Connecting theory and simulation with experiment for the study of diffusion in nanoporous solids

Nanoporous solids are ubiquitous in chemical, energy, and environmental processes, where controlled transport of molecules through the pores plays a crucial role. They are used as sorbents, chromatographic or membrane materials for separations, and as catalysts and catalyst supports. Defined as materials where confinement effects lead to substantial deviations from bulk diffusion, nanoporous materials include crystalline microporous zeotypes and metal–organic frameworks (MOFs), and a number of semi-crystalline and amorphous mesoporous solids, as well as hierarchically structured materials, containing both nanopores and wider meso- or macropores to facilitate transport over macroscopic distances. The ranges of pore sizes, shapes, and topologies spanned by these materials represent a considerable challenge for predicting molecular diffusivities, but fundamental understanding also provides an opportunity to guide the design of new nanoporous materials to increase the performance of transport limited processes. Remarkable progress in synthesis increasingly allows these designs to be put into practice. Molecular simulation techniques have been used in conjunction with experimental measurements to examine in detail the fundamental diffusion processes within nanoporous solids, to provide insight into the free energy landscape navigated by adsorbates, and to better understand nano-confinement effects. Pore network models, discrete particle models and synthesis-mimicking atomistic models allow to tackle diffusion in mesoporous and hierarchically structured porous materials, where multiscale approaches benefit from ever cheaper parallel computing and higher resolution imaging. Here, we discuss synergistic combinations of simulation and experiment to showcase theoretical progress and computational techniques that have been successful in predicting guest diffusion and providing insights. We also outline where new fundamental developments and experimental techniques are needed to enable more accurate predictions for complex systems.

Pulsed field gradient NMR diffusion measurement in nanoporous materials

Labeling in diffusion measurements by pulsed field gradient (PFG) NMR is based on the observation of the phase of nuclear spins acquired in a constant magnetic field with purposefully superimposed field gradients. This labeling does in no way affect microdynamics and provides information about the probability distribution of molecular displacements as a function of time. An introduction of the measuring principle is followed by a detailed description of the ranges of measurements and their limitation. Particular emphasis is given to an explanation of possible pitfalls in the measurements and the ways to circumvent them. Showcases presented for illustrating the wealth of information provided by PFG NMR include a survey on the various patterns of concentration dependence of intra-particle diffusion and examples of transport inhibition by additional transport resistances within the nanoporous particles and on their external surface. The latter information is attained by combination with the outcome of tracer exchange experiments, which are shown to become possible via a special formalism of PFG NMR data analysis. Further evidence provided by PFG NMR concerns diffusion enhancement in pore hierarchies, diffusion anisotropy and the impact of diffusion on chemical conversion in porous catalysts. A compilation of the specifics of PFG NMR and of the parallels with other measurement techniques concludes the paper.

Comparing Geometry and Chemistry When Confined Molecules Diffuse in Monodisperse Metal-Organic Framework Pores

The monodisperse pore structure of MOFs (metal-organic frameworks) is advantageous for investigating how porosity influences diffusion. Here we report translational and rotational diffusion using fluorescence correlation spectroscopy and time-correlated single-photon counting, using the three-dimensional pores of the zeolitic-like metal-organic framework family. We compare the influence of size and electric charge as well as dependence on pore size that we controlled through postsynthetic cation-exchange modifications. Charge-charge interactions with the MOF appeared to produce transient adsorption, manifested as a relatively fast and a slower diffusion process, but diffusants without net electric charge displayed a single diffusion process. Obtained from this family of guest molecules selected to be fluorescent, these findings suggest potentially useful general design rules to predict how pore size, guest size, and host-guest interaction control guest mobility within nanopores. With striking fidelity, diffusion coefficient scales with the ratio of cross-sectional areas of diffusant and host pores when charge is taken into account.

Transport properties of hierarchical micro–mesoporous materials

This work provides an overview of different experimental techniques of diffusion measurements in porous materials and discusses transport properties of several classes of hierarchically organized micro-mesoporous materials.

Diffusion and Molecular Exchange in Hollow Core-Shell Silica Nanoparticles

The diffusion behavior of small molecules in hollow core-shell nanocapsules was studied using pulsed field gradient NMR. By purposefully selecting the liquid saturating the hollow core and the porous shell and the solvent between the nanocapsules, two different situations corresponding to the excluded and admitted molecular exchange between the intra- and intercapsule liquids at the external boundary of the nanoparticles were covered. In the former case, corresponding to the reflective boundary condition for the molecules approaching the nanocapsule boundary, restricted diffusion in the complex pore space formed by the hollow core and the mesoporous shell was observed. The time-dependent diffusivities measured in the experiment were inter-related with the geometry of the intraparticle pore space. The thus assessed structural information was found to be in a good agreement with that provided by electron microscopy. In the case of the molecular exchange occurring between the two pools of molecules in the nanocapsules and between them, the diffusive dynamics of only the molecules remaining in the nanocapsules during the entire observation times was studied.

Soft matter in hard confinement: phase transition thermodynamics, structure, texture, diffusion and flow in nanoporous media

Spatial confinement in nanoporous media affects the structure, thermodynamics and mobility of molecular soft matter often markedly. This article reviews thermodynamic equilibrium phenomena, such as physisorption, capillary condensation, crystallisation, self-diffusion, and structural phase transitions as well as selected aspects of the emerging field of spatially confined, non-equilibrium physics, i.e. the rheology of liquids, capillarity-driven flow phenomena, and imbibition front broadening in nanoporous materials. The observations in the nanoscale systems are related to the corresponding bulk phenomenologies. The complexity of the confined molecular species is varied from simple building blocks, like noble gas atoms, normal alkanes and alcohols to liquid crystals, polymers, ionic liquids, proteins and water. Mostly, experiments with mesoporous solids of alumina, gold, carbon, silica, and silicon with pore diameters ranging from a few up to 50 nm are presented. The observed peculiarities of nanopore-confined condensed matter are also discussed with regard to applications. A particular emphasis is put on texture formation upon crystallisation in nanoporous media, a topic both of high fundamental interest and of increasing nanotechnological importance, e.g. for the synthesis of organic/inorganic hybrid materials by melt infiltration, the usage of nanoporous solids in crystal nucleation or in template-assisted electrochemical deposition of nano structures.

Influence of Structural Heterogeneity on Diffusion of CH4 and CO2 in Silicon Carbide-Derived Nanoporous Carbon

We investigate the influence of structural heterogeneity on the transport properties of simple gases in a Hybrid Reverse Monte Carlo (HRMC) constructed model of silicon carbide-derived carbon (SiC-DC). The energy landscape of the system is determined based on free energy analysis of the atomistic model. The overall energy barriers of the system for different gases are computed along with important properties, such as Henry constant and differential enthalpy of adsorption at infinite dilution, and indicate hydrophobicity of the SiC-DC structure and its affinity for CO₂ and CH₄ adsorption. We also study the effect of molecular geometry, pore structure and energy heterogeneity considering different hopping scenarios for diffusion of CO₂ and CH₄ through ultramicropores using the Nudged Elastic Band (NEB) method. It is shown that the energy barrier of a hopping molecule is very sensitive to the shape of the pore entry. We provide evidence for the influence of structural heterogeneity on self-diffusivity of methane and carbon dioxide using molecular dynamics simulation, based on a maximum in the variation of self-diffusivity with loading. A comparison of the MD simulation results with self-diffusivities from quasi-elastic neutron scattering (QENS) measurements and, with macroscopic uptake-based low-density transport coefficients, reveals the existence of internal barriers not captured in MD simulation and QENS experiments. Nevertheless, the simulation and macroscopic uptake-based diffusion coefficients agree within a factor of 2-3, indicating that our HRMC model structure captures most of the important energy barriers affecting the transport of CH₄ in the nanostructure of SiC-DC.

The Mechanism of Pseudomorphic Transformation of Spherical Silica Gel into MCM-41 Studied by PFG NMR Diffusometry

The pseudomorphic transformation of spherical silica gel (LiChrospher^® Si 60) into MCM-41 was achieved by treatment at 383 K for 24 h with an aqueous solution of cetyltrimethylammonium hydroxide (CTAOH) instead of hexadecyltrimethylammonium bromide (CTABr) and NaOH. The degree of transformation was varied via the ratio of CTAOH solution to initial silica gel rather than synthesis duration. The transformed samples were characterized by N₂ sorption at 77 K, mercury intrusion porosimetry, X-ray diffraction (XRD) and scanning electron microscopy (SEM). Thus, MCM-41 spheres with diameters of ca. 12 μm, surface areas >1000 m² g⁻¹, pore volumes >1 cm³ g⁻¹ and a sharp pore width distribution, adjustable between 3.2 and 4.5 nm, were obtained. A thorough pulsed field gradient nuclear magnetic resonance (PFG NMR) study shows that the diffusivity of n-heptane confined in the pores of the solids passes through a minimum with progressing transformation. The final product of pseudomorphic transformation to MCM-41 does not exhibit improved transport properties compared to the initial silica gel. Moreover, the PFG NMR results support that the transformation occurs via formation and subsequent growth of domains of <1 μm containing MCM-41 homogeneously distributed over the volume of the silica spheres.

Diffusion Study by IR Micro-Imaging of Molecular Uptake and Release on Mesoporous Zeolites of Structure Type CHA and LTA

The presence of mesopores in the interior of microporous particles may significantly improve their transport properties. Complementing previous macroscopic transient sorption experiments and pulsed field gradient NMR self-diffusion studies with such materials, the present study is dedicated to an in-depth study of molecular uptake and release on the individual particles of mesoporous zeolitic specimens, notably with samples of the narrow-pore structure types, CHA and LTA. The investigations are focused on determining the time constants and functional dependences of uptake and release. They include a systematic variation of the architecture of the mesopores and of the guest molecules under study as well as a comparison of transient uptake with blocked and un-blocked mesopores. In addition to accelerating intracrystalline mass transfer, transport enhancement by mesopores is found to be, possibly, also caused by a reduction of transport resistances on the particle surfaces.

Capillary condensation in one-dimensional irregular confinement

A lattice-gas model with heterogeneity is developed for the description of fluid condensation in finite sized one-dimensional pores of arbitrary shape. Mapping to the random-field Ising model allows an exact solution of the model to be obtained at zero-temperature, reproducing the experimentally observed dependence of the amount of fluid adsorbed in the pore on external pressure. It is demonstrated that the disorder controls the sorption for long pores and can result in H2-type hysteresis. Finite-temperature Metropolis dynamics simulations support analytical findings in the limit of low temperatures. The proposed framework is viewed as a fundamental building block of the theory of capillary condensation necessary for reliable structural analysis of complex porous media from adsorption-desorption data.

Diffusion in hierarchical mesoporous materials: applicability and generalization of the fast-exchange diffusion model

Transport properties of cyclohexane confined to a silica material with an ordered, bimodal pore structure have been studied by means of pulsed field gradient nuclear magnetic resonance. A particular organization of the well-defined pore structure, composed of a collection of spatially ordered, spherical mesopores interconnected via narrow worm-like pores, allowed for a quantitative analysis of the diffusion process in a medium with spatially ordered distribution of the fluid density for a broad range of the gas-liquid equilibria. The measured diffusion data were interpreted in terms of effective diffusivities, which were determined within a microscopic model considering long-range molecular trajectories constructed by assembling the alternating pieces of displacement in the two constituting pore spaces. It has further been found that for the system under study, in particular, and for mesoporous materials with multiple porosities, in general, this generalized model simplifies to the conventional fast-exchange model used in the literature. Thus, not only was justification of the applicability of the fast-exchange model to a diversity of mesoporous materials provided, but the diffusion parameters entering the fast-exchange model were also exactly defined. The equation resulting in this way was found to nicely reproduce the experimentally determined diffusivities, establishing a methodology for targeted fine-tuning of transport properties of fluids in hierarchical materials with multiple porosities.

Role of stringlike, supramolecular assemblies in reentrant supernematic liquid crystals

Using a combination of isothermal-isobaric Monte Carlo and microcanonical molecular dynamics we investigate the relation between structure and self-diffusion in various phases of a model liquid crystal using the Gay-Berne-Kihara potential. These molecules are confined to a mesoscopic slit pore with atomically smooth substrate surfaces. As reported recently [seeM. G. Mazza et al., Phys. Rev. Lett. 105, 227802 (2010)], a reentrant nematic (RN) phase may form at sufficiently high pressures and densities. This phase is characterized by a high degree of nematic order and a substantially enhanced self-diffusivity in the direction of the director n that exceeds that of the lower-density nematic and an intermittent smectic-A phase by about an order of magnitude. Here we demonstrate that the unique transport behavior in the RN phase may be linked to a confinement-induced packing effect that causes the formation of supramolecular, stringlike conformations. The strings consist of several molecules traveling in the direction of n as individual "trains" consisting of chains of molecular "cars."

Solid state NMR of porous materials : zeolites and related materials

Solid state NMR spectroscopy applied to the science of crystalline micro- and mesoporous silica materials over the past 10 years is reviewed. A survey is provided of framework structure and connectivity analyses from chemical shift effects of various elements in zeolites including heteroatom substitutions, framework defects and pentacoordinated silicon for zeolites containing fluoride ions. New developments in the field of NMR crystallography are included. Spatial host-guest ordering and confinement effects of zeolite-sorbate complexes are outlined, with special emphasis on NMR applications utilizing the heteronuclear dipolar interaction. The characterization of zeolite acid sites and in situ NMR on catalytic conversions is also included. Finally, the motion of extra-framework cations is investigated in two tutorial cases of sodium hopping in sodalite and cancrinite.

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.

In situ study on molecular diffusion phenomena in nanoporous catalytic solids

As an omnipresent phenomenon in nature, diffusion is among the rate-determining processes in many technological processes. This is in particular true for catalytic conversion in nanoporous materials. We provide a critical review of the possibilities of exploring diffusion phenomena over microscopic dimensions in such media by direct experimental observation. By monitoring the probability distribution of molecular displacements as a function of time, the pulsed field gradient technique of NMR (PFG NMR) records the rate of molecular re-distribution. By varying the observation time, PFG NMR is thus able to trace even hierarchies of transport resistances as occurring, e.g., in catalyst particles in the form of binder-compacted assemblages of zeolite crystallites. Alternatively, and complementary to this information, interference microscopy (IFM) and IR microscopy (IRM) are able to follow the evolution of intracrystalline concentration profiles during uptake and release. This allows, in particular, an accurate quantification of the transport resistances on the surface of the individual crystallites and of the probability that reactant molecules from the gas phase, upon colliding with the external surface, are able to penetrate through such "surface barriers" into the crystal bulk phase. Being able to distinguish between different molecular species, IRM is able to record the evolution of intracrystalline concentration profiles even during multi-component adsorption and catalytic reactions (169 references).