Modelling of the pore size distribution of ultrafiltration membranes

A dilute aqueous solution of polydisperse neutral dextrans was used to determine the sieving properties (flux and rejection) of porous polyacrylonitrile membranes. Gel ermeation chromatography was used to measure the solute mole and concentration in the permeate. From these data, rejection coefficients were calculated as a function of solute molecular size. A mathematical model was then developed to relate the flux and solute rejection to pore size distribution and the total number of pores, based upon the assumption that solute rejection was the result of purely geometric considerations. As a first approximation, a solute molecule was considered either too large to enter a membrane pore, or if it entered, its concentration in the permeate from that pore, as well as the solvent flux through the pore, were not affected. This model also considered the effects of steric hindrance and hydrodynamic lag on the convection of solute through a membrane. The shape and sharpness of pore size distributions were found to be useful in comparisons of ultrafiltration membranes.     

Effect of the pore geometry in the characterization of the pore size distribution of activated carbons

In this work, the characterization of Activated Carbons (AC) by using the independent pore models is discussed, with special emphasis on the issue of how the assumed pore geometry can affect the resulting Pore Size Distribution (rPSD) and on the problem of the unicity of the PSD when different probe molecules are used in adsorption experiments. A theoretical test was performed using virtual solids based in the so-called Mixed Geometry Model (MGM) (Azevedo et al. 2010). The MGM uses a kernel of adsorption isotherms generated by GCMC for different pore sizes and two pore geometries: slit and triangular. The adsorption isotherms of a virtual MGM solid were fitted with both the traditional Slit Geometry Model (SGM) and the Mixed Geometry Model (MGM). It is demonstrated that, by assuming a different pore geometry model from that of the real sample, different PSDs may be obtained by fitting adsorption isotherms of different probe gases. Finally, experimental results are shown which both point toward the MGM as an acceptable extension of the SGM and confirm that the MGM is a closer representation of the actual porous structure of most activated carbons.     

Determination of pore size distribution in hollow fibre membranes

Isopropanol displacement under nitrogen pressure was used for the determination of pore size distribution in microfiltration polypropylene hollow fibres. Applying various assumptions about gas transport process two completely different characteristics of pore sizes were obtained. To verify these results an analysis of SEM images of the investigated membrane was conducted concerning its porous structure (pore diameters, surface occupied by pores). According to the SEM analysis the mean coverage of membrane surface by pore entrances should be about 20% of total area. For the distribution which accounted for pore evacuation according to Young–Laplace equation with contact angle θ=67° surprisingly dense coverage amounting to over 70% of total surface (by calculated total pore number over 10¹³ per m²) was predicted. Results for the distribution which accounted for gas bubble formation at the membrane surface (equivalent to θ=0°) fit into the expected range of pore numbers and membrane coverages (about 10¹¹ per m² and about 10%, respectively). It is concluded that the mechanism of bubble formation, determined by an actual pressure, liquid surface tension and pore size, is the crucial process while the value of contact angle θ does not play any role in the determination of pore size distribution.     

Nanoporous metals with controlled multimodal pore size distribution

A simple two-step dealloying strategy is described to make free-standing metal membranes with hierarchical porous architecture. This structure has a bimodal pore size distribution composed of large porosity channels and small porosity channel walls, where each pore size can be tailored independently of the others. A new gas-phase electroless plating technique was also developed here that could be used to uniformly fill porous structures with pore size as small as 10 nm.     

Inorganic salt hydrates as cryoporometric probe materials to obtain pore size distribution

The depression of the melting temperature of Zn(NO3)2.6H2O was used to obtain the pore size distributions in controlled pore glasses. Measured by 1H NMR, the average value of the temperature depression DeltaT and the known average pore size yield K=DeltaT.d approximately 116 K.nm as the material-dependent factor for Zn(NO3)2.6H2O in the Gibbs-Thompson equation. The melting temperature is close to room temperature. Hence, this salt hydrate and some related other ones are better materials than water (K approximately 50 K.nm) for cryoporometric studies of systems with hydrophilic pores. The data also provide 46 mN/m for the solid-liquid surface tension of this salt hydrate.     

Tailoring pore size and pore size distribution of kidney dialysis hollow fiber membranes via dual-bath coagulation approach

Polyethersulfone (PES) hollow fiber membranes for kidney dialysis application were prepared by the dry-jet wet-spinning method. A dual-coagulation bath technology was first time employed for fabricating the kidney dialysis membranes with a tight inner skin and loose outer supporting layer structure. A weak coagulant isopropanol (IPA) was served as the first external coagulation bath, while water as the second bath. Experiments demonstrate their advantages of better controlling both inner and outer skin morphology. The as-spun fibers have a higher mean effective pore size (μ_p), pure water permeation flux (PWP) and molecular weight cut-off (MWCO) with an increase in N-methyl-2-pyrrolidone (NMP) percentage in bore fluid (i.e., internal coagulant). After being treated in 8000 ppm NaOCl solution for 1 day, fibers show larger pore sizes and porosity in both inner and outer surfaces, and thinner inner and outer layers than their as-spun counterparts. Among them, the bleached fibers spun with 50 wt.% NMP in bore fluid have the MWCO (43 kDa) and PWP (40 × 10⁻⁵ L m⁻² Pa⁻¹ h⁻¹) suitable for kidney dialysis application. Based on SEM observations and solute rejection performance, the further heat treated fibers in an aqueous solution is found to be an effective way to fine tune membranes morphology and MWCO for kidney dialysis application. The solute rejection performance data of the hollow fiber membranes spun with 55 wt.% NMP in bore fluid after heat treated at 90 °C in water for 2 h were found to be very appropriate for the kidney dialysis application.     

Synthesis of FDU-1 silica with narrow pore size distribution and tailorable pore entrance size in the presence of sodium chloride

Ordered silicas with large (9-15 nm), uniform, cagelike mesopores were synthesized under acidic aqueous conditions from tetraethyl orthosilicate in the presence of sodium chloride using poly(ethylene oxide)-poly(butylene oxide)-poly(ethylene oxide) triblock copolymer B50-6600 (EO39BO47EO39, Dow Chemicals) as a supramolecular template. Except for the use of NaCl in our case, the synthesis mixture composition was the same as that originally reported by Zhao et al. for the synthesis of FDU-1 silica, which was later shown to exhibit a cubic close-packed (Fm3m) structure with stacking faults related to the occurrence of hexagonal close-packed stacking sequences. The copolymer-templated silicas were formed at room temperature and in most cases were subjected to the hydrothermal treatment at 373 or 393 K. The calcined materials were characterized using small-angle X-ray scattering (SAXS) and nitrogen and argon adsorption at 77 K. SAXS patterns were generally similar to those reported for FDU-1 silica, indicating the cubic close-packed (Fm3m) structure, but the presence of stacking faults characteristic of a hexagonal close-packed structure cannot be precluded. The addition of the salt was found to significantly narrow the pore size distributions and to improve the uniformity of entrances to the cagelike mesopores, whereas the pore diameter, specific surface area, and pore volume were similar (in most cases slightly lower) to those for FDU-1 silicas obtained in the absence of NaCl. The materials synthesized in the presence of NaCl also appeared to have better resolved SAXS patterns. The feasibility of tailoring the pore cage diameter (from approximately 9.5 to 14.5 nm) and pore entrance diameter (from below 4 to approximately 8 nm) simply by adjusting the hydrothermal treatment temperature and time was demonstrated, indicating that these simple and convenient ways of structural design of cagelike mesopores are operative in the case of syntheses in the presence of inorganic salts.     

On the methods of pore size distribution and the multilayer thickness

Summary Pore-volume-area distribution was achieved for a wide variety of silica gels, employing six computational procedures, and using botht-estimates ofCranston andInkley, andButt. The mathematical induction procedure outlined byRoberts proved to be as good as the more exact procedures. S _eum values evaluated by employing theButt estimate of multilayer thickness were exclusively higher than SBET data. The tabulatedt-values ofCranston andInkley yielded relatively reasonableS _eum results than those obtained above but are also higher thanS _BET. An overlooked parameter was suggested to be partly the cause of the divergence between the varioust-evaluations, namely: the thickness of a monolayer. Recalculation of all data, and using a value of 4.132 Å for the adsorbed monolayer, in botht-estimates ofLippens andButt, resulted in an excellent agreement betweenS _eum andS _BET in most of the cases. It is suggested that the multilayer thickness, as determined from adsorption data on non-porous solids, might not necessarily be of value in pore-size distribution analysis.     

Determination of the pore connectivity and pore size distribution and pore spatial distribution of porous chromatographic particles from nitrogen sorption measurements and pore network modelling theory

The pore connectivity, pore size distribution and pore spatial distribution of the porous structure of native and silanized silica particles were determined by matching the experimental nitrogen sorption data with the theoretical results obtained from pore network model simulations. The agreement between theory and experiment is found to be good. The results clearly indicate that the deposition of the silane layer to the pore surfaces of the native silica particles produces a silanized silica particle with a mean pore diameter and pore connectivity smaller than that of the native silica particle. Furthermore, the evaluation of the pore diffusivity of ribonuclease under unretained conditions shows that the lower values of the pore connectivity found in the samples of silanized silica particles, when compared with the values of the pore connectivity obtained for the native silica particles, increase the diffusional mass transfer resistance within the porous structure of the silanized silica particles.     

Error introduced into determination of pore size distribution by size exclusion chromatography

Simultaneous use of large standard molecules and small particles of the product examined gives rise to errors in pore size determination by size exclusion chromatography. This error is calculated for packings of spherical particles, thus making corrections possible.     

Analysis of pore size distribution by 2H NMR

The pore size distributions of four controlled pore glasses with mean diameters ranging from ca. 7.9 to 23.9 nm were analysed by measuring the (2)H NMR signals from the liquid fraction of confined benzene-d(6) and cyclohexane-d(12) as a function of temperature, in steps of ca. 0.1-1 K. The liquid and solid components of the adsorbates were distinguished, on the basis of the spin-spin relaxation time T(2), by employing a spin-echo sequence. The experimental intensity curves of the non-frozen liquids are well represented by a sum of two error functions. The observed melting point depressions are well represented by the simplified Gibbs-Thompson equation DeltaT = k(p)/R where R is the pore radius and k(p) is a characteristic property of the adsorbate. The k(p) value mainly affects the position of the pore size distribution curve, i.e., the mean pore radius, while the slope of the intensity curve determines the width of the distribution curve. In practice, the NMR method can only be used to determine pore sizes with reasonable accuracy in the mesoporous range unless liquids undergoing larger melting point depressions than the ones investigated so far can be found.     

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     

Modification of crystallinity and pore size distribution in coagulated cellulose films

In this study the effects of altering the coagulation medium during regeneration of cellulose dissolved in the ionic liquid 1-ethyl-3-methylimidazolium acetate, were investigated using solid-state NMR spectroscopy and NMR cryoporometry. In addition, the influence of drying procedure on the structure of regenerated cellulose was studied. Complete conversion of the starting material into regenerated cellulose was seen regardless of the choice of coagulation medium. Coagulation in water predominantly formed cellulose II, whereas coagulation in alcohols mainly generated non-crystalline structures. Subsequent drying of the regenerated cellulose films, induced hornification effects in the form of irreversible aggregation. This was indicated by solid-state NMR as an increase in signal intensity originating from crystalline structures accompanied by a decrease of signal intensity originating from cellulose surfaces. This phenomenon was observed for all used coagulants in this study, but to various degrees with regard to the polarity of the coagulant. From NMR cryoporometry, it was concluded that drying induced hornification generates an increase of nano-sized pores. A bimodal pore size distribution with pore radius maxima of a few nanometers was observed, and this pattern increased as a function of drying. Additionally, cyclic drying and rewetting generated a narrow monomodal pore size pattern. This study implies that the porosity and crystallinity of regenerated cellulose can be manipulated by the choice of drying condition.     

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.     

Oscillations in the capacitance of a nanopore containing an electrolyte due to pore width and nonzero size ions

Jiang, Jin, and Wu (Nano Lett., 11 (2011), pp. 5373-5377) have reported the results of a density functional theory (DFT) study of the capacitance of a nanopore containing an electrolyte consisting of charged hard spheres of equal diameter and charge. They find that the capacitance of the nanopore and electrolyte oscillates. The 'period' of the oscillations is of the order of the ionic diameter. Intuitively, the capacitance should tend to zero when the pore diameter is too small to accommodate the electrolyte ions and, thus, would contains no charge. A superficial glance at their table of contents figure might lead one to think that this does not occur in their calculation. However, they do not report results for exceedingly small pore diameters. In order to gain insight into their results, the Poisson-Boltzmann (PB) theory, which does not account for the diameter, is examined briefly. Unsurprisingly, the PB capacitance decreases monotonically to zero as the pore diameter decreases. The effect of a nonzero ion diameter is included in a semi-empirical manner by appealing to the results of the mean spherical approximation (MSA). The resulting capacitance oscillates and is qualitatively similar to the DFT results; it is zero at small pore diameter.     

Adsorption-induced deformation of microporous carbons: pore size distribution effect

We present a thermodynamic model of adsorption-induced deformation of microporous carbons. The model represents the carbon structure as a macroscopically isotropic disordered three-dimensional medium composed of stacks of slit-shaped pores of different sizes embedded in an incompressible amorphous matrix. Adsorption stress in pores is calculated by means of Monte Carlo simulations. The proposed model reproduces qualitatively the experimental nonmonotonic dilatometric deformation curve for argon adsorption on carbide-derived activated carbon at 243 K and pressure up to 1.2 MPa. The elastic deformation (contraction at low pressures and swelling at higher pressures) results from the adsorption stress that depends strongly on the pore size. The pore size distribution determines the shape of the deformation curve, whereas the bulk modulus controls the extent of the sample deformation.     

Lorentzian line shape due to an inhomogeneous size distribution without relaxation

In this paper, it is demonstrated that a Lorentzian line shape is predicted for a resonance interacting with a bath of equally spaced levels, even in the sparse, low density of states limit, if one performs an inhomogeneous average over the position of the bath states relative to the bright state. The implication for the spectroscopy of molecules in helium nanodroplets and possibly other samples with a significant size distribution is that coupling of excitations to phonons can lead to Lorentzian shaped transitions entirely from inhomogeneous broadening with no population relaxation in the sample.     

Polyester scaffolds with bimodal pore size distribution for tissue engineering

This paper presents a method for the preparation of porous poly(L-lactide)/poly[(L-lactide)-co-glycolide] scaffolds for tissue engineering. Scaffolds were prepared by a mold pressing-salt leaching technique from structured microparticles. The total porosity was in the range 70-85%. The pore size distribution was bimodal. Large pores, susceptible for osteoblasts growth and proliferation had the dimensions 50-400 microm. Small pores, dedicated to the diffusion of nutrients or/and metabolites of bone forming cells, as well as the products of hydrolysis of polyesters from the walls of the scaffold, had sizes in the range 2 nm-5 microm. The scaffolds had good mechanical strength (compressive modulus equal to 41 MPa and a strength of 1.64 MPa for 74% porosity). Scaffolds were tested in vitro with human osteoblast-like cells (MG-63). It was found that the viability of cells seeded within the scaffolds obtained using the mold pressing-salt leaching technique from structured microparticles was better when compared to cells cultured in scaffolds obtained by traditional methods. After 34 d of culture, cells within the tested scaffolds were organized in a tissue-like structure. Photos of section of macro- and mesoporous PLLA/PLGA scaffold containing 50 wt.-% of PLGA microspheres after 34 d of culture. Dark spots mark MG-63 cells, white areas belong to the scaffold. The specimen was stained with haematoxylin/eosin. Bar = 100 microm.     

Quantitative information on pore size distribution from the tangents of comparison plots

The comparison plot obtained from the nitrogen adsorption data has a similar shape to that of the curve of accumulating pore volume of a solid. The intrinsic nature of this relation is discussed. It is known that the derivatives of the accumulating pore volume with respect to the pore size are the pore size distribution (PSD) of the solid. Thus, the tangent curve of the comparison plot can display, at least qualitatively, the PSD of a solid, over a wide range of pore sizes (from approximately 1 to 50 nm) because the comparison plot is applicable to both micropores and mesopores. Quantitative pore structure information can be derived from the comparison plots by establishing a relationship between the t value and the pore size from the samples with uniform pore structure and known pore sizes, such as MCM-41 and alumina pillared clay samples. A calculation procedure to derive quantitative PSD from the comparison plots is suggested, giving reasonable results. This study proposes concise and reliable methods based on the comparison plots to derive information on pore structure in porous solids.