The Finite Pore Volume GAB Adsorption Isotherm Model as a Simple Tool to Estimate a Diameter of Cylindrical Nanopores

The finite pore volume Guggenheim–Anderson–de Boer (fpv-GAB) adsorption isotherm model has been considered as a simple tool which not only enables us to analyze the shape of isotherms theoretically, but also provides information about pore diameter. The proposed methodology is based on the geometrical considerations and the division of the adsorption space into two parts: the monolayer and the multilayer space. The ratio of the volumes of these two spaces is unambiguously related to the pore diameter. This ratio can be simply determined from the N₂ adsorption isotherm by its fitting with the use of fpv-GAB model. The volume ratio is equal to the ratio of the adsorption capacities in the monolayer and the multilayer—two of the best-fit parameters. The suggested approach has been verified using a series of isotherms simulated inside ideal carbon nanotubes. The adsorption data for some real adsorbents has also been used during tests. The studies performed have proven that diameters estimated with the use of the proposed method are comparable with the geometrical sizes or diameters published by others and based on the application of more sophisticated methods. For pores wider than 3 nm, the relative error does not exceed a few percent. The approach based on the fpv-GAB model reflects well the differences in pore sizes for the series of materials. Therefore, it can be treated as a convenient tool to compare various samples.     

Adsorption behavior of BSA in microfiltration with porous glass membrane

The fouling mechanism during dead-end microfiltration of bovine serum albumin (BSA) with porous glass membrane was investigated from the point of BSA adsorption onto the pore surface of membrane under the condition of pH 5.0 and ionic strength 0.01. The location of BSA retention was confirmed by comparing the filtration performance between dead-end mode and cross-flow mode. During the dead-end microfiltration BSA was retained only by the adsorption on the pore surface. The adsorption was irreversible and of multilayer type, which consists of the adsorption on clean pore surface, i.e. the primary adsorption, and that on preadsorbed pore surface, i.e. the secondary one. The adsorption isotherm was high affinity type. The adsorption rate was proportional to the feed rate of BSA, and the proportional coefficient was dependent on the adsorption process. The flux decline was correlated quantitatively with the amount of adsorbed BSA from the pore radius narrowing model by adsorption.     

A New Method of Calculating Pore Size Distribution: Analysis of Adsorption Isotherms of N(2) and CCl(4) for a Series of MCM-41 Mesoporous Silicas

The adsorption isotherms of N(2) gas at 77 K and CCl(4) vapor at 283.1(5), 298.1(5), and 308.1(5) K were measured for six samples of the mesoporous silicas having uniform cylindrical pores (MCM-41). The pore radii of the six samples (r(p)), which were evaluated from the alpha(s) plots of the N(2) isotherms, were 1.13, 1.29, 1.50, 1.65, 1.90, and 2.53 nm. The CCl(4) adsorption isotherms show that the capillary condensation occurs at the very narrow P/P(0) range. The core radii of the six adsorbents (r(c)), which were estimated from a comparison plot of the CCl(4) isotherm, were 0.90, 1.01, 1.28, 1.37, 1.60, and 2.17 nm. In the comparison plot, the standard CCl(4) isotherm for nonporous silica was used as the reference isotherm. It has been clarified that the Polanyi adsorption potential of capillary condensation is proportional to the reciprocal of the core radii: RT ln(P(0)/P)=5.37r(c)(-1) nm(-1), ln(P(0)/P)=2.17r(c)(-1) nm(-1) at 298.1(5) K, [A]. The statistical thickness of adsorbed CCl(4) on the curved surface (t((pore))), which was estimated from the difference between the pore radii and the core radii, was given by Eq. [B]: t((pore))=0.188+0.336(P/P(0))+0.382(P/P(0))(2) nm [B], (0.08

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Estimating the pore size distribution of activated carbons from adsorption data of different adsorbates by various methods

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Restricted diffusion in silica particles measured by pulsed field gradient NMR

The restricted diffusion coefficient of water through porous silica is measured by pulsed field gradient (PFG) NMR as a function of loading in order to develop a model for self-diffusion at full pore filling in sol-gel-made porous silica particles. This model describes the pore or intraparticle diffusion coefficient as a function of particle porosity, tortuosity, and the steric hindrance applied on the molecules by the pore space. The particle morphology is characterized by nitrogen adsorption and an appropriate tortuosity model is chosen in comparison with literature data. To characterize the material, NMR relaxation and diffusion studies at different degrees of pore filling were carried out in relation to the silica/water adsorption isotherm.     

Microstructure and Porosity of Silica Xerogel Monoliths Prepared by the Fast Sol-Gel Method

An adaptation of the fast sol-gel method to the synthesis of xerogel monoliths using tetramethoxysilane (TMOS) as the alkoxide precursor is described in this paper. The procedure involves running the reaction at 70–80°C in an open vessel, which accelerates hydrolysis and condensation and reduces the amount of liquid by expelling excess methanol through outdistillation. This procedure yields crack-free monoliths. The porosity and microstructure of these xerogel monoliths were studied by using N₂ adsorption and desorption and scanning electron microscopy (SEM). The SEM data show that the solid skeletal phase has a globular morphology with particles, 20–40 nm in diameter, arranged into agglomerates a few hundred nm in diameter. The microstructure of the acid-catalyzed xerogel is a consolidation of these agglomerates. The isotherm data show these xerogels to be microporous. In contrast, the base-catalyzed xerogel has a hierarchical morphology with the clusters of agglomerates organized into larger clusters approaching 1 m in diameter. An analysis of the isotherm data shows these xerogels to be less microporous with a narrow distribution of mesopores having an average diameter of 50 Å.     

Equilibrium adsorption in cylindrical mesopores: a modified Broekhoff and de Boer theory versus density functional theory

This paper presents a thermodynamic analysis of capillary condensation phenomena in cylindrical pores. Here, we modified the Broekhoff and de Boer (BdB) model for cylindrical pores accounting for the effect of the pore radius on the potential exerted by the pore walls. The new approach incorporates the recently published standard nitrogen and argon adsorption isotherm on nonporous silica LiChrospher Si-1000. The developed model is tested against the nonlocal density functional theory (NLDFT), and the criterion for this comparison is the condensation/evaporation pressure versus the pore diameter. The quantitative agreement between the NLDFT and the refined version of the BdB theory is ascertained for pores larger than 2 nm. The modified BdB theory was applied to the experimental adsorption branch of adsorption isotherms of a number of MCM-41 samples to determine their pore size distributions (PSDs). It was found that the PSDs determined with the new BdB approach coincide with those determined with the NLDFT (also using the experimental adsorption branch). As opposed to the NLDFT, the modified BdB theory is very simple in its utilization and therefore can be used as a convenient tool to obtain PSDs of all mesoporous solids from the analysis of the adsorption branch of adsorption isotherms of any subcritical fluids.     

Characterization and adsorption properties of diatomaceous earth modified by hydrofluoric acid etching

This work was a study of the chemical modification of diatomaceous earth (DE) using hydrofluoric acid (HF) solution. Under the experimental conditions investigated, it was found that HF under controlled conditions significantly etched inward into the interior of the existing pore structure in the clay mineral due to its high content of silica, leaving a framework possessing a larger BET surface area (ca. 10 m2 g(-1)) in comparison with that (ca. 4 m2 g(-1)) of its precursor (i.e., DE). Further, the results indicated that the HF concentration is a more determining factor in creating more open pores than other process parameters (temperature, holding time, and solid/liquid ratio). This observation was also in close agreement with the examinations by the silicon analysis, scanning electron microscopy, X-ray diffraction, and Fourier transform infrared spectroscopy. The adsorption kinetics and the adsorption isotherm of methylene blue onto the resulting clay adsorbent can be well described by a pseudo-second-order reaction model and the Freundlich model, respectively.     

Analysis of diffusion models for protein adsorption to porous anion-exchange adsorbent

The ion-exchange adsorption kinetics of bovine serum albumin (BSA) and gamma-globulin to an anion exchanger, DEAE Spherodex M, has been studied by batch adsorption experiments. Various diffusion models, that is, pore diffusion, surface diffusion, homogeneous diffusion and parallel diffusion models, are analyzed for their suitabilities to depict the adsorption kinetics. Protein diffusivities are estimated by matching the models with the experimental data. The dependence of the diffusivities on initial protein concentration is observed and discussed. The adsorption isotherm of BSA is nearly rectangular, so there is little surface diffusion. As a result, the surface and homogeneous diffusion models do not fit to the kinetic data of BSA adsorption. The adsorption isotherm of gamma-globulin is less favorable, and the surface diffusion contributes greatly to the mass transport. Consequently, both the surface and homogeneous diffusion models fit to the kinetic data of gamma-globulin well. The adsorption kinetics of BSA and gamma-globulin can be very well fitted by parallel diffusion model, because the model reflects correctly the intraparticle mass transfer mechanism. In addition, for both the favorably bound proteins, the pore diffusion model fits the adsorption kinetics reasonably well. The results here indicate that the pore diffusion model can be used as a good approximate to depict protein adsorption kinetics for protein adsorption systems from rectangular to linear isotherms.     

Molecular dynamic calculation of isotherms of alcohol adsorption in model pores of active carbon

The molecular dynamics method is used to calculate the isotherms of vapor adsorption in a model pore of active carbon taking into account the microheterogeneous structure of the adsorbent. In the calculations, an additional volume with a nonadsorbing surface (bulk phase) is added to the adsorbing volume of a slitlike model pore with a diameter and a width of ≈3 and 0.7 nm, respectively, and the trajectories of molecules are determined. After the dynamic equilibrium is established, the equilibrium numbers of molecules in the adsorption and bulk phases are estimated through averaging over time. Varying the total number of molecules, the necessary number of points is obtained for an isotherm, which, employing the theory of volume filling of micropores, is presented in ordinary coordinates and can be compared with the experimental data.     

Modeling of Adsorption in Finite Cylindrical Pores by Means of Density Functional Theory

Adsorption of argon at its boiling point in finite cylindrical pores is considered by means of the non-local density functional theory (NLDFT) with a reference to MCM-41 silica. The NLDFT was adjusted to amorphous solids, which allowed us to quantitatively describe argon adsorption isotherm on nonporous reference silica in the entire bulk pressure range. In contrast to the conventional NLDFT technique, application of the model to cylindrical pores does not show any layering before the phase transition in conformity with experimental data. The finite pore is modeled as a cylindrical cavity bounded from its mouth by an infinite flat surface perpendicular to the pore axis. The adsorption of argon in pores of 4 and 5 nm diameters is analyzed in canonical and grand canonical ensembles using a two-dimensional version of NLDFT, which accounts for the radial and longitudinal fluid density distributions. The simulation results did not show any unusual features associated with accounting for the outer surface and support the conclusions obtained from the classical analysis of capillary condensation and evaporation. That is, the spontaneous condensation occurs at the vapor-like spinodal point, which is the upper limit of mechanical stability of the liquid-like film wetting the pore wall, while the evaporation occurs via a mechanism of receding of the semispherical meniscus from the pore mouth and the complete evaporation of the core occurs at the equilibrium transition pressure. Visualization of the pore filling and empting in the form of contour lines is presented.     

Insights into adsorption of uncharged macrocyclic complexes into a Nafion Film: Adsorption characteristics and analysis of tetraphenylporphyrine zinc(II)

Tetraphenylporphine zinc(II) (ZnTPP) was found to be adsorbed from its CH2Cl2 solution into a Nafion (Nf) film. The characteristics of the adsorption of ZnTPP into the Nf film were studied using a visible absorption spectroscopic technique. The initial rate (v0, mol cm(-2) s(-1)) for uptake of ZnTPP was saturated with increasing ZnTPP concentration (c0, M) in the solution. This kinetic profile was analyzed in terms of a Michaelis-Menten model considering preequilibrium of ZnTPP adsorption between the solution and the outer layer of the Nf film, followed by diffusion to an inner bulk region, giving a maximum diffusion reflux of v(max) = (2.2 +/- 0.2) x 10(-13) mol cm(-2) s(-1). This is different from the kinetics for the Nf/phthalocyanine zinc(II) (ZnPc) film, which gives a linear plot of v(0) vs c(0). This can be explained by the relatively slow diffusion of ZnTPP in the film compared to that of ZnPc because of steric factors: ZnTPP contains bulky tetraphenyl moieties attached perpendicular to a porphyrin ring, whereas ZnPc has higher planarity. The isotherm for the adsorption of ZnTPP into the Nf film was analyzed using a Langmuir isotherm equation, yielding an equilibrium constant of (3.6 +/- 1.1) x 10(6) M(-1) and a saturated amount of adsorbed ZnTPP of (1.8 +/- 0.1) x 10(-9) mol cm(-2), suggesting monolayer adsorption of ZnTPP on the hydrophobic polymer network interfacial with hydrophilic transport channels without significant intermolecular overlap. This is in contrast to the multilayer adsorption mode suggested for the ZnPc adsorption. The tetraphenyl moieties could prevent the stacking of ZnTPP for multilayer adsorption.     

Evaluation of pore size distribution in boundary region of micropore and mesopore using gas adsorption method

This paper discusses an accurate method of pore size distribution evaluation in boundary regions of micropores and mesopores using the gas adsorption process on the basis of the capillary condensation theory, which is liable to be underestimated with the existing BJH and DH methods. A typical nitrogen adsorption isotherm for highly ordered mesoporous silica, which has cylindrical pores with diameter smaller than 4 nm, is considered to be type IV and it is well known for the steep increase of the amount adsorbed through capillary condensation in the region of the relative pressure P/P0 smaller than 0.4. In calculating the distribution of the pore size from the change of the amount adsorbed due to capillary condensation, it is important to accurately predict both the multilayer thickness t of the adsorbed nitrogen molecules and the critical radius rc where capillary condensation occurs. It is necessary to consider the curvature of the adsorption layer-gas phase interface when predicting the multilayer thickness t of nitrogen adsorbed within the pore of highly ordered mesoporous silica. Revision of the Kelvin equation is also required when rc is to be predicted. While the predicted value of t based on the Broekhoff and de Boer theory is matched well with the value of t which is actually measured using highly ordered mesoporous silica, and the predicted value of rc based on the GTKB-Kelvin-cylindrical equation that has been revised considering the effect of the interfacial curvature on the interfacial tension of the adsorption layer-gas phase interface is matched with the value of rc which is actually measured using highly ordered mesoporous silica. A combination method of the Broekhoff and de Boer equation and the GTKB-Kelvin-cylindrical equation is proposed as a means of accurately evaluating, from the nitrogen adsorption isotherm, the pore size distribution in the highly ordered mesoporous silica in boundary region of micropore and mesopore. The proposed new method of pore size evaluation features high accuracy and offers the convenience of obtaining the pore size distribution without repeated calculations by employing the same algorithm as DH method. The pore size predicted by the Halsey equation and the Kelvin equation of the conventional DH method is about 20% smaller than the pore size predicted by the newly proposed evaluation method.     

A Stefan-Maxwell Model of Single Pore Pressurization for Langmuir Adsorption of Gas Mixtures

A model based on the application of the Maxwell-Stefan approach has been used to describe the dynamics of intraparticle transport (pore diffusion, surface diffusion and convection) in a single pore during and after a pressurization process. The model was first compared with the model proposed by Taqvi and Levan (Adsorption, 2, 299–309 (1996)) for a linear adsorption isotherm. The effect of several parameters (pressurization rate, adsorption capacity, bulk gas-phase mole fraction, adsorption affinity and pore radius) was studied, evaluating the relative importance of each mass-transport mechanism in different conditions. A binary mixture of an inert and an adsorbable component was considered first, extending the analysis of the pore radius effect to a ternary mixture. In general, surface diffusion is dominant with very low pore radius, whereas gas-phase fluxes dominate in a large pore. However, depending on the value of the bulk gas-phase mole fraction (which is related to the surface coverage level through the adsorption equilibrium isotherm), the equilibrium and rate parameters, and the surface to volume ratio, surface diffusion cannot be always neglected for large pores. More generally, system non-linearity can switch the dominant mechanism and create fronts.     

Mesoporous Titanium-containing Silicas. Synthesis and geometrical characteristics

Mesoporous titanium-containing silicas with TiO₂ contents from 1 up to 70 mol% were prepared. The obtained samples have been characterized by the powder X-ray diffraction data, the diffuse reflectance infrared Fourier transform method, and nitrogen adsorption at 77 K. Specific surface area, total pore volume, distribution pore volume on pore sizes were determined from nitrogen adsorption isotherm for synthesized titanosilicas. This revised version was published online in August 2006 with corrections to the Cover Date.     

Local structure of a phase-separating binary mixture in a mesoporous glass matrix studied by small-angle neutron scattering

The mesoscopic structure of the binary system isobutyric acid + heavy water (D(2)O) confined in a porous glass (controlled-pore silica glass, mean pore width ca. 10 nm) was studied by small-angle neutron scattering at off-critical compositions in a temperature range above and below the upper critical solution point. The scattering data were analyzed in terms of a structure factor model similar to that proposed by Formisano and Teixeira [Eur. Phys. J. E 1, 1 (2000)], but allowing for both Ornstein-Zernike-type composition fluctuations and domainlike structures in the microphase-separated state of the pore liquid. The results indicate that the phase separation in the pores is shifted by ca. 10 K and spread out in temperature. Microphase separation is pictured as a transition from partial segregation at high temperature, due to the strong preferential adsorption of water at the pore wall, to a tube or capsule configuration of the two phases at low temperatures, depending on the overall composition of the pore liquid. Results for samples in which the composition of the pore liquid can vary with temperature due to equilibration with extra-pore liquid are consistent with this picture.     

Comparison of the adsorption equilibrium of a few low-molecular mass compounds on a monolithic and a packed column in reversed-phase liquid chromatography

Adsorption isotherm data were acquired by frontal analysis for several low-molecular mass compounds (3-phenyl 1-propanol, 4-tert.-butylphenol, butylbenzene, and butyl benzoate) on a classical packed column and a monolithic column using methanol-water RP-HPLC conditions. These columns have similar characteristics (C18-bonded silica, close specific surface areas and bonding densities). In each case, the isotherm model best accounting for the data was the same on both columns. The solute polarity determines the class of this model. For the two -OH compounds it was a Langmuirian adsorption isotherm. The hydrocarbon data were best modeled by an anti-Langmuir convex-downward isotherm model. The adsorption data for the aromatic ester exhibited a nearly linear behavior, depending on the methanol concentration of the mobile phase. A slightly convex downward isotherm was obtained at high methanol concentrations while the best fitting was obtained with a liquid-solid extended multilayer B.E.T. isotherm model at low concentrations. The validation of these models is discussed in detail. In all cases, similar values of the adsorption-desorption constants were found, underlining the closeness of the adsorption energies on both columns. By contrast, the adsorption capacity of the monolithic column was found to be approximately 1.4 greater than that of the packed column in spite of the close values of the surface areas of the silica in both columns.