A new method to determine pore size and its volume distribution of porous solids having known atomistic configuration

A simple method, based on Monte Carlo integration, is presented to derive pore size and its volume distribution for porous solids having known configuration of solid atoms. Because pores do not have any particular shape, it is important that we define the pore size in an unambiguous manner and the volume associated with each pore size. The void volume that we adopt is the one that is accessible to the center of mass of the probe particle. We test this new method with porous solids having well defined pores such as graphitic slit pores and carbon nanotubes, and then apply it to obtain the pore volume distribution of complex solids such as disordered solids, rectangular pores, defected graphitic pores, metal organic framework and zeolite.     

A novel and consistent method (TriPOD) to characterize an arbitrary porous solid for its accessible volume,accessible geometrical surface area and accessible pore size

We present an improved Monte Carlo integration method to calculate the accessible pore size distribution of a porous solid having known configuration of solid atoms. The pore size distribution obtained with the present method is consistent with the accessible volume and the accessible geometric surface area presented in previous publications (Do and Do, in J. Colloid Interface Sci. 316(2):317–330, 2007; Do et al. in Adsorpt. J., 2010). The accessible volume, accessible geometrical surface area and the pore size distribution method construct an unambiguous and robust single framework to characterize porous solids. This framework is based on the derivation of the space accessible to the center of mass of a probe molecule. The accessible pore size presented is an absolute quantity in the sense that a zero value is possible. We present the entire framework of this characterization method and compare the improved method with the one presented previously for a set of porous solids such as graphitic slit pores, defective slit pores, bundle of carbon nanotubes, zeolite and some metal organic frameworks.     

The role of accessibility in the characterization of porous solids and their adsorption properties

This paper addresses the role of accessibility for adsorption in porous solids on the adsorption properties including Henry constant, adsorption isotherms and isosteric heat of adsorption. The relevant parameters are the accessible volume, the accessible geometrical surface area and the accessible pore size and its associated volume. This concept will be demonstrated to be important and calls for the need to consider adsorption characteristics in the most coherent and consistent manner. It is particularly reinforced by the limitations inherent in the conventional ways in determining the void volume, surface area and pore size. We provide a number of examples to support this; the challenge that faces us is the development of consistent experimental procedures to determine these accessible quantities. We define the accessible pore size as the size of the largest sphere that rests on three closest solid atoms in such a manner that any probe particle residing in that sphere would have a non-positive solid-fluid potential energy. For each accessible pore size there is an associated accessible pore volume, giving rise to a new accessible pore size distribution (APSD). This is distinct from the classical pore size distribution commonly used in the literature, and in our definition of accessible pore size, a zero pore size is possible. It is also emphasized that the accessible quantities that we introduce here are dependent on the choice of molecular probe, which is entirely consistent with the concept of molecular sieving.     

Effect of cellulase on the pore structure of bead cellulose

An enzymatic treatment with cellulases fromTrichoderma viride was investigated in its effect on the pore structure of different types of bead cellulose. One objective of this study was to establish a suitable procedure for combined enzymatic treatment and solvent exchange that would restore the original pore structure which the beads had before drying without causing major losses in mechanical stability. Another aim was to further increase the accessible pore space and internal surface area for separation of large molecular weight compounds with regard to Chromatographic applications. Finally, an attempt was made to extend the findings for unsubstituted beads to the derivatives carboxymethyl (CM) and diethylaminoethyl (DEAE) cellulose beads. The enzymatically treated samples were characterized by microscopic methods and porosity measurements such as mercury porosimetry, nitrogen sorption and size exclusion chromatography. It was found that under controlled conditions the low-porosity surface layer of dried beads could be removed making the internal pore space accessible without reducing the resistance to deformation of the beads. Additionally, a shift in pore size distribution towards larger pores was observed. Supplementary swelling treatments in solvents of high swelling power could substantially restore the former porosity of the dried beads but did not enhance the accessibility to the cellulases to a considerable extent. Internal pore volume and surface area of the derivatives were dramatically increased in the case of DEAE upon enzymatic hydrolysis, however, at the expense of mechanical stability, whereas CM was found to be less affected.     

Interpretation of water isotherm hysteresis for an activated charcoal using stochastic pore networks

Water vapor adsorption equilibria on activated carbons typically exhibit hysteresis. The size and shape of the hysteresis loop which separates the adsorption and desorption branches is a strong function of the pore size and interconnectivity of the pores. Neither conventional pore filling models nor statistical thermodynamics approaches provide a means for predicting the extent of hysteresis from only adsorption measurements. This work uses the Kelvin Equation in conjunction with the structural concept of a stochastic pore network to describe measured water isotherms on BPL carbon. Using a pore segment distribution function determined from the adsorption branch, it is shown that totally random assemblies underestimate the extent of hysteresis. It is possible, however, to closely fit the measured BPL-water hysteresis loop using a patchy heterogeneity in which a proportion of the larger pores are preferentially located on the exterior, mid-range pores are concentrated in a sub-surface layer and some large pores form shielded voids behind much smaller pores.Nomenclature p vapor phase partial pressure of sorbate - p _sat saturation vapor pressure of sorbate - R gas constant - r pore radius - T absolute temperature - t adsorbed layer thickness - V _L molar volume of adsorbed phase - surface tension - contact angle     

Appropriate volumes for adsorption isotherm studies: the absolute void volume, accessible pore volume and enclosing particle volume

In adsorption studies the choice of an appropriate void volume in the calculation of the adsorption isotherm is very crucial. It is often taken to be the apparent volume as determined by the helium expansion experiments. Unfortunately this method has difficulties especially when dealing with microporous solids, in which adsorption of helium might become significant at ambient temperatures. The amount adsorbed is traditionally obtained as the excess amount and the term "excess" refers to the excess over the amount occupying the apparent volume that has the same density as the bulk gas density. This could give rise to the maximum in the plot of excess amount versus pressure under supercritical conditions, and in some cases giving negative excess. Such behavior is difficult to analyze because the excess amount is not amenable to any classical thermodynamic treatments. In this paper we will present a method to determine the absolute void volume, and in that sense this volume is independent of temperature and adsorbate. The volume that is accessible to the centers of gas molecules is also investigated, and it is called the accessible volume. This volume depends on the choice of adsorbate, and it is appropriate to use this volume to calculate the pore density because we can assess how dense the adsorbed phase is. In the quest to determine the "absolute" adsorption isotherm so that a thermodynamics analysis can be applied, it is necessary to introduce the concept of "enclosing" volume, which is essentially the volume that encloses all solid particles, including all void spaces in them. The amount adsorbed is defined by the number of molecules residing in this volume. Having these volumes, we can derive the geometrical accessible void volume inside the particle and the solid volume, from which the particle and solid densities can be calculated.     

Comparison between different presentations of pore size distribution in porous materials

The different presentations of the pore size distribution derived from the gas adsorption method and the mercury porosimetry are connected with some problems. This concerns especially the use of the logarithmically differential pore volume distribution. The incorrect application of this distribution to bimodal pore systems involves the danger of an apparent overemphasizing of larger pores. This effect may also occur using the incremental pore size distribution in case the experimental point spacing considerably increases towards the larger pore radii. The pore volume density distribution defined as the linear derivative of the cumulative pore volume curve with respect to the pore radius has been found the most convenient form among the various kinds of pore volume distribution presentations. It has been shown that the direct comparison between this distribution and the logarithmically differential pore volume distribution is not allowed. Nevertheless, there is a clear connection between these definitions for the pore size distribution so that they are completely equivalent.     

Modeling liquid porosimetry in modeled and imaged 3-D fibrous microstructures

In this paper, an analysis to distinguish the geometric and porosimetric pore size distributions of a fibrous material is presented. The work is based on simulating the intrusion of nonwetting fluid in a series of 3-D fibrous microstructures obtained from 3-D image reconstruction or virtual geometries mathematically generated according to the properties of the media. We start our study by computing the pore size distribution of two typical hydroentangled nonwoven materials and present a theoretical model for their geometric pore size distributions based on Poisson line network model of the fibrous media. It is shown that the probability density function of the geometric pore size distribution can be approximated by a two-parametric Gamma distribution. We also study connectivity of the pore space in fibrous media by computing and comparing the accessible and allowed pore volumes in the form access function graphs. It is shown that the so-called ink-bottle effect can significantly influence the fluid intrusion in a porous material. The pore space connectivity of a homogeneous fibrous media is observed to be a function of thickness, solid volume fraction (SVF), and fiber diameter. It is shown that increasing the materials' thickness or SVF, while other properties are kept constant, reduces the pore space connectivity. On the other hand, increasing the fiber diameter enhances the connectivity of the pores if all other parameters are fixed. Moreover, modeling layered fibrous microstructures; it is shown that the access function graphs can be used to detect the location of the bottle neck pores in a layered/composite porous material.     

Comparison between different presentations of pore size distribution in porous materials

The different presentations of the pore size distribution derived from the gas adsorption method and the mercury porosimetry are connected with some problems. This concerns especially the use of the logarithmically differential pore volume distribution. The incorrect application of this distribution to bimodal pore systems involves the danger of an apparent overemphasizing of larger pores. This effect may also occur using the incremental pore size distribution in case the experimental point spacing considerably increases towards the larger pore radii. The pore volume density distribution defined as the linear derivative of the cumulative pore volume curve with respect to the pore radius has been found the most convenient form among the various kinds of pore volume distribution presentations. It has been shown that the direct comparison between this distribution and the logarithmically differential pore volume distribution is not allowed. Nevertheless, there is a clear connection between these definitions for the pore size distribution so that they are completely equivalent. Received: 15 May 1998 / Revised: 8 October 1998 / Accepted: 10 October 1998     

Packing effects on argon and methanol adsorption inside graphitic cylindrical and slit pores: a GCMC simulation study

Using Grand Canonical Monte Carlo simulation, we have studied the effects of confinement on argon and methanol adsorption in graphitic cylindrical and slit pores. Linear chain, zigzag and incomplete helical packing are observed for argon adsorption in cylindrical pores. However, for methanol adsorption different features appear because the electrostatic interactions favour configurations that maximize the hydrogen bonding among methanol molecules. We have found zigzag chains with hydrogen-bonded structures for methanol adsorption in cylindrical and slit pores. To investigate how dense the adsorbed phase is and how many molecules could be packed per unit physical volume of the solid, we consider two different definitions of pore density; one based on the physical volume and the other on the accessible volume. That based on accessible volume gives a measure of the fluid density, while that based on the physical volume gives a measure of how much adsorbate can be stored per unit volume of the adsorbent. It is found that the adsorbate is denser in cylindrical pores, but that slit pores can pack more molecules per unit solid volume. We also discuss the effects on the isosteric heat of argon and methanol of pore size, pore geometry and loading.     

Effect of hypercrosslinking conditions on pore size distribution and efficiency of monolithic stationary phases

Three dihalogenic solvents differing in the length of alkyl chain (1,2‐dichloroethane, 1,4‐dichlorobutane, and 1,6‐dichlorohexane) with three Friedel–Crafts alkylation catalysts varying in reactivity (AlCl₃, FeCl₃, and SnCl₄) have been used to prepare hypercrosslinked poly(styrene‐co‐vinylbenzyl chloride‐co‐divinylbenzene) columns. Hydrodynamic characteristics as well as column efficiency and mass transfer resistance were tuned by the combination of swelling solvent and alkylation reaction catalyst in the modification mixture. The column swelled in 1,6‐dichlorohexane and hypercrosslinked in the presence of AlCl₃ provided the highest column efficiency and enabled fast isocratic separations of small molecules in a RP mode. To uncover factors controlling the efficiency of hypercrosslinked monolithic columns, we have studied pore volume distribution of prepared columns. We found that column efficiency increases with the higher pore volume of pores smaller than 2 nm.     

On the cavitation and pore blocking in slit-shaped ink-bottle pores

We present GCMC simulations of argon adsorption in slit pores of different channel geometry. We show that the isotherm for an ink-bottle pore can be reconstructed as a linear combination of the local isotherms of appropriately chosen independent unit cells. Second, depending on the system parameters and operating conditions, the phenomena of cavitation and pore blocking can occur for a given configuration of the ink-bottle pore by varying the geometrical aspect ratio. Although it has been argued in the literature that the geometrical aspects of the system govern the evaporation mechanism (either cavitation or pore blocking), we here put forward an argument that the local compressibility in different parts of the ink-bottle pore is the deciding factor for evaporation. When the fluid in the small neck is strongly bound, cavitation is the governing process, and molecules in the cavity evaporate to the surrounding bulk gas via a mass transfer mechanism through the pore neck. When the pore neck is sufficiently large, the system of neck and cavity evaporates at the same pressure, which is a consequence of the comparable compressibility between the fluid in the neck and that in the cavity. This suggests that local compressibility is the measure of cohesiveness of the fluid prior to evaporation. One consequence that we derive from the analysis of isotherms of a number of connected pores is that by analyzing the adsorption branch or the desorption branch of an experimental isotherm may not lead to the correct pore sizes and the correct pore volume distribution.     

Control of pore size and pore uniformity in films based on self‐assembling block copolymers

Blends of self‐assembling polystyrene‐block‐poly(4‐vinyl pyridine) (PS‐b‐P4VP) diblock‐copolymers and poly(4‐vinyl pyridine) (P4VP) homopolymers were used to fabricate isoporous and nanoporous films. Block copolymers (BCP) self‐assembled into a structure where the minority component forms very uniform cylinders, while homopolymers, resided in the core of the cylinders. Selective removal of the homopolymers by ethanol immersion led to the formation of well‐ordered pores. In films without added homopolymer, just immersion in ethanol and subsequent swelling of the P4VP blocks was found to be sufficient to create pores. Pore sizes were tuned between 10 and 50 nm by simply varying the homopolymer content and the molecular weight of the block‐copolymer. Uniformity was lost when the average pore size exceeded 30 nm because of macrophase separation. However, preparation of films from low M_W diblock copolymers showed that it is possible to have excellent pore size control and a high porosity, while retaining a low pore size distribution. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 1568–1579     

Grand canonical Monte Carlo simulation of argon adsorption at the surface of silica nanopores: effect of pore size, pore morphology, and surface roughness

Argon adsorption (77 K) in atomistic silica nanopores of various sizes and shapes has been studied by means of grand canonical Monte Carlo simulations (GCMC). We discuss the effects of confinement (pore size), pore morphology (ellipsoidal, hexagonal, constricted pore), and surface texture (rough/smooth) on the thickness variation of the adsorbed film with pressure onto the disordered inner surface of porous materials (usually called t-plot or t-curve). We show that no confinement effect occurs when the diameter of the regular cylindrical pore is larger than 10 nm. For pores smaller than 6 nm, we find that the film thickness increases as the pore size decreases. We show that the adsorption isotherm in the rough pore can be described as the sum of an adsorbed amount similar to that found for a smooth pore (of the same radius) and a constant contribution due to atoms "trapped" in the infractuosities of the rough surface which act as a microporous texture. Simulation snapshots for Ar adsorption in hexagonal and ellipsoidal smooth pores indicate that at low pressures the gas/adsorbate interface retains memory of the pore shape and becomes cylindrical prior to the capillary condensation of the fluid in the pore. The film thickness in the hexagonal pore is close to that obtained for a cylindrical pore having a similar dimension. By contrast, we find that the film thickness for an ellipsoidal pore is always larger than that for an equivalent cylindrical pore (having the same length and volume but a circular section). We show that this effect strengthens as the pore size decreases and/or the pore asymmetry increases. Ar adsorption in a cylindrical constricted pore shows that the presence of the narrower part considerably modifies the adsorption mechanism. Finally, we report GCMC simulations of Ar adsorption (77 K) on a plane silica reference substrate for different intermolecular potentials. We discuss the effect of the interaction on the shape of the adsorption isotherm and compare our results with experiments.     

Light transmission study for determination of pore size distribution in microporous membranes

A new method combining light transmission and bubble pressure has been tried to determine the pore size distribution of microporous membranes. The method is based on a simple phenomenon that the light transmitivity through an opaque porous membrane increases as its pores are filled with a transparent liquid. The pore size distributions of Durapore HVHP and Sartorius PTFE membranes were tested using the new method and the results were compared with those obtained by the fluid displacement and the mercury intrusion methods.     

Localization of the copper-containing component in the pore volume of zeolite ZSM-5

The distribution of the copper-containing component in the pore volume of zeolite ZSM-5 has been investigated by H₂ and N₂ adsorption at 77 K and IR spectroscopy. Samples were synthesized by ion exchange and incipient wetness impregnation. Copper-containing clusters are mostly located on the surface of the mesopores formed by packed zeolite nanocrystallites. This causes partial blocking of the volume of microporous channels for N₂ molecules, but these channels remain accessible for H₂ molecules. It has been deduced that no considerable amount of copper located in the structural channels of the zeolite. According to IR spectroscopic data, the sorption of copper ions in the Cu/ZSM-5 catalysts takes place on extraframe-work aluminum, which forms Al-OH-Al bridges and terminal Al-OH groups, and on terminal Si-OH groups located on the zeolite crystal surface.     

Why the pore geometry model could affect the uniqueness of the PSD in AC characterization

In this paper we discuss why the pore geometry can affect the unicity of the pore size distribution (PSD) of a given activated carbon (AC) sample, when different probe gases are used in adsorption measures. In order to characterize the solid sample we used grand canonical Monte Carlo simulation and the independent pore model with slit or triangular pore geometry, focusing our analysis on the possibility of representing the adsorptive processes of a triangular pore of defined size by means of a combination of slit pores of different sizes. This representation is tested on experimental adsorption data of N₂ (77 K) on AC samples and acceptable results were obtained. Finally, we have performed a theoretical test, which consisted of analyzing a virtual porous solid with this approach and different probe gases (N₂ at 77 K and CO₂ at 273 K), showing that the differences between the pore representations can cause differences between the solid representations for the adsorptive properties, for these different gases. The analysis presented here can be extended to other pore geometries and other adsorbates, and provide arguments to further explain results presented in our previous paper, which refers to cases when different adsorbates yield different PSDs for a given sample and the same pore geometry model.     

Nondestructive technique for the characterization of the pore size distribution of soft porous constructs for tissue engineering

Polymer scaffolds tailored for tissue engineering applications possessing the desired pore structure require reproducible fabrication techniques. Nondestructive, quantitative methods for pore characterization are required to determine the pore size and its distribution. In this study, a promising alternative to traditional pore size characterization techniques is presented. We introduce a quantitative, nondestructive and inexpensive method to determine the pore size distribution of large soft porous solids based on the on the displacement of a liquid, that spreads without limits though a porous medium, by nitrogen. The capillary pressure is measured and related to the pore sizes as well as the pore size distribution of the narrowest bottlenecks of the largest interconnected pores in a porous medium. The measured pore diameters correspond to the narrowest bottleneck of the largest pores connecting the bottom with the top surface of a given porous solid. The applicability and reproducibility of the breakthrough technique is demonstrated on two polyurethane foams, manufactured using the thermally induced phase separation (TIPS) process, with almost identical overall porosity (60-70%) but very different pore morphology. By selecting different quenching temperatures to induce polymer phase separation, the pore structure could be regulated while maintaining the overall porosity. Depending on the quenching temperature, the foams exhibited either longitudinally oriented tubular macropores interconnected with micropores or independent macropores connected to adjacent pores via openings in the pore walls. The pore size and its distribution obtained by the breakthrough test were in excellent agreement to conventional characterization techniques, such as scanning electron microscopy combined with image analysis, BET technique, and mercury intrusion porosimetry. This technique is suitable for the characterization of the micro- and macropore structure of soft porous solids intended for tissue engineering applications. The method is sensitive for the smallest bottlenecks of the largest continuous pores throughout the scaffold that contributes to fluid flow.     

Large pore volume mesoporous aluminum oxide synthesized via nano-assembly

A new nano-assembly approach has been proposed for the preparation of macropore volume mesoporous aluminum oxide supports. Secondary nano-assembly and a frame structure mechanism for large pore volume mesoporous supports have been proposed. In a primary nano-assembly supersoluble micelle, aluminum hydroxide nanoparticles were precipitated in situ in surfactants with a volume balance (VB) less than 1, followed by secondary nano-assembly in linear and cylindrical shapes. The secondary nano-assembly of cylindrical aluminum hydroxides was calcined to form nano cylindrical aluminum oxides. For the formation of macropore volume mesoporous supports, we utilized a frame structure mechanism of mesoporous support, in which the exterior surface of the carrier may not be continuous. This macropore volume support has been used for the hydrotreatment of a residual oil catalyst, which possesses the following physical characteristics: pore volume 1.8–2.7 mL·g⁻¹, specific surface area 180–429 m²·g⁻¹, average pore diameter 17–57 nm, average pore diameter more than 10 nm (81%–94%), porosity 87%–93%, and crush strength 7.7–25 N·mm⁻¹.