase transitions quite in general. Though further complicating the measuring procedure and the correlation between experimental observation and the underlying mechanisms, diffusion hysteresis is doubtlessly among the new options provided by diffusion studies for gaining deeper insight into the structure and dynamics of complex porous systems.

Cooperative Dynamics of Highly Entangled Linear Polymers within the Entanglement Tube.

We present a quantitative comparison of the dynamic structure factors from unentangled and strongly entangled poly(butylene oxide) (PBO) melts. As expected, the low molecular weight PBO displays Rouse dynamics, however, with very significant subdiffusive center-of-mass diffusion. The spectra from high molecular weight entangled PBO can be very well described by the dynamic structure factor based on the concept of local reptation, including the Rouse dynamics within the tube and allowing for non-Gaussian corrections. Comparing quantitatively the spectra from both polymers leads to the surprising result that their spectra differ only by the contribution of classical Rouse diffusion for the low molecular weight melt. The subdiffusive component is common for both the low and high molecular weight PBO melts, indicating that in both melts the same interchain potential is active, thereby supporting the validity of the Generalized Langevin Equation approach.

Kinetics of Guest-Induced Structural Transitions in Metal-Organic-Framework MIL-53(Al)-NH2 Probed by High-Pressure Nuclear Magnetic Resonance.

A nuclear magnetic resonance (NMR) study of a pore opening in amino-functionalized metal-organic framework (MOF) MIL-53(Al) in response to methane pressure variation is presented. Variations of both NMR signal intensities and transversal relaxation rates for methane are found to reveal hysteretic structural transitions in the MOF material, which are smeared out over broad pressure ranges. Experiments with pressure reversals upon an incomplete adsorption/desorption gave deeper insight into the microscopic transition mechanisms. These experiments have unequivocally proven that the non-stepwise pore opening/closing transitions observed in the experiments are governed by a distribution of the opening/closing pressures over different MOF crystallites, for example, due to a distribution of the crystal sizes or shapes. The slow kinetics of the structural transitions measured in the hysteresis regime revealed a complex free energy landscape for the phase transition process.

Monitoring molecular mass transfer in cation-free nanoporous host crystals of type AlPO-LTA.

Micro-imaging is employed to monitor the evolution of intra-crystalline guest profiles during molecular adsorption and desorption in cation-free zeolites AlPO-LTA. The measurements are shown to provide direct evidence on the rate of intra-crystalline diffusion and surface permeation and their inter-relation. Complemented by PFG NMR and integral IR measurements, a comprehensive overview of the diffusivities of light hydrocarbons in this important type of host materials is provided.

Intracrystalline diffusion in mesoporous zeolites.

Specially synthesized extra-large crystallites of zeolite LTA with intentionally added mesoporosity are used for an in-depth study of guest diffusion in hierarchical nanoporous materials by the pulsed field gradient NMR technique. Using propane as a guest molecule, intracrystalline mass transfer is demonstrated to be adequately described by a single effective diffusivity resulting from the weighted average of the diffusivities in the two (micro- and meso-) pore spaces. Gas-kinetic order-of-magnitude estimates of the diffusivities are in satisfactory agreement with the experimental data and are thus shown to provide a straightforward means for predicting and quantifying the benefit of hierarchically structured nanoporous materials in comparison with their purely microporous equivalent.

Exploration of molecular dynamics during transient sorption of fluids in mesoporous materials.

In recent years, considerable progress has been made in the development of novel porous materials with controlled architectures and pore sizes in the mesoporous range. An important feature of these materials is the phenomenon of adsorption hysteresis: for certain ranges of applied pressure, the amount of a molecular species adsorbed by the mesoporous host is higher on desorption than on adsorption, indicating a failure of the system to equilibrate. Although this phenomenon has been known for over a century, the underlying internal dynamics responsible for the hysteresis remain poorly understood. Here we present a combined experimental and theoretical study in which microscopic and macroscopic aspects of the relaxation dynamics associated with hysteresis are quantified by direct measurement and computer simulations of molecular models. Using nuclear magnetic resonance techniques and Vycor porous glass as a model mesoporous system, we have explored the relationship between molecular self-diffusion and global uptake dynamics. For states outside the hysteresis region, the relaxation process is found to be essentially diffusive in character; within the hysteresis region, the dynamics slow down dramatically and, at long times, are dominated by activated rearrangement of the adsorbate density within the host material.

On the Comparative Analysis of Different Phase Coexistences in Mesoporous Materials.

Alterations of fluid phase transitions in porous materials are conventionally employed for the characterization of mesoporous solids. In the first approximation, this may be based on the application of the Kelvin equation for gas-liquid and the Gibbs-Thomson equation for solid-liquid phase equilibria for obtaining pore size distributions. Herein, we provide a comparative analysis of different phase coexistences measured in mesoporous silica solids with different pore sizes and morphology. Instead of comparing the resulting pore size distributions, we rather compare the transitions directly by using a common coordinate for varying the experiment's thermodynamic parameters based on the two equations mentioned. Both phase transitions in these coordinates produce comparable results for mesoporous solids of relatively large pore sizes. In contrast, marked differences are found for materials with smaller pore sizes. This illuminates the fact that, with reducing confinement sizes, thermodynamic fluctuations become increasingly important and different for different equilibria considered. In addition, we show that in the coordinate used for analysis, mercury intrusion matches perfectly with desorption and freezing transitions.

Guest diffusion in interpenetrating networks of micro- and mesopores.

Pulsed field gradient NMR is applied for monitoring the diffusion properties of guest molecules in hierarchical pore systems after pressure variation in the external atmosphere. Following previous studies with purely mesoporous solids, also in the material containing both micro- and mesopores (activated carbon MA2), the diffusivity of the guest molecules (cyclohexane) is found to be most decisively determined by the sample "history": at a given external pressure, diffusivities are always found to be larger if they are measured after pressure decrease (i.e., on the "desorption" branch) rather than after pressure increase (adsorption branch). Simple model consideration reproduces the order of magnitude of the measured diffusivities as well as the tendencies in their relation to each other and their concentration dependence.

How to compare diffusion processes assessed by single-particle tracking and pulsed field gradient nuclear magnetic resonance.

Heterogeneous diffusion processes occur in many different fields such as transport in living cells or diffusion in porous media. A characterization of the transport parameters of such processes can be achieved by ensemble-based methods, such as pulsed field gradient nuclear magnetic resonance (PFG NMR), or by trajectory-based methods obtained from single-particle tracking (SPT) experiments. In this paper, we study the general relationship between both methods and its application to heterogeneous systems. We derive analytical expressions for the distribution of diffusivities from SPT and further relate it to NMR spin-echo diffusion attenuation functions. To exemplify the applicability of this approach, we employ a well-established two-region exchange model, which has widely been used in the context of PFG NMR studies of multiphase systems subjected to interphase molecular exchange processes. This type of systems, which can also describe a layered liquid with layer-dependent self-diffusion coefficients, has also recently gained attention in SPT experiments. We reformulate the results of the two-region exchange model in terms of SPT-observables and compare its predictions to that obtained using the exact transformation which we derived.

Entropy-driven enhanced self-diffusion in confined reentrant supernematics.

We present a molecular dynamics study of reentrant nematic phases using the Gay-Berne-Kihara model of a liquid crystal in nanoconfinement. At densities above those characteristic of smectic A phases, reentrant nematic phases form that are characterized by a large value of the nematic order parameter S≃1. Along the nematic director these "supernematic" phases exhibit a remarkably high self-diffusivity, which exceeds that for ordinary, lower-density nematic phases by an order of magnitude. Enhancement of self-diffusivity is attributed to a decrease of rotational configurational entropy in confinement. Recent developments in the pulsed field gradient NMR technique are shown to provide favorable conditions for an experimental confirmation of our simulations.

Charge transport and diffusion of ionic liquids in nanoporous silica membranes.

Charge transport in 1-hexyl-3-methylimidazolium hexafluorophosphate ionic liquid in oxidized nanoporous silicon membranes is investigated in a wide frequency and temperature range by a combination of Broadband Dielectric Spectroscopy (BDS) and Pulsed Field Gradient Nuclear Magnetic Resonance (PFG NMR). By applying the Einstein-Smoluchowski relations to the dielectric spectra, diffusion coefficient is obtained in quantitative agreement with independent PFG NMR measurements. More than 10-fold systematic decrease in the effective diffusion coefficient from the bulk value is observed in hydrophilic silica nanopores. A model assuming a reduced mobility at the pore-matrix interface is shown to provide a quantitative explanation for the remarkable decrease of effective transport quantities (such as diffusion coefficient, dc conductivity and consequently, the dielectric loss) of the ionic liquid in non-silanized membranes. This approach is supported by the observation that silanization of porous silica membranes results in a significant increase of the effective diffusion coefficient, which approaches the value for the bulk liquid.