Log-normal Particle Size Distribution Research Articles

Moments of particle size distributions under sequential breakage with applications to species abundance

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.

An electro-optic birefringence fine-particle sizer

When subjected to pulsed electric fields, particles in suspension undergo orientational order. With the majority of materials this is accompanied by the induction of birefringence into the medium. An apparatus is described in which the birefringence is generated, measured and analysed for suspensions of sub-micron sized particles when subjected to electric pulses of short duration. By varying the amplitude of the applied field, data are accumulated and analysed via an average rotary diffusion coefficient for the particles to yield an equivalent log-normal distribution of particle sizes for polydisperse samples. The method is extremely fast and is able to handle particles of non-spherical shape in terms of any assumed geometry. Illustrative data are given for the nearly-spherical particles of polytetrafluoroethylene in water.

Clay particle sizing by electrically-induced birefringence

AbstractUnder the influence of a pulsed field, dilute clay sols become birefringent as the particles undergo orientational order. The rate of decay of the birefringence on removal of the field is characteristic of the particle geometry. Measurement of the decay rates under two specific experimental conditions provides sufficient information from which the particle-size distribution can be evaluated in terms of a two-parameter function. Experimental data are reported and analysed in terms of a log-normal distribution of particle sizes for attapulgite (rods), kaolinite (discs) and halloysite (ellipsoids) sols and compared with success to electron microscopic data. The ability of the method to determine size distributions in terms of the major dimensions of the clay particles, rather than those of the often used equivalent sphere, is highlighted.

On the products of lognormal and cumulative lognormal particle size distributions

The output distributions of particle size which would result when an input aerosol having a lognormal size distribution is passed through a size classifier (e.g. impactor) or control device (e.g. cyclone) having a cumulative lognormal penetration function are investigated theoretically. A computer program is presented for determining these (output) product distributions, their geometric mean sizes ( a′ 8 ), and geometric standard deviations (GSD'). The program was used for forty-five cases having different input aerosol size distributions and different functions of penetration vs particle size. These results are tabulated and used to form regression equations to facilitate predicting a′ 8 and GSD' for conditions not tabulated. The output geometric mean size ( a′ 8 ) is always smaller than the input ( a′ 8 ) and was highly correlated with it. The output aerosol geometric standard deviation (GSD') was also highly correlated with the input aerosol GSD and was never larger. Although the product size distributions are not lognormal, they are approximately so in many instances.

The relative importance of atmospheric sulfates and nitrates in visibility reduction

Atmospheric visibility is largely determined by the concentration, size distribution and chemical composition of the ambient aerosol. In the present paper, total extinction coefficients were calculated for mixed NH 4NO 3-(NH 4) 2SO 4 aerosols of two different compositions under various humidity conditions. Pertinent thermodynamic and optical properties were measured at 25° C and incorporated into a computer model for droplet growth and Mie calculations. Log-normal particle size distributions with mass median diameters characteristic of the Los Angeles aerosol were assumed throughout the study. The results show that the extent of light extinction by aerosol particles depends on whether the aerosol is present as a multicomponent solution droplets of identical composition (internal mixture) or as a mixture of single-salt aerosols (external mixture|. The extinction coefficients calculated for externally mixed sulfate — nitrate aerosols compare remarkably well with published statistical correlations derived from the Los Angeles visibility data. In addition, it is shown that, for externally mixed sulfate-nitrate aerosols, the sulfate indeed contributes overwhelmingly to visibility reduction, a finding in agreement with general observations for atmospheric aerosols.

Tarnishing kinetics of powders with lognormal particle size distributions

Equations are derived for the reaction kinetics of powders in which a particle of size D reacts according to dFD/d(tz)=(kz/βD)(1 –βF)m and which have lognormal, truncated lognormal or composite lognormal particle size distributions. This leads to a method for estimating such kinetics for powders with arbitrary particle size distribution.

Effects of particulate complex refractive index and particle size distribution variations on atmospheric extinction and absorption for visible through middle ir wavelengths

A comprehensive sensitivity study has been made using Mie theory to determine the effect of realistic variations in values of real and imaginary parts of the complex index of refraction on volume extinction and absorption coefficients for a wide range of log normal particle size distributions (defined by geometric mean radius r(g) and geometric standard deviation sigma(g)). Wavelengths lambda from the visible (0.55 microm) through the middle ir (10.6 microm) were considered. Extinction is independent of the complex index to within 20% for the majority of realistic particle size distributions, providing lambda < 2 microm. However, changes in extinction by up to an order of magnitude are caused by realistic variations in refractive indexes for 2 microm </= lambda </= 10.6 microm, with the real index being more important in affecting extinction than the imaginary index. Similar changes are caused by variations in particle size distribution for values of refractive indexes typical of atmospheric constituents. For bimodal size distributions representative of desert aerosols, values of the complex refractive index that result in minimum and maximum extinction coefficients are given. Absorption is generally less dependent on size distribution than is extinction and is not, in general, linear with the imaginary index, especially for broad particle distributions.

On the determination of log-normal particle-size distributions using half widths and detectabilities of scattered laser power spectra

We present the theoretical curves needed to define log-normal particle-size distributions from the measured power spectra of scattered laser radiation. (i) We show how to make estimates of the particle sizes of monodisperse systems using only the measured half-widths of the observed, scattered laser-power spectra. This technique has been employed by us before on flat-flame burners and does not involve the optical properties of the scattering centers. Number densities of particles may also be determined readily for monodisperse systems but require use of the index of refraction for the scattering centers. (ii) We define procedures for making estimates of the count-median diameters and standard deviations in log-normal size distributions using (a) measured half-widths of scattered laser-power spectra and (b) measured ratios of detectabilities determined for a given laser at two different scattering angels. We find that it is not possible to achieve adequate sensitivity for experimental utilization of this technique. (iii) We describe techniques for estimating the count-median diameters and standard deviations in log-normal size distributions using (a) measured half-widths of scattered laser-power spectra and (b) measured ratios of detectabilities determined at the same or at different scattering angles for two different laser wavelengths. Our calculations indicate that these procedures will also not yield adequate sensitivities for the determinations of particle-size distributions. (iv) The use of two linearly polarized beams [one polarized in the plane defined by the direction of laser-beam propagation and the vertical direction (1) and the other defined by the direction of laser-beam propagation and the horizontal (2)] in making half-width and detectability measurements has been studied. Although acceptable sensitivities for the determination of σ g and ♂r are achievable for selected particle-size ranges, the use of polarized laser radiation will not always lead to satisfactory characterization of log-normal particle-size distributions. Experiments involving detectabilities do not offer clear-cut advantages over those involving half-widths insofar as measurement sensitivities are concerned. (v) We describe procedures for determination of the total number of particles contained in the log-normal particle-size distributions using measured absolute values for the detectabilities observed for the power spectra of scattered laser radiation, after prior definition of the count-median diameters and standard deviations in log-normal size distributions.

Measurements of particle size distribution parameters in ferrofluids

A method is presented by means of which (for a ferrofluid with a lognormal distribution of particle size) it is possible to determine the standard deviation (σ) and median particle diameter (D <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</inf> ) from the room temperature magnetisation curve. The method has been applied to several commercial ferrofluids, containing ferrite particles, and to fluids consisting of cobalt particles in toluene prepared by the method of Hess and Parker. The particle size distributions have also been measured by using an electron microscope. It has been found that the magnetic data gives estimates of the 'magnetic size distribution' parameters D <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vm</inf> and σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> , which differed from the 'physical size distribution' parameters D <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vp</inf> and σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> obtained from electron microscope data. It is found that D <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vp</inf> > D <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vm</inf> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma_{p} &lt; \sigma_{m}</tex> . These observations are consistent with the results of Kaiser and Miskolczy and Granqvist and Buhrman.

New Method for Characterizing Dissolution Properties of Drug Powders

Various multiparticulate dissolution models that assume a log-normal particle-size distribution are fitted by nonlinear least-squares regression to data from the dissolution of micronized glyburide. Estimates of parameters describing the effective initial particle-size distribution are obtained, together with estimates of the specific dissolution rate parameter. A dissolution equation based on an ideal, untruncated, log-normal distribution, with the single particles dissolving according to the Hixson-Crowell cube root law, is the best model. The dissolution behavior of glyburide can be well described by this model in terms of the specific dissolution rate parameter and one other parameter accounting for the distribution effect. The estimation of these two parameters represents a more exact way of describing the dissolution characteristics of drug powders than previous approaches. The method should be of interest in the quality control of drugs that may cause bioavailability problems because of dissolution rate-limited absorption.

The effect of a particle size distribution on the coercovity and remanence of a fine particle system

Experimental measurements of the coercivity and remanence of a fine particle system have been interpreted as arising from a lognormal particle size distribution. Good agreement is obtained between theory and experiment with KV p/ kT a constant ≈13, where K is the anisotropy constant, and V p the critical volume for superparamagnetic behaviour.

Model Systems for Dissolution of Finely Divided (Multisized) Drug Powders

Several reports have dealt with model systems for the dissolution of log-normally distributed powders. A numerical solution is presented for the previously published Higuchi-Hiestand equations for a log-normal particle-size distribution. This solution was realized via an application of the System 360 Continuous System Modeling Program (CSMP). The resulting computer-aided calculations were utilized in the comparisons between the Higuchi-Hiestand model and other existing models. These comparisons provided a basis for the development of an adaptation of the Nielsen moving sphere model for log-normally distributed powders. The adaptation of the Nielsen equation for multisized particle systems is suggested as being potentially useful for treating experimental data where hydrodynamic effects must be considered.

Log-normal size distributions from magnetization measurements on small superconducting Al particles

Field-dependent magnetization of superconducting fine Al particles has been measured down to 0.5 K with a SQUID system. The results can be explained in considerable detail by the existence of a log-normal distribution of particle size.

Dissolution Patterns of Polydisperse Powders: Oxalic Acid Dihydrate

AbstractThe dissolution of oxalic acid dihydrate crystals of log-normal particle-size distribution in 0.1 N HCI was studied. A biphasic cube root dependence was found; the slopes of the initial cube root plots were consistent with theory based on dissolution of isometric, isotropic particles where assumptions were made of: (a) sink conditions, (b) particle-size-independent solubility, and (c) particle-size-independent film thickness of adsorbed liquid layers.

Interpretation of diffusion battery data

The graphical method by which the parameters of a lognormal distribution are estimated from experimental data on the penetration of aerosols through a diffusion battery has been simplified by consolidating the numerous families of theoretical penetration curves into a single family showing penetration as a function of medium diffusion parameter for several values of standard deviation. Curves and tables of data are given for batteries having ducts of either rectangular or circular cross-section. Empirical relationship between particle diameter and diffusivity, with which observed distribution statistics can be converted to statistics for lognormal particle size distributions, are provided.

Fragmentation of Uranium Dioxide after Molten Uranium Dioxide-Sodium Interaction

A series of 30 runs were performed to gain information on the fragmentation of molten UO2when dropped into a bath of liquid sodium. In this experiment, hollow cylindrical UO2 pellets were heated with electric current fed to a tungsten rod, passed through to center hole. The resulting molten UO2 dripped down its fall into the sodium bath. High-speed cinematography (500~2,000 frames/sec) was utilized to observe the fragmentation of the UO2 droplets upon its impact with the sodium surface. Records were taken of the changes in pressure and temperature induced in the sodium bath by the impact of the U2 droplets. The physical characteristics of the solid U2 particles resulting from the quench were examined for particle size distribution, particle shape and surface condition, as well as microstructure, to gain an insight into the mechanism of fragmentation. It is concluded from the study that, for the purpose of Fuel Coolant Interaction (FCI) analysis, the U2 particles can be assumed to be spherical in shape and to have log-normal particle-size distribution. The density function in terms of particle weight, representing the finest particle-size distribution, can be expressed on the basis of presently available empirical data, by , D: Diameter of U2 particle in microns.

Small angle X-ray scattering investigation of platinum metal dispersions on alumina catalysts

The metal particle size diameters and particle size distributions of platinum supported on alumina catalysts have been investigated using small angle X-ray scattering techniques. The interfering background scattering from the micropores of the alumina support was eliminated by adsorption of small concentrations of CH2I2 and C4H9I, organic liquids of electron density similar to that of the alumina. Although CH2I2 is twice as effective as C4H9I in reducing the pore scattering from the η-alumina support, care must be exercised to prevent an excess of organic material which will cause unwanted scattering. The primary parameters derived from the scattering of X-rays by the platinum crystallites are RG, the Guinier radius, and RP, the Porod radius. These radii are expressed as the ratio of moments of a distribution which is related to σ2 and μ, the variance and geometric mean, respectively, of a log-normal particle size distribution function. Using this simple procedure, we find that both the mean particle diameter and the particle size distribution of an aged 0.62 wt % Pt on alumina catalyst have shifted to larger sizes. In the experimental intensity curves of a 0.35 wt % Pt catalyst and that of the fresh 0.62 wt % Pt catalyst, shoulders are observed that are more or less pronounced. This behavior indicates interparticle scattering effects, suggesting the presence of small clusters of platinum particles approximately 200 Å in diameter before any agglomeration has ocurred.

Sampling distributions of surface and mass statistics of a log-normal distribution when estimated by the method of weighted frequencies

Theoretical sampling distributions have been derived for surface and mass statistics when they are estimated from a random sample of a log-normal particle size distribution by weighting the frequencies observed at a given size with the corresponding diameter raised to the appropriate power. The theoretical results, which predict a systematic bias of statistics calculated in this way, are compared with computer-generated “experimental” distributions. A method is given for deriving theoretical correction factors with which to adjust the observed median diameters and the adjusted distributions are shown to be very similar to those observed for median diameters calculated from the Hatch-Choate equation. The significance of the results with respect to other distributions is discussed.

Carbide growth in a molybdenum TZC alloy

A study of the rate of growth of the carbide phase in a molybdenum-TZC alloy has been investigated over the range 1200 to 1500° C. A constant log-normal particle size distribution was maintained over the time-temperature spectrum explored. Particle growth kinetics were found which could be related to either diffusion or interface controlled processes. It is concluded that quantitative metallographic techniques are inadequate to distinguish between these mechanisms in complex alloy systems.

Spray-Congealed Formulations of Sulfaethidole (SETD) and Waxes for Prolonged-Release Medication: Effect of Modifiers

An investigation was undertaken to examine the effects of the addition of certain substances—viz., low molecular weight polyethylene, ethylcellulose 20 cps., and glyceryl ester of hydrogenated rosin, to carnauba wax, hydrogenated castor oil, and synthetic wax-like ester, on the in vitro dissolution patterns of spray-congealed particles of sulfaethidole (SETD) and wax-modifier combinations. The pneumatically atomized and congealed particles of the drug-wax-modifier suspensions were observed to possess log-normal particle-size distributions. These were characteristic for a particular wax. The addition of low molecular weight polyethylene was observed to increase the particle size significantly for all three wax formulations. The in vitro dissolution patterns appear to be second order, being functions of the effective surface area and the weight of drug available for dissolution. The dissolution patterns of SETD from the three wax formulations were observed to be characteristic for each wax and dissolution medium. The addition of low molecular weight polyethylene was observed to retard the dissolution rates of SETD from hydrogenated castor oil and synthetic wax-like ester formulations. The addition of ethylcellulose tended to increase the dissolution rate from carnauba wax and synthetic wax-like ester formulations. The addition of glyceryl ester of hydrogenated rosin tended to decrease the dissolution rate of the drug from carnauba wax and hydrogenated castor oil formulations.