A discretization method for computing chain length distributions

The method of Kumar and Ramkrishna is a numerical technique to solve population balance equations (PBEs) by discretization while preserving two moments of the distribution. When the method is used to calculate chain length distributions in polymerization reactions, the polydispersity, which depends on the first three moments of the distribution, cannot be estimated correctly. This work presents a modification of the method that allows to preserve three moments and thus calculate the polydispersity correctly, independently of the number of grid points. The modified method is applied to a model of controlled radical polymerization via RAFT and compared with the original one.     

Bimodal pore size distributions for carbons: experimental results and computational studies

It is well known that the bimodal shape of the pore size distribution (PSD) curves is typical for various carbonaceous materials (of different origin and/or treated thermally or chemically). A systematic investigation of this effect has been discussed using the Nguyen and Do method with proposed recently the ASA algorithm. A series of numerically generated adsorption isotherms (N(2), T=77 K) and experimental data were analyzed. We investigated various possible situations related to the shape of the PSD curves, i.e., the intensity of the both peaks, their mutual location and the vanishing of one of them. Moreover, the problem in the similarity of the local adsorption isotherms from the range of pore widths corresponding to the gap between peaks is discussed. The analysis of obtained results (as well as published by others) shows that the bimodal shape of the pore size distributions is a characteristic feature for many adsorbents possessing even a small amount of micropores. It is shown that this feature results from the similarity of the local adsorption isotherm in the range of the pore widths for which the gap between peaks (related to the primary and secondary micropore filling mechanism) exists.     

Influence of diffusion on pore size distributions determined by xenon porometry

Xenon porometry is a new method for characterization of porous materials. In this method, the material is immersed in a medium, and its properties are studied by means of 129Xe NMR spectra of xenon dissolved in the sample. The method is particularly suitable for the determination of pore size distribution of the material, since the spectra display two signals whose chemical shift is dependent on the pore size. A prerequisite for an accurate determination is the fact that the diffusion of xenon between different pores is slow enough. The diffusion is studied in this work using two-dimensional exchange spectroscopy (2-D EXSY). The spectra measured as a function of the mixing time imply that the exchange is really slow as compared with the NMR time scale, and therefore the distribution of the resonance frequencies indeed represents the pore size distribution.     

Ion dynamics into different pore size distributions in supercapacitors under compression

Compressing supercapacitor(SCs) electrode is essential for improving the energy storage characteristics and minimizing ions’ distance travel, faradaic reactions, and overall ohmic resistance. Studies comprising the ion dynamics in SC electrodes under compression are still rare. So, the ionic dynamics of five aqueous electrolytes in electrodes under compression were studied in this work for tracking electrochemical and structural changes under mechanical stress. A superionic state is formed when ...     

An optical method for determining size distributions of water droplets

A method for determining aerosol size distributions by single laser-shot single droplet cavity enhanced Raman scattering (CERS) is presented. Droplets are illuminated with the tripled output from a Nd:YAG laser at 355 nm and the CERS fingerprint acquired with a spectrograph and CCD. Droplets with radii in the range 10–50 μm are probed. The extension of this to the determination of a distribution of droplet sizes is illustrated. We suggest that the CERS signature from water could be used to determine droplet size while the observation of Raman scattering from other constituents could be used to identify trace chemical constituents within water droplets.     

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.     

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     

Are pore size distributions in microfiltration membranes measurable by two-phase flow porosimetry?

The issue of evaluating equivalent pore diameter distributions in membrane microfilters from gas-liquid (g-l) porosimetry data has been critically examined. Experiments performed with one isotropic and one composite anisotropic membrane in both possible orientations revealed conspicous dependence of the obtained (g-l) porosimetry peaks on imposed pressure ramp rates, p. Interference of this kinetic effect can be eliminated from the measured data by extrapolation to p = 0. The ramp rate effect is most likely caused by tortuous pore length distribution, and relatively long times required for liquid expulsion. For two experiments, the observed effects of p could be reconciled with predictions of the Schlesinger-Bechhold theory [Bechold et al., Kolloid Z., 55 (1931) 172–198]. The data obtained with the thin top layer of the composite membrane facing intruding air directly did deviate somewhat from the theory. Pores characterized by (g-l) porosimetry are likely of the “throat type”, and their size distribution is considerably more narrow than that obtained for the “node-type” pores by SEM-image analysis [Zeman and Denault, J. Membrane Sci., 71 (1992) 221–231]. A single bivariate distribution function was constructed for these two distinct pore populations. Flow-weighted or number fraction distributions can be calculated from the extrapolated porosimetry data. For narrow ranges of “throat” diameters, these distributions are fairly similar.     

Cation control of pore and channel size in cage-based metal-organic porous materials

Porous materials resembling zeolites that are composed of organic and inorganic building units were synthesized and characterized. Control of pore and channel size was achieved by using different-sized cations. The metal-assembled, anionic cage molecule, Co(4)1(2)(8-), with a hydrophobic cavity and four carboxylate rich arms, was used as a structural unit for the formation of materials with pores and channels. When assembled into a solid material with dications (Mg(2+), Ca(2+), Sr(2+), and Ba(2+)), Co(4)1(2)(8-) arranges into sheets of cages linked together by cations. The series of materials based on Co(4)1(2)(8-) and containing alkaline earth cations was characterized using X-ray crystallography. The magnesium material packs with cages close together, has small channels, and has cation-carboxylate linkages in three dimensions. The calcium material has cages packed with voids between them and has 5 x 10 A channels and 10 x 21 A pores. The strontium and barium materials also pack with voids between the cages and similarly to each other. They have 11 x 13 A and 11 x 11 A channels and 10 x 27 A and 9 x 27 A pores, respectively. Each of these materials has many (20-50) solvent water molecules associated with each cage. The associated water can be removed from and adsorbed by the materials. The heat of water binding has been measured to be -52 kJ/mol (Mg(4)Co(4)1(2)); -47 kJ/mol (Ca(4)Co(4)1(2)); -48 kJ/mol (Sr(4)Co(4)1(2)); -49 kJ/mol (Ba(4)Co(4)1(2)).     

Modified liquid displacement method for determination of pore size distribution in porous membranes

A new method called constant pressure liquid displacement method (CPLM) was developed and tested to measure the pore size distribution of porous membranes. The permeability, defined as a ratio of the flow rate to the pressure applied, used to be assumed constant either for a conventional liquid displacement method or for a bubble point method, leading to the erroneous interpretation of the pore size distribution. However, it was possible to eliminate such an assumption by measuring the flow rates experimentally at a standard low pressure through the pores penetrated with a permeating liquid according to the proposed method. The pore size distribution for a hydrophobic PVDF membrane was successfully measured by the CPLM and compared with those measured by two different methods such as the conventional liquid displacement method and the mercury intrusion method.     

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.     

An improved method for determining zeta potential and pore conductivity of porous materials

An improved method based on streaming potential and streaming current was proposed to determine zeta potential and surface conductance of porous material simultaneously. In the electrokinetic generation mode, a resistor is connected to the generator and by measuring the voltage drop across resistors with different resistance, a true streaming current can be determined. The zeta potential and surface conductivity can be obtained simultaneously from their relation to streaming potential and streaming current. The electrode and ion concentration polarization effects during the measurement were also discussed. The resistance from channel ends to electrodes, which has typically been ignored in the literature, was shown to have a significant influence on the calculated zeta potential and surface conductance. Ignorance of this resistance would lead to underestimation of both zeta potential and surface conductance values.     

Influence of the pore size of reversed phase materials on peptide purification processes

The influence of the pore size of a chromatographic reversed phase material on the adsorption equilibria and diffusion of two industrially relevant peptides (i.e. a small synthetic peptide and insulin) has been studied using seven different reversed phase HPLC materials having pore sizes ranging from 90 Å to 300 Å. The stationary phase pore size distribution was obtained by inverse size exclusion measurement (iSEC). The effect of the pore size on the mass transfer properties of the materials was evaluated from Van Deemter experiments. It has been shown that the lumped mass transfer coefficient increases linearly with the average pore size. The Henry coefficient and the impurity selectivity were determined in diluted conditions. The saturation capacity of the main peptides was determined in overloaded conditions using the inverse method (i.e. peak fitting). It was shown that the adsorption equilibria of the peptides on the seven materials is well described by a surface-specific adsorption isotherm. Based on this a lumped kinetic model has been developed to model the elution profile of the two peptides in overloaded conditions and to simulate the purification of the peptide from its crude mixture. It has been found that the separation of insulin from its main impurity (i.e. desamido-insulin) was not affected by the pore size. On the other hand, in the case of the synthetic peptide, it was found that the adsorption of the most significant impurity decreases with the pore size. This decrease is probably due to an increase in silanol activity with decreasing pore size.     

Quasi-equilibrium nitrogen adsorption gravimetry : comparison with volumetry for the determination of surface areas and pore size distributions

The comparison involves one gravimetric technique and two volumetric ones (either conventional or quasi-static). The samples used are a bronze powder (0.1 m²g⁻¹ only), a graphite powder (c.a. 10 m²g⁻¹, with a phase change of the nitrogen monolayer used as a sensitive thermometer) and two mesoporous silica gels (pore size c.a., 5.0 and 11.5 nm). Relative advantages of either technique are pointed out.     

Improvement of the Kruk-Jaroniec-Sayari method for pore size analysis of ordered silicas with cylindrical mesopores

In this work, the X-ray diffraction structure modeling was employed for analysis of hexagonally ordered large-pore silicas, SBA-15, to determine their pore width independently of adsorption measurements. Nitrogen adsorption isotherms were used to evaluate the relative pressure of capillary condensation in cylindrical mesopores of these materials. This approach allowed us to extend the original Kruk-Jaroniec-Sayari (KJS) relation (Langmuir 1997, 13, 6267) between the pore width and capillary condensation pressure up to 10 nm instead of previously established range from 2 to 6.5 nm for a series of MCM-41 and to improve the KJS pore size analysis of large pore silicas.     

Sieving variations due to the choice in pore size distribution model

The effect of the choice of the standard probabilistic model to describe the pore size distribution was theoretically studied on predicting membrane performance parameters, area average water flux and area average membrane sieving coefficient. Preliminary discrete pore size distributions were generated from rejection profiles of dextran and PEG for 10,000, 30,000 and 100,000 molecular weight cutoff (MWCO) polysulfone and cellulose acetate membranes. The standard probability distribution functions (PDF), gamma, lognormal, normal, Weibel and Rayleigh were used to fit the resulting pore size distribution data. It was observed that the area averaged sieving coefficients are sensitive to the choice of the PDF. These results implied that an uncertainty in the choice of distribution in describing the membrane morphology could lead to a propagated uncertainty in predicting overall membrane performance.     

Determination of pore/protein size via electrophoresis and slit sieve model

We used electrophoresis for three purposes: (i) estimation of the mean pore size of polyacrylamide gels via measuring electrophoretic mobility of globular proteins of known sizes in combination with simple sieve (cylindrical and slit) models; (ii) determination of the average size of protein molecules (native or denatured) by the use of the same models; (iii) monitoring the changes in molecular dimensions of proteins in the course of their denaturation. Both models yield results that are in good agreement with those found via the more elaborate techniques (considering the principal differences involved). The approach provides a direct and convenient way of monitoring the variations in protein sizes during the course of their denaturation in gels having a gradient of denaturants, and possibly the number of conformational states involved in the process, a facet that is quite unique and useful. The simpler slit model seems to yield better results in the latter case and is moreover supported by the recently reported data on electrophoresis of DNA molecules through the 1 microm slits of a microbrush matrix made of micropillars arranged in a hexagonal lattice.     

Effect of pore size of packing materials on molecular-weight separation by temperature gradient interaction chromatography

Temperature gradient interaction chromatography (TGIC) is an interactive polymer chromatography technique varying the column temperature during the elution in a programmed manner to control the solute retention. In the present paper, the effect of the pore size of packing materials on the molecular-weight separation of polystyrene and poly(methyl methacrylate) standard samples by TGIC was studied by using the columns (octadecyl modified silica) with different pore size (100, 300 and 1000 Å) and eluent mixture of CH₂Cl₂/CH₃CN. By rising temperature gradient, both polymers were separated by molecular weight from lower to higher. It became clear that each sample elutes out earlier as the pore size is larger. These experimental results could be explained by the theory based on the scaling concept of Gorbunov and Skvortsov.     

A method for the determination of accessible surface area,pore volume,pore size and its volume distribution for homogeneous pores of different shapes

We develop a novel method to determine the accessible pore volume, the accessible pore size and its distribution for pores having homogeneous surfaces but taking an arbitrary shape. The accessible pore volume is essentially the volume space that is accessible to the centre of an adsorbate molecule, while the accessible pore size is defined by the largest sphere that can be accommodated in the accessible space. The size of this sphere depends on the point in the accessible volume that we select. The accessible pore size is therefore, a local variable and this means that even a geometrically simple pore can possess many sizes. Each local accessible pore size is associated with a local accessible pore volume and the relationship between this pore volume and pore size is called the accessible pore size distribution. In this paper, we illustrate this methodology with a number of model pores ranging from simple to complex geometry and present the analytical accessible pore size distribution.