Log-normal Pore Size Distribution Research Articles

Understanding dextran retention data for hollow fiber ultrafiltration membranes

Dextran retention tests have become a standard method for characterizing ultrafiltration (UF) membranes. However, most quantitative analyses of dextran transport have been performed with flat sheet membranes using simple stirred cells in which the transmembrane pressure is nearly uniform across the membrane. The objective of this work was to develop a more fundamental understanding of the factors governing dextran retention tests for large pore size hollow fiber ultrafiltration membranes suitable for use in bioprocessing applications, e.g., the purification of vaccines. Experiments were performed using experimental hollow fiber membranes provided by GE Healthcare. Data were analyzed using a theoretical model that accounts for: (1) the axial variation in transmembrane pressure, and in turn the filtrate flux, due to the pressure drop associated with flow through the fiber lumens, (2) the axial variation in the bulk mass transfer coefficient due to the growth in the concentration polarization boundary layer, and (3) the effects of dextran polarization on the local filtrate flux. The intrinsic sieving coefficients were evaluated using available hydrodynamic models assuming a log-normal pore size distribution. The measured dextran retention coefficients were a strong function of the permeate and feed flow rates. For example, the retention coefficient of a 1000 kDa dextran fraction decreased from R = 0.41 to less than 0.02 as the permeate flow rate was increased from 3 to 18 mL/min at a feed flow rate of 120 mL/min due to concentration polarization in the hollow fiber module. The experimental data were in excellent agreement with model calculations over the entire range of dextran molecular weights and flow conditions. The results provide important insights into the proper design and interpretation of dextran retention tests for hollow fiber ultrafiltration membranes.

Three-dimensional imaging of pore structures inside low-κ dielectrics

The three-dimensional reconstruction of a porous low-dielectric constant film (κ=2.5), resolving pores as small as 1 nm, was achieved using annular dark-field scanning transmission electron tomography, enabling quantitative measurements of the pore morphologies and size distribution. Most large pores were elliptical. Together with log-normal pore-size distribution, this suggests pore coalescence during the material’s growth. Ellipsometric porosimetry indicates a high degree of interconnectivity between pores. Tomography shows the material exhibits little large-scale pore connectivity, thus placing an upper limit on the size of the interconnections at below 1 nm. Systematic errors in the tomographic and ellipsometric size distributions appear to be largely complementary.

Effect of molecular shape on rejection of uncharged organic compounds by nanofiltration membranes and on calculated pore radii

We examined the effects of molecular shape on the rejection of uncharged organic compounds by nanofiltration membranes and on pore radii calculated for those membranes. Molecular shape was approximated by a rectangular parallelepiped; molecular length ( L) and molecular width ( MWd) were used as shape parameters. The steric partition factor of a solute was calculated from L and MWd, and the Stokes radius and diffusivity were estimated from MWd. Twenty-four alcohols and polyhydric alcohols were used as probe compounds, and pore radius was estimated by both non-spherical and spherical models within a uniform pore-size model or log-normal pore-size distribution model. The uniform pore/non-spherical model gave better correspondence between the calculated and observed rejections for NTR-729HF membranes, and the calculated pore radius (0.634 nm) was larger than that from the spherical model (0.387 nm). Similar results were obtained from previous rejection data. In the case of the non-spherical model, employing a log-normal pore-size distribution model did not improve the correspondence between observed and calculated rejections. The logarithm of the product of calculated solute permeability and pore length log( B* Δ x) was correlated linearly with MWd. Molecular width is thus a major factor controlling solute permeation.

The structural porosity in soil hydraulic functions - a review

Products of biological processes are the dominant factor of soil structure formation in A horizons, while in B horizons their role is less expressed. Soil structure influences dominantly the structural domain of the pore system in bimodal soils thus affecting soil hydraulic functions. The form of soil hydraulic functions depends upon the pore size distribution and generally upon configuration of the soil pore system. We used the functions derived for the lognormal pore size distribution and modified them to bi-modal soils. The derived equations were tested by experimental data of catalogued soils. The procedure leads to the separation of two mutually different domains of structural and matrix pores. The value of the pressure head (potential) separating the two domains is not constant and varies in a broad range. For each domain we obtained its water retention function and unsaturated hydraulic conductivity function. The separation of hydraulic functions for the two domains is a key problem in the solution of preferential flow and in controlling lateral flow between the structural and matrix domains. Water retention function is fully physically based while the conductivity function still keeps fitting parameters, too. Their simple relationship to tortuosity and pores connectivity was not confirmed. Since they differ substantially for matrix and structural domains, we suppose that there exists a great difference in configuration of porous systems in structural and matrix domains. The use of uniform fitting conductivity parameters for the whole range of pores is not justifiable.

Soil porous system changes quantified by analyzing soil water retention curve modifications

Soil water retention curves (SWRCs) relate soil water pressure head ( h) to soil water content ( θ) and can also be used to find information regarding soil pore distribution. To analyze SWRCs in relation to pore size distribution (PSD), changes due to wetting and drying (W–D) cycles were studied in three different tropical soils (Geric Ferralsol, GF; Eutric Nitosol, EN; Rhodic Ferralsol, RF), using three different treatments: T 0, the control with samples not submitted to W–D cycles; T 3, samples submitted to three consecutive W–D cycles; T 9, samples submitted to nine consecutive W–D cycles. Log-normal PSD equations for each treatment were obtained using the S-theory. For the GF soil, the pressure heads separating structural and matrix domains ( h s) were 17.7, 12.2 and 14.7 kPa for T 0, T 3, and T 9, respectively. These values are equivalent to pore radia of 8.4 μm ( T 0), 12 μm ( T 3), and 10 μm ( T 9). For the RF soil, h s values were 8.5 kPa ( T 0), 20.5 kPa ( T 3), and 15.1 kPa ( T 9), equivalent to radia of 18 μm ( T 0), 7.3 μm ( T 3), and 9.9 μm ( T 9); and finally, for the EN soil, h s were 18.1 kPa ( T 0), 9.1 kPa ( T 3), and 13.5 kPa ( T 9), equivalent to radia of 8.2 μm ( T 0), 16 μm ( T 3), and 11 μm ( T 9). It was found that the soil structure presented important changes in PSD due to W–D cycles for all the investigated soils. It was also observed that the W–D cycles increased the S inf (slope of SWRC) value for the GF soil for all treatments; S inf did not substantially change in all treatments for the EN soil; S inf decreased between T 0 and T 3, and T 0 and T 9 for the RF soil. According to the S-theory, it is possible to infer that W–D cycles improved the soil structure of GF, made the RF soil structure worse and did not substantially change the EN soil structure.

'Parking' and packing problems in particulate materials processing

One-dimensional 'parking' simulations are used to lay a foundation for analysis of packing problems encountered in particulate materials processing. Parking simulations involve a random sequence of segment space filling steps with simultaneous fragmentation of the remaining unoccupied space; analogous to how automobiles are parked along a curb or particles are placed in a die. The present study captures both the geometric and kinetic changes associated parking simulations. The results are generalised to three-dimensional particle structures. For example, data are extracted from parking simulations to show how powder filling into a container generates a lognormal pore size distribution. The results are useful in understanding powder packing problems such as optimisation of bimodal powder packing, powder reaction rates, rearrangement effects on packing density and polymer (binder or lubricant) pyrolysis.

Characterization of meso- and macroporous ceramic membranes in terms of flux measurement: A moment-based analysis

This work addresses the determination of unimodal log-normal pore size distributions (PSDs) in porous symmetric membranes through a convenient selection of a set of statistical moment generating functions. Relevant information concerning such distributions can be obtained by performing three independent permeation experiments: (1) pure Knudsen single-gas diffusion permeance, (2) pure liquid permeability, and (3) non-hindered diffusion permeance of a target solute. In the special case of porous asymmetric UF, NF and MF ceramic membranes, the contribution of the support to the overall mass transfer tends to increase for permeation processes governed by lower moments of the PSD. For such membranes, a mean diameter of the top layer and the thickness-to-porosity ratio of the support can be accurately determined, which constitute two structural parameters relevant for a preliminary membrane selection. The application of moment theory following the guidelines of Mason allows the determination of upper and lower bounds on the cumulative PSD of active layers from measured moments.

Binding Aedes aegypti densonucleosis virus to ion exchange membranes

Experimental and numerical results for binding Aedes aegypti densonucleosis virus (AeDNV) using anion and cation exchange membranes are presented. AeDNV particles are adsorbed by anion and cation exchange membranes providing the virus particles and membranes are oppositely charged. Q membranes which are strongly basic anion exchangers were the most effective. Dynamic and static capacities for Q membranes were found to be similar. A numerical model is proposed which assumes a log normal pore size distribution. By estimating the required parameters from static binding experiments, the model may be used to calculate the breakthrough curve for virus adsorption.

The influence of uniaxial compression upon pore size distribution in bi-modal soils

Transport of water, of dissolved and non-dissolved chemicals, of gas, the rooting of plants and the conditions for all soil biota are influenced by the soil porous system and by the soil pore size distribution. We have analyzed the change of the soil porous system due to the short time compression in ranges from 0 to 300 kPa in Entisols, Alfisols, and Vertisols. The procedure was based upon the evaluation of the soil water retention curves. The equation of log-normal pore size distribution in bi-modal soils was applied for the analysis. The obtained physical parameters describe the soil porous system and its change due to the compression. The minimum pressure head h A between two peaks of pore size distribution represents the boundary between the structural and matrix domains of the soil porous system. The value of h A was in broad ranges from h A = −136 up to h A = −585 cm. Pressure head corresponds to equivalent pore radius from r = 10.9 to r = 2.5 μm. The rate of change of h A due to the applied compression was not the same in all examined soils. h A decreased in some soil taxons and in other soils it slightly increased when the compression stress rose. It follows that the boundaries between the soil pore categories cannot be taken as a fixed value for all soils and for all compression stresses. The increase of the compression stress caused a decrease of the total porosity, but the decrease of the structural porosity was much more expressed. The ratio of the decrease of structural porosity due to compression was in ranges between 0.91 and 0.48. The change of the matrix porosity has not the same tendency in all soils. In majority of instances it was increasing, but in two soils it was slightly decreasing with the increase of compression stress. The characteristics of the pore size distribution did not change in a uniform way either. The pores size distribution curve is more flat and more broadly distributed due to compression in the structural domain. The tendency is opposite in the matrix domain. A generally valid rule on the change of the pore size distribution due to compression does not exist for all soil taxons. The change was great in both domains of soils with a low stability of aggregates. The change was relatively small in soils with a well-developed structure where it was more related to the structural domain. The log-normal model does not fit in all instances to pore size distribution in the structural domain. A profound study of the soil micromorphology may result in a better description of the soil porous system and its internal change during compression.

Coarsening in a Viscous Matrix

When a viscous material is three-dimensionally constrained (as when a gel or slurry of glass is infiltrated into the interstices of a particle packing), heat treatment leads to coarsening of the pore-size distribution, rather than densification. The smaller pores generate greater capillary pressure; therefore, they shrink and force the large pores to expand. In this article, the kinetics of coarsening are analyzed, and sample calculations are performed for a body with a lognormal distribution of pore sizes.

Permeability and selectivity analysis for ultrafiltration membranes

One of the major challenges facing end-users of ultrafiltration membranes is the difficulty in comparing membrane products provided by different manufacturers and made from different polymeric or ceramic materials. In this short communication, we examine the trade-off between permeability and selectivity for different ultrafiltration membranes using data for bovine serum albumin. Results for a number of different ultrafiltration membranes fall along, or below, an “upper bound” that reflects the current state-of-the-art in commercial ultrafiltration membranes, analogous to the Robeson Plot used to analyze the performance of gas separation membranes. The shape of this upper bound is consistent with a theoretical analysis of solute and solvent transport through a membrane composed of a parallel array of cylindrical pores having a log–normal pore size distribution. These results provide a framework that can be used to analyze the performance of new ultrafiltration membranes, an approach that was demonstrated using data for a prototype negatively charged ultrafiltration membrane.

Soil hydraulic properties as related to soil structure

Soil structure is one of the main attributes of soil quality. It influences the soil pore system and through it all parameters of equations describing the transport processes in soil. Soil pore size distribution is either log-normal, or the log-normal distribution is an acceptable approximation. Kosugi [Water Resour. Res. 30 (1994) 891] has applied the log-normal distribution to data on the soil water retention curve and has replaced the empirical equation by the physically-based model. Log-normal pore size distributions were used for the estimation of the unsaturated hydraulic conductivity function. This paper deals with the application of Kosugi's model to the description of the retention curve and of the unsaturated conductivity function in soils exhibiting a bi-modal distribution of pores, typical for all soil horizons having a certain type of structure. The domain of structural pores and the domain of matrix pores are defined in these structured soils. Results from this study show that two hydraulic functions, the retention curve and the unsaturated conductivity are well described by models based on a log-normal distribution of pores in both the structural and the matrix domains of the soil porous system. The exponents α, β defined as parameters of the unsaturated conductivity are different for the structural and matrix domains. It is therefore, assumed that the configuration of soil pores including the tortuosity differs in these domains. The destruction of soil structure changes distinctly the configuration of soil pores which is demonstrated by the change of parameters α, β Consequently, the saturated and unsaturated conductivity is changed, too.

Linking hydraulic conductivity and tortuosity parameters to pore space geometry and pore-size distribution

To enable a link between the pore size distribution of the soil measured in two dimensions and the hydraulic conductivity, a factor, which includes the connectivity and tortuosity of the pore network in three dimensions, is needed. In order to increase confidence in the models which predict hydraulic conductivity, this paper aims to gain insight into relationship between tortuosity and connectivity parameters used in statistical models of hydraulic conductivity and soil structural form measurements derived with image analysis. The analysis was based on a dataset of the hydraulic and soil structural properties of 19 Vertisol samples from Eastern Australia. The soil structural data were measured on large (200 mm height and 150 mm diameter) cores at 5–7 depth slices. Several soil structural form parameters, including a pore size distribution, were calculated using an image analysis program ( Solicon v2.1). The hydraulic conductivity model developed by Kosugi [Soil Sci. Soc. Am. J. 63 (1999) 270] based on the lognormal pore size distribution was fitted to measured hydraulic conductivities. Hydraulic conductivity was strongly related to porosity and a measure of connectivity in the horizontal direction (pore genus). The tortuosity parameters α and β from the Kosugi [Soil Sci. Soc. Am. J. 63 (1999) 270] model were related to the mean pore size, with increasing mean pore size indicating decreasing tortuosity of the sample.

Mathematical modeling of influence of porous structure a membrane on its vapour-conductivity in the process of membrane distillation

Membrane distillation (MD) is a perspective method of separation and concentration of inorganic aqueous solutions. The driving force of MD is temperature difference on both sides of a hydrophobic porous membrane, which separates two aqueous solutions with different temperatures and concentrations. The coefficient of vapor conductivity is one of more important parameters of water vapor transfer through hydrophobic microporous membrane. The influence of membrane porous structure on vapor conductivity coefficient was studied theoretically using the pore network as a model of porous medium. The pore networks were considered as a long and straight cylindrical capillaries arranged around the bonds of two- or three-dimensional lattices with fixed coordination number z for lognormal pore size distribution. The effective-medium theory (EMT) and smooth field approximation (SFA) were used for describing of mass-transfer through the porous membrane. It was shown that the vapor conductivity coefficient value decreases with increasing of the pore dispersion d and decreasing of mean radius porous « rå, grows with increasing of coordination number z. As a result, MD process is more effective using the porous membrane with narrow pore size distribution.

Modelling of membrane nanofiltration—pore size distribution effects

The effects of log-normal pore size distributions on the rejection of uncharged solutes and NaCl at hypothetical nanofiltration membranes have been assessed theoretically. The importance of pore radius-dependent properties such as solvent viscosity and dielectric constant is increased by the introduction of a pore size distribution in calculations. However, the effect of porewise variation in viscosity is less apparent when considered at a defined applied pressure rather than at a defined flux, showing a further advantage of basing theoretical analysis of nanofiltration in terms of applied pressure. Truncated pore size distributions gave better agreement than full distributions with experimental rejection data for a Desal-DK nanofiltration membrane. Such truncation is in agreement with the findings of atomic force microscopy (AFM). Analysis of uncharged solute rejection data alone could not give useful information about membrane pore size distribution. Neither could such a distribution be obtained quantitatively directly from AFM images. However, use of the shape of the distribution obtained by AFM in conjunction with experimental rejection data for an uncharged solute allows calculation of corrected distributions. Importantly, incorporation of such a corrected pore size distribution in calculations of NaCl rejection gave better agreement with experimental data, compared to calculations assuming uniform pores, at high pressure, the region of industrial interest.

Simultaneous scaling of soil water retention and unsaturated hydraulic conductivity functions assuming lognormal pore-size distribution

Using simultaneous scaling, soil spatial variability of hydraulic functions can be described from a single set of scaling factors. The conventional scaling approach is based on empirical curve fitting, without paying much attention to the physical significance of the scaling factors. In this study, the concept of simultaneous scaling of the soil water retention and unsaturated hydraulic conductivity functions is applied to a physically based scaling theory. In this approach, it is assumed that soils are characterized by a lognormal pore-size distribution, which leads directly to lognormally distributed scaling factors. To test this concept, a total of 143 undisturbed soil samples were collected from two soil depths (25 and 50 cm), with each depth divided into two subsets based on the median soil capillary pressure head value, as determined from the lognormal pore-size distribution assumption. Moreover, the theory was compared with the conventional simultaneous scaling method. Both the conventional and physically based simultaneous scaling method performed equally well for all four subsets, as determined from the reduction in weighted root mean squared residual (WRMSR) values after scaling. We showed that the theoretical interpretation of the lognormal scaling factor distribution was applicable to simultaneous scaling of soil hydraulic functions.

Synthesis and Related Kinetics of Nanocrystalline Ni by Hydrogen Reduction of NiO

The present study has attempted to elucidate the formation mechanism of nanocrystalline (nc) Ni by hydrogen reduction of fine NiO powder in terms of related kinetics aspects. So, the kinetics and related mechanism of hydrogen reduction of NiO were investigated on the basis of structure modification of the NiO powder during reaction. The ball-milled NiO agglomerate powder having 20 nm in grain size and a log-normal pore size distribution was used for study. The non-isothermal reduction study showed that the nano-agglomerate NiO underwent a two-step reduction process which is presumably due to a chemical reaction at lower temperatures and a diffusion controlled process at higher temperatures. The activation energy for the nano-agglomerate NiO was 85.4 kJ/mol for lower temperatures and 105.1 kJ/mol for higher temperatures. The value for lower temperatures is consistent with that of as-received NiO of 85.6 kJ/mol. Such higher activation energy for higher temperatures can be attributed to the retardation of the reduction process by the change in the reduction mechanism from the chemical reaction to the diffusion process. Conclusively, the structure change during the reduction is believed to be responsible for the change in the reduction mechanism.

Reduction in Mean Glomerular Pore Size Coincides with the Development of Large Shunt Pores in Patients with Diabetic Nephropathy

The glomerular size selectivity was determined by neutral dextran clearance sieving technique and plasma cystatin C levels in two groups of patients with long-standing type I diabetes mellitus and different stages of albuminuria but normal glomerular filtration rate and in a group of healthy controls. The sieving characteristics of the glomerular filter were determined using a mathematical model of log normal pore size distribution. Patients with albuminuria above 200 μg/min exhibited a significant increase of cystatin c plasma concentrations and a significant reduction in mean glomerular filtration slit size. Only these patients exhibited large, unrestrictive pores in the glomerular filter (shunt). The plasma cystatin c levels correlated significantly with 26-angstrom neutral dextran plasma levels in microalbuminuric patients and in patients with albuminuria above 200 μg/ min. We conclude that a reduction in average pore size of the glomerular filter does not occur earlier than the development of large shunt pores. The renal clearance of cystatin c in patients with proteinuric diabetic nephropathy but a normal glomerular filtration rate is reduced due to its molecular size.

Microstructural development during final-stage sintering of nanostructured zirconia based ceramics

The microstructural evolution of nanostructured zirconia based ceramics during pressureless final-stage sintering in vacuum was followed using X-ray diffraction, high resolution scanning electron microscopy and small-angle neutron scattering (SANS). The ultra-fine zirconia and yttria powders were synthesized by the inert gas condensation technique, the ultra-fine alumina by chemical vapor condensation. Powder compacts of ZrO 2 and dispersion mixed 5 mol% Y 2O 3 partially stabilized ZrO 2 and ZrO 2+14 wt.% Al 2O 3 were sintered at temperatures between 1000 and 1200°C for 2 h. Densification and concurrent grain growth as a function of temperature revealed a significant suppression of grain growth in the zirconia/alumina composite. Due to a homogeneous distribution of the alumina second phase in the zirconia matrix nearly full density and grain sizes <40 nm were achieved. Changes in the microstructure of monoclinic zirconia sintered at T>1050°C were detected by SANS and could be attributed to the detrimental effects of the monoclinic to tetragonal martensitic phase transformation. The log-normal pore size distributions were also evaluated for the ceramics.

Padé approximants for the acoustical properties of rigid frame porous media with pore size distributions

Expressions for the viscosity correction function, and hence bulk complex impedance, density, compressibility, and propagation constant, are obtained for a rigid frame porous medium whose pores are prismatic with fixed cross-sectional shape, but of variable pore size distribution. The low- and high-frequency behavior of the viscosity correction function is derived for the particular case of a log-normal pore size distribution, in terms of coefficients which can, in general, be computed numerically, and are given here explicitly for the particular cases of pores of equilateral triangular, circular, and slitlike cross-section. Simple approximate formulae, based on two-point Padé approximants for the viscosity correction function are obtained, which avoid a requirement for numerical integration or evaluation of special functions, and their accuracy is illustrated and investigated for the three pore shapes already mentioned.