Fact-checked by Grok 2 weeks ago

Particle-size distribution

Particle-size distribution (PSD) is a set of numbers or a mathematical function that characterizes the relative amounts of particles present in a powder, granular material, or dispersion according to their sizes, often expressed as the equivalent spherical diameter of particles that settle at the same velocity.^[1] It can be represented as a number distribution (based on particle count), volume distribution, mass distribution, or surface area distribution, providing a comprehensive profile of particle dispersity within a sample.^[2] PSD analysis is fundamental across disciplines, as it quantifies how particle sizes range from nanometers to millimeters, influencing the overall behavior and performance of particulate systems.^[3] The importance of PSD lies in its direct impact on material properties and process outcomes; for instance, in pharmaceuticals, narrower distributions enhance drug dissolution rates, bioavailability, and content uniformity in dosage forms like tablets and aerosols.^[4] In materials science and engineering, PSD governs flowability, packing density, mechanical strength, reactivity, and thermal/electrical conductivity of powders used in ceramics, metal processing, and composites.^[1] Additionally, in environmental and aerosol science, it affects particle settling, filtration efficiency, and air quality control, with lognormal or bimodal distributions commonly observed in natural and industrial dispersions.^[2] Variations in PSD can lead to inconsistencies in product quality, such as emulsion instability or reduced therapeutic efficacy, underscoring the need for precise control during manufacturing.^[3] PSD is measured using standardized techniques tailored to particle size ranges and material types, including sieving for coarse particles (20 µm to 125 mm), laser light diffraction for broad ranges (0.04 µm to 8000 µm), gravitational sedimentation for fine particles (50 nm to 300 µm), and microscopy methods like scanning electron microscopy for high-resolution imaging (0.1 µm to 1000 µm).^[1] These methods rely on principles such as light scattering, electrical sensing, or image analysis, with accuracy depending on factors like refractive index, dispersion stability, and representative sampling to avoid errors from agglomeration or shape irregularities.^[3] Calibration with reference materials, such as NIST Standard Reference Materials, ensures reliability, enabling applications in quality control, regulatory compliance, and research across industries.^[1]

Fundamentals

Definition

Particle-size distribution (PSD) refers to the statistical representation of the relative amounts of particles present in a sample across a range of sizes, typically expressed as a function of particle diameter or volume.^[5] It describes the proportion of particles by number, volume, or mass within specified size intervals, providing a measure of the sample's polydispersity unless the material is perfectly monodisperse, where all particles are identical in size.^[5] Particle size itself is often defined in terms of an equivalent spherical diameter, which is the diameter of a sphere that has the same volume, mass, or other relevant property as the actual particle.^[5] For non-spherical particles, such as those found in irregular shapes like needles or plates, this equivalent diameter serves as a standardized metric to approximate dimensions.^[5] PSD thus quantifies the inherent heterogeneity in materials like powders, suspensions, aerosols, and granular substances, where variations in size influence physical and chemical behaviors.^[1] The concept of PSD originated in the early 20th century within soil science, where it was used to classify soil components such as sand, silt, and clay based on settling rates in fluids, and in metallurgy to evaluate powder properties for industrial processes.^[1] Key developments included the work of Andreasen in the 1920s, who introduced methods for constructing cumulative particle size curves through sedimentation analysis.^[6] Basic graphical representations of PSD include histograms, which depict frequency distributions showing the number or volume of particles at discrete size intervals; density plots, illustrating the probability density function for continuous size variations; and cumulative distribution functions (CDFs), which plot the cumulative percentage of particles smaller than a given size.^[5]^[7] These visuals provide an intuitive overview of the distribution's shape and spread.^[5]

Nomenclature

In particle size analysis, the fundamental measure of particle size is the particle diameter (d), defined as the diameter of a hypothetical sphere that exhibits the same physical behavior as the actual particle in a given measurement context, known as the equivalent spherical diameter.^[8] Key characteristic diameters include the median diameter (d_{50} or x_{50}), which represents the particle size at which 50% of the cumulative distribution lies below and 50% above, dividing the population into equal halves by volume, number, or other weighting.^[9]^[10] Percentile notations use subscripts to indicate the cumulative percentage, such as d_{10} for the size below which 10% of particles fall and d_{90} for the size below which 90% fall; these are widely used to describe the range and tails of distributions.^[11]^[12] Various mean diameters quantify central tendency, with the arithmetic mean (number-weighted, D[1,0]) calculated as the simple average of individual particle diameters, suitable for number-based distributions. The geometric mean accounts for logarithmic scaling in skewed distributions, defined as the exponential of the average logarithm of diameters, while the volume-weighted mean (D[4,3] or De Brouckere mean) emphasizes larger particles by weighting by volume, given by \sum n_i d_i^4 / \sum n_i d_i^3, where n_i is the number of particles of diameter d_i.^[10]^[13] The modal diameter denotes the size class with the highest frequency, corresponding to the peak in the differential distribution curve.^[14] Distribution descriptors include the span, a width metric calculated as (d_{90} - d_{10}) / d_{50}, providing a normalized measure of spread independent of the median size.^[12] The uniformity coefficient (C_u), often d_{60}/d_{10}, assesses the relative breadth for granular materials, with values near 1 indicating high uniformity.^[15] Skewness quantifies asymmetry, positive for right-tailed distributions (finer particles dominant) and negative for left-tailed ones (coarser particles dominant), calculated via the third standardized moment.^[16]^[17] Particle sizes are commonly expressed on logarithmic scales due to the frequent log-normal nature of distributions, with units spanning nanometers (nm) for ultrafine particles to micrometers (μm) or millimeters (mm) for coarser ones.^[18] These notations follow international conventions outlined in the ISO 9276 series, first established in the late 1990s (e.g., ISO 9276-1:1998) and updated through the 2020s (e.g., ISO 9276-2:2014, ISO 9276-1:2025), standardizing terminology for moments, means, and graphical representations to ensure interoperability in analysis.^[19]^[20]^[10]

Significance and Applications

General Importance

Particle size distribution (PSD) profoundly impacts the physical properties of particulate materials, dictating behaviors such as flowability, packing density, reactivity, and dissolution rates. Narrower PSDs typically enhance flowability by minimizing variations in particle interactions and reducing frictional forces between particles, leading to more predictable handling in bulk systems.^[21] In contrast, broader or optimized PSDs, such as bimodal distributions, improve packing density by allowing smaller particles to fill voids between larger ones, potentially increasing density by up to 35% in cementitious materials.^[22] Smaller particles within a distribution elevate surface area-to-volume ratios, accelerating chemical reactivity in processes like catalysis or pozzolanic reactions and boosting dissolution rates through enhanced solvent contact.^[23]^[24] The economic and process implications of PSD control are substantial in manufacturing, where uniform distributions ensure consistent quality and minimize inefficiencies. Non-uniform PSDs can cause variations in product performance, higher energy demands during size reduction, and waste from off-specification batches; for instance, in cement production, suboptimal PSDs increase grinding costs and reduce output efficiency.^[25] Tailored PSD engineering in such processes can cut material usage by 20-25% without compromising strength, yielding direct savings in production costs and environmental impact.^[26] Similarly, in pigment manufacturing, precise PSD management optimizes dispersion and coverage, preventing defects that lead to rework and material loss.^[14] PSD's interdisciplinary relevance underscores its role as a fundamental parameter in processes like rheology, filtration, and sedimentation. In rheology, PSD breadth influences suspension viscosity and shear-thinning effects, with narrower distributions often yielding higher shear sensitivity in dense slurries.^[27] For filtration, particle size variations affect cake permeability and overall throughput, as heterogeneous distributions can increase resistance and clogging risks.^[28] In sedimentation, PSD determines particle settling dynamics, where polydisperse systems exhibit complex velocity profiles that impact separation efficiency in environmental and industrial flows.^[29] In powder systems, PSD width directly modulates compressibility and bulk density; for example, shifting from a uniform to an optimized distribution can enhance initial packing and reduce compressibility variability, with reported bulk density improvements of 20-35% depending on material type and process conditions.^[22]^[30]

Environmental and Health Contexts

Particle size distribution (PSD) plays a critical role in classifying particulate matter (PM) in environmental monitoring, where PM is categorized based on aerodynamic diameter: PM10 includes particles up to 10 μm that can enter the upper respiratory tract, PM2.5 encompasses fine particles up to 2.5 μm capable of penetrating deep into the lungs, and PM1 refers to submicron particles up to 1 μm that pose risks similar to or greater than PM2.5 due to their ability to reach alveolar regions.^[31]^[32] Fine particles smaller than 2.5 μm are particularly hazardous, as they can penetrate the lung's alveolar walls, leading to respiratory diseases such as asthma exacerbations, chronic obstructive pulmonary disease, and increased cardiovascular risks including stroke and heart attacks.^[31]^[33] The PSD of airborne particles significantly influences the performance of pollution control technologies, such as electrostatic precipitators (ESPs) and high-efficiency particulate air (HEPA) filters, which exhibit varying collection efficiencies based on particle size. ESPs achieve higher efficiencies for larger particles (e.g., >1 μm) due to enhanced charging and collection mechanisms, often reaching 95-99% removal for particles above 1 μm, while struggling with submicron sizes below 0.5 μm where diffusion dominates.^[34] HEPA filters, by contrast, are designed to capture at least 99.97% of particles around 0.3 μm—the most penetrating size—with efficiency increasing for both smaller (via diffusion) and larger (via impaction and interception) particles.^[35] In the human respiratory tract, PSD determines deposition patterns: particles in the 0.5-1 μm range experience significant deposition (up to approximately 50% total under moderate breathing conditions) primarily through sedimentation and impaction in the tracheobronchial and alveolar regions, facilitating direct exposure to sensitive lung tissues.^[36] In atmospheric aerosols, PSD is essential for understanding radiative forcing, as the scattering and absorption of solar radiation vary with particle size, composition, and distribution, contributing to climate effects estimated at -0.5 to -1.0 W/m² globally from anthropogenic aerosols.^[37] Ultrafine particles smaller than 0.1 μm, often formed via nucleation processes, play a key role in cloud nucleation by serving as cloud condensation nuclei (CCN), altering cloud droplet size distributions and reflectivity, which can enhance or suppress precipitation and influence regional climate patterns.^[38] Recent developments in the 2020s, including the U.S. Environmental Protection Agency's (EPA) 2024 revision lowering the annual PM2.5 standard to 9.0 μg/m³ (which remains in effect as of November 2025, pending a reconsideration process announced in March 2025), have heightened emphasis on advanced monitoring techniques, such as real-time PSD analysis in urban areas to better capture size-specific pollution dynamics and inform targeted interventions for air quality improvement.^[39]^[40]

Industrial and Scientific Applications

In materials engineering, particle size distribution (PSD) plays a critical role in optimizing the properties of ceramics, metals, and composites during processing. For ceramics, bimodal PSDs enhance sintering efficiency by improving powder packing density and reducing porosity, leading to higher flexural strength and lower shrinkage compared to unimodal distributions. In metal and composite fabrication, such as powder injection molding, bimodal distributions promote denser green bodies and final sintered parts. In binder jetting additive manufacturing, studies have shown improvements of up to 8.2% in packing density, 10.5% in flowability, and 4.0% in sintered density for bimodal mixtures versus single-mode powders.^[41] These optimizations are particularly valuable for achieving enhanced toughness in composites, where fine particles fill voids between larger ones, thereby improving mechanical integrity without compromising overall sinterability. In the pharmaceutical industry, PSD directly influences drug bioavailability and formulation performance, with narrow distributions ensuring uniform dissolution rates and consistent therapeutic effects. For oral solids, coarser particles may slow dissolution and reduce absorption, while finer ones accelerate it, necessitating PSD control to meet bioequivalence standards under 21 CFR Part 320, which recognizes particle size as a key determinant of bioavailability. The U.S. Food and Drug Administration (FDA) mandates PSD characterization in specifications for new drug substances and products per the Q6A guidance, requiring testing at release and acceptance criteria for critical applications like injectables, where particles exceeding 10 µm or 25 µm limits can pose risks, as outlined in USP <788>. This requirement, emphasized in FDA guidelines from the early 2000s and reaffirmed in subsequent updates, ensures product quality and safety for injectable formulations.^[42] Nanotechnology leverages precise PSD control for nanoparticles under 100 nm, where size variations induce quantum effects that alter optical, electrical, and catalytic properties, enabling tailored applications in drug delivery and catalysis. In drug delivery systems, nanoparticles with controlled PSDs (typically 1–100 nm) optimize biodistribution, cellular uptake, and reduced toxicity, as smaller sizes enhance tumor penetration but may increase unintended accumulation in healthy tissues. For catalysts, PSD tuning below 10 nm maximizes surface area and activity, with quantum confinement effects influencing reactivity, as seen in gold nanoparticles for CO oxidation.^[43] These advancements, detailed in reviews of precision nanoparticle engineering, underscore PSD's role in mitigating toxicity while boosting efficacy in biomedical and industrial catalysis. In food processing and agriculture, PSD analysis ensures product quality and environmental management. During flour milling, controlled PSD yields uniform particle sizes that affect baking performance, with whole wheat flours exhibiting bimodal distributions peaking at 20–25 µm for fine fractions and larger aggregates up to 100 µm, influencing dough rheology and starch gelatinization. Roller milling produces narrower PSDs than stone milling, promoting consistent texture and nutritional retention in end products like bread. In agriculture, soil PSD is essential for predicting erosion rates, as finer particles (e.g., silt and clay fractions) increase sediment transport and deposition, with median diameters (d50) serving as key predictors in dynamic models of runoff and wind erosion. This informs soil management practices to minimize farmland degradation on regions like the Loess Plateau. As of 2025, emerging applications include AI-assisted PSD optimization in precision milling for improved food functionality.^[44]

Types of Distributions

Frequency and Cumulative Distributions

Particle size distributions are commonly represented using frequency distributions, which describe the proportion of particles within specific size intervals or bins. These distributions can be based on number (counting individual particles equally), volume (weighting by particle volume, proportional to the cube of the diameter), or mass (equivalent to volume for uniform density materials). For instance, a number-based frequency distribution highlights the prevalence of smaller particles, while a volume- or mass-based one emphasizes larger particles due to their greater contribution to the total.^[45] Frequency distributions are typically visualized as histograms, with particle size along the x-axis (often on a logarithmic scale for wide-ranging sizes) and the percentage or fraction in each bin along the y-axis. Cumulative distributions provide an alternative representation by plotting the accumulated fraction of particles as a function of size, offering a smoother curve that facilitates percentile calculations such as the median (D50) or the sizes below which 10% (D10) or 90% (D90) of the material lies. The undersize cumulative distribution, denoted Q(x), shows the fraction of particles smaller than size x, rising from 0 to 1 as x increases, while the oversize distribution (1 - Q(x)) depicts the fraction larger than x, often used in contexts like sieving where retention is key. These are plotted with size on the x-axis (linear or logarithmic) and cumulative fraction (0 to 100%) on the y-axis, allowing easy assessment of distribution breadth and skewness.^[45] The relationship between frequency and cumulative distributions is integral, enabling conversion between them for data analysis. Specifically, the frequency density function f(x), which represents the relative amount of material at size x, is the derivative of the undersize cumulative distribution:

f(x) = \frac{dQ(x)}{dx}

Conversely, the cumulative can be obtained by integrating the frequency distribution. This mathematical linkage allows histograms to be transformed into cumulative curves and vice versa, aiding in the comparison of data from different measurement techniques.^[45] In practice, cumulative distributions are often plotted on probability graph paper to linearize certain empirical forms, such as log-normal distributions, where the logarithm of size versus the cumulative probability yields a straight line for easier parameter estimation.^[45]

Common Distribution Shapes

Particle size distributions (PSDs) are commonly classified based on the number of peaks in their frequency curves, distinguishing monomodal distributions, which exhibit a single prominent peak, from multimodal distributions featuring two or more distinct peaks. Monomodal distributions often arise in controlled processes yielding relatively uniform particle populations, such as the milling of pharmaceutical powders to produce narrow, Gaussian-like profiles for polystyrene microspheres or fat emulsions in the 200–300 nm range.^[46] In contrast, multimodal distributions are prevalent in heterogeneous systems, including colloidal assemblies where aggregation creates bimodal patterns with peaks at primary particle sizes (0.1–4 μm) and flocculi (4–20 μm), or in mixed latex suspensions measured by laser diffraction.^[47] These multiple modes reflect subpopulations formed by varying formation mechanisms, complicating analysis in techniques like dynamic light scattering for aggregated gold nanoparticles.^[48] Symmetry in PSD shapes further characterizes their form, with symmetric distributions displaying balanced tails around the central tendency, akin to ideal normal curves where mean, median, and mode coincide, as seen in some spherical particle systems like quartz beads or constrained pharmaceutical fits.^[46] Skewed distributions, however, predominate in natural materials, often positively skewed with elongated fine tails due to sorting processes; for instance, fluvial sediments and soils frequently show right-skewed profiles, where 8 of 13 soil samples exhibited non-normal asymmetry across size fractions like sand.^[49]^[50] In river sands and aggregates, broad, skewed multimodal shapes emerge from bimodal gravel-fine mixtures, with the median size dividing the distribution amid peaks in sand-to-pea gravel ranges.^[51] Log-normal distributions, a skewed monomodal form common in aerosols and powders, arise from multiplicative growth or breakup processes that yield Gaussian patterns on a logarithmic scale, with geometric standard deviations around 2.0 observed in atmospheric particles and nano-scale fumed silica.^[52]^[7] Process factors significantly influence these shapes: grinding in ball mills can narrow distributions to monomodal by reducing larger particles, while wet conditions promote finer, less skewed outcomes compared to dry grinding; agglomeration broadens PSDs into multimodal forms by clustering primaries into secondary particles, as in crystal formation; and classification via air classifiers refines shapes toward uniform monomodal profiles by separating sizes based on aerodynamic response.^[53]^[54]^[55] Cumulative representations of these shapes, such as sigmoidal curves for monomodal versus stepped profiles for multimodal, aid in visualizing overall spread.^[46]

Sampling Methods

Principles of Representative Sampling

Representative sampling is essential for accurately determining particle-size distribution (PSD) in heterogeneous materials, where variability arises from differences in particle size, shape, composition, and spatial distribution. Pierre Gy's Theory of Sampling (TOS) provides the foundational framework, identifying the fundamental sampling error (FSE) as the irreducible variance due to the material's inherent heterogeneity, particularly influenced by particle size differences.^[56] The FSE variance is given by the equation:

V(FSE) = K \frac{d_N^3}{M_S}

where M_S is the sample mass, d_N is the nominal (maximum) particle size, and K is a heterogeneity constant incorporating factors such as the granulometric factor g (reflecting PSD variability, typically 0.25 for broad distributions), shape factor f (0.2–0.5), mineralogical composition factor c, and liberation factor l.^[56] This formula quantifies how larger particles disproportionately increase sampling variance, emphasizing the need to either reduce d_N through comminution or increase M_S to achieve desired precision (e.g., relative standard deviation of 5–15%).^[57] Representativeness requires that the sample mirrors the population's PSD across all size fractions, free from systematic biases introduced by segregation, where particles separate by size, density, or shape during handling (e.g., finer particles settling or coarser ones rolling to the surface).^[1] To ensure this, sampling must follow "golden rules" such as capturing material in motion across the entire stream in short, equal increments, using tools like flat-edged scoops to avoid preferential selection of coarse particles, and incorporating randomization to minimize operator bias.^[1] Segregation can be mitigated by thorough mixing prior to sampling or by employing riffler devices that divide streams proportionally, thereby preserving the original size fractions.^[1] Statistical considerations dictate minimum sample sizes to achieve acceptable error levels, often scaled to the largest particle size for adequate representation of rare large particles. In practice, standards like ASTM D75 specify minimum masses based on nominal maximum particle size (e.g., 25–50 kg for aggregates up to 75 mm), ensuring statistical reliability for PSD analysis.^[1] For microscopy-based methods, counting approximately 2000 particles yields ±5% precision at 90% confidence, highlighting the need for larger samples in heterogeneous lots.^[1] Challenges arise particularly with cohesive or sticky particles in suspensions, where agglomeration alters effective PSD by clumping fines, leading to underrepresentation of smaller sizes and increased variance.^[1] Such materials exhibit poor flowability (e.g., particles <2 µm), complicating random extraction and promoting segregation; mitigation involves dispersants (e.g., sodium polyacrylate) and ultrasonication (e.g., 3-minute cycles at 30 W) to achieve well-dispersed states without fracturing or overheating.^[1] In Gy's framework, these issues amplify the compositional and distributional heterogeneity components of K, necessitating pre-sampling homogenization to restore representativeness.^[57]

Collection Techniques

Collection techniques for particle-size distribution (PSD) analysis focus on acquiring representative samples from various media, such as powders, soils, suspensions, and aerosols, while adhering to established protocols to ensure accuracy. These methods are designed to capture the inherent variability in particle populations without introducing bias, often guided by the need to minimize sampling errors as outlined in representative sampling principles, including international standards like ISO 14488 for particulate materials sampling and sample splitting.^[58]^[1] Dry sampling is commonly employed for granular materials like powders and soils, where techniques such as scooping, riffling, and rotary sample division are utilized to obtain subsamples. Scooping involves manually extracting portions from a bulk pile using a shovel or scoop, ensuring even distribution to avoid segregation of larger particles. Riffling divides the sample by passing it through a series of chutes that alternate the flow into multiple bins, promoting homogeneity. Rotary sample dividers rotate the material through a spinning mechanism to split it evenly into sectors, which is particularly effective for fine powders to achieve precise subsampling. These methods are detailed in guidelines for optimizing sampling plans in dry particulate systems.^[59] Wet sampling addresses suspensions and slurries, where dilution or centrifugation prevents agglomeration and facilitates representative extraction. In slurry dilution, the sample is dispersed in a liquid medium, often with dispersants like sodium hexametaphosphate, to reduce concentration and enable uniform mixing before subsampling. Centrifugation separates particles by size and density, allowing collection of specific fractions for PSD evaluation, though care is taken to avoid altering the original distribution. For aerosols, cascade impactors collect particles by inertial impaction across multiple stages with decreasing nozzle sizes, fractionating them into size bins from coarse to fine (typically 0.5–10 µm). These impactors, such as multi-stage designs operating at controlled flow rates, provide size-resolved samples for subsequent analysis.^[60]^[61] In-situ sampling occurs directly at the source, such as probe insertion into pipes or real-time emission monitors, contrasting with laboratory methods that involve transported samples. Probe sampling in pipelines uses isokinetic probes to extract particles at the same velocity as the flow, minimizing distortion in size distribution for suspended solids in gases or liquids. Real-time monitors, often deployed in emission stacks, enable continuous collection without relocation, preserving transient PSD characteristics. Laboratory sampling, by comparison, relies on transported aliquots but risks alteration due to settling or evaporation during transit.^[62]^[63] Standardized procedures ensure consistency across applications, with ASTM D4057 providing guidelines for manual sampling of petroleum and related products, including coning and quartering for semi-solids and solids. Coning and quartering involves forming a conical pile, flattening it, and discarding alternate quarters to reduce sample volume while maintaining representativeness; this method was updated in the 2022 revision (D4057-22) of the standard to incorporate modern equipment and safety considerations. These standards emphasize equipment calibration and procedural rigor to support reliable PSD studies.^[64]

Measurement Techniques

Sieving and Mechanical Methods

Sieving methods represent one of the oldest and most straightforward mechanical techniques for determining particle-size distribution (PSD), particularly effective for coarser granular materials. In sieve analysis, a sample is passed through a stack of test sieves featuring progressively smaller apertures, typically ranging from 10 mm down to 45 μm, constructed from woven wire cloth or perforated metal plates compliant with standards such as ISO 3310-1. The assembly, including a lid and receiver pan, is subjected to mechanical agitation—such as horizontal circular motion, tapping, or vertical throw-action sieving—to facilitate particle passage based on their effective diameter relative to the mesh openings.^[65] This process fractionates the material by mass, with particles retained on each sieve corresponding to specific size intervals, allowing for the separation of aggregates like soils, aggregates, or powders.^[66] Wet sieving adapts this approach for fine, cohesive, or moisture-sensitive materials that tend to agglomerate during dry processing, enhancing dispersion and reducing clogging. The sample is wetted with a liquid (often water) and sieved using a spray nozzle or overflow to rinse particles through the stack until the effluent runs clear, after which residues are dried and weighed.^[65] This method is particularly useful for samples containing clays or organic matter, as the liquid medium minimizes adhesion forces and improves throughput for particles down to approximately 20 μm.^[67] Procedures for wet sieving are outlined in ISO 2591-1, which specifies controlled liquid addition to avoid altering particle properties. Despite their simplicity and cost-effectiveness, sieving methods have notable limitations, especially for finer particles. Accuracy diminishes below 50 μm due to sieve blinding—where particles wedge into apertures causing incomplete separation—and electrostatic or cohesive forces that hinder passage.^[67] Overloading sieves (e.g., exceeding 200 g per 200 mm diameter sieve for 2 mm apertures) or insufficient agitation can lead to sieving losses greater than 1%, necessitating repetition.^[65] The ISO 2591-1 standard from 1988 addresses these issues by defining apparatus, sieving durations (e.g., until less than 0.1% mass passes in 1 minute), and maximum particle sizes to prevent damage, but it recognizes that results are influenced by material friability and humidity. The primary output from sieving is mass-based fractions retained on each sieve, expressed as percentages of the total sample mass, which can be plotted as frequency distributions or converted to cumulative PSD curves for further analysis.^[65] For particles finer than 50 μm, sieving is often complemented by sedimentation techniques to achieve a complete PSD profile.^[1]

Sedimentation and Elutriation

Sedimentation methods for determining particle-size distribution (PSD) rely on the gravitational settling of particles in a fluid, where the terminal settling velocity is governed by Stokes' law. This law describes the velocity v of a spherical particle as v = \frac{(\rho_p - \rho_f) g d^2}{18 \eta}, where \rho_p is the particle density, \rho_f is the fluid density, g is gravitational acceleration, d is the particle diameter, and \eta is the fluid viscosity.^[68] By measuring the concentration of particles at various depths or times during settling, the PSD can be constructed from the cumulative distribution of settling velocities. These techniques are particularly suited for particles in the mid-range sizes, typically 1–100 μm, as larger particles settle too quickly and smaller ones too slowly for practical measurement.^[69] In gravimetric sedimentation, such as the Andreasen pipette method, a dilute suspension of particles (around 1% by weight) is prepared in a liquid medium, allowed to settle in a vertical tube, and aliquots are withdrawn at predetermined depths and times using a pipette to measure the mass of settled material.^[70] This incremental sampling assumes spherical particles and laminar flow conditions, enabling calculation of the percentage of particles finer than specific sizes based on Stokes' law. Volumetric setups, like those using X-ray absorption to monitor suspension density continuously without sampling, provide automated equivalents, improving precision and reducing manual error in PSD determination.^[68] Elutriation classifies particles by introducing a counterflow of air or liquid that opposes gravitational settling, separating fractions based on whether their terminal velocity exceeds the fluid velocity. In cyclone elutriators, particles enter a conical chamber where swirling fluid motion enhances separation; finer particles follow the upward flow and exit, while coarser ones settle or are collected separately, allowing fractionation into size classes.^[69] Devices like the Cyclosizer use multiple cyclones in series with liquid elutriation to achieve rapid PSD analysis across discrete size ranges.^[71] For non-spherical particles, which deviate from ideal settling behavior, corrections to Stokes' law incorporate shape factors such as aspect ratios—the ratio of particle length to width—to adjust the effective diameter and predict actual velocities more accurately.^[72] These adjustments are essential for materials like soils or minerals, where irregular shapes can alter settling rates by up to 30–50% compared to spheres.^[69] Sedimentation techniques originated in the 1920s for analyzing soil and cement particle sizes, with the Andreasen pipette method introduced by A.H.M. Andreasen in 1928 as a standardized gravimetric approach.^[73] Modern advancements, including automated X-ray sedimentation systems like the SediGraph introduced in the 1970s and refined through the 2000s with autosamplers, have enhanced throughput and reproducibility for industrial applications.^[74]

Optical and Laser-Based Methods

Optical and laser-based methods for particle-size distribution (PSD) analysis rely on the interaction of light with particles to infer size and, in some cases, shape information, enabling rapid, non-destructive measurements suitable for a wide range of materials. These techniques encompass laser diffraction, which examines light scattering patterns, and image-based approaches, such as microscopy and dynamic imaging, which directly visualize particles. They are particularly valued in industries like pharmaceuticals, materials science, and environmental monitoring for their speed and ability to handle both dry powders and liquid dispersions.^[75] Laser diffraction is a cornerstone of these methods, where a laser beam illuminates a dilute suspension or aerosol of particles, producing a scattering pattern that detectors capture across various angles. The underlying principle is Mie theory, which mathematically describes how electromagnetic waves scatter from spherical particles, correlating the scattering angle inversely with particle size—larger particles scatter light at smaller angles, while smaller ones scatter at larger angles. Developed by Gustav Mie in 1908, this theory provides an exact solution for scattering by homogeneous spheres and is implemented in software to invert the measured diffraction pattern into a volume-based PSD.^[76] Instruments like the Malvern Mastersizer series apply this approach, achieving measurements over a broad range from approximately 0.1 μm to 3 mm with high resolution and repeatability.^[77] These systems typically require sample dilution to minimize multiple scattering, ensuring accurate single-scattering assumptions inherent to Mie theory.^[75] Photoanalysis, often termed image analysis in modern contexts, complements laser diffraction by providing direct morphological data alongside size distribution through high-resolution imaging. Static methods use optical or electron microscopy to capture particle images, followed by automated software that segments and measures features like Feret diameters or projected areas to derive PSDs, excelling in revealing shape irregularities that diffraction methods assume as spherical. Dynamic image analysis extends this by flowing particles through an illuminated cell, where a high-speed camera records silhouettes against a backlight, allowing real-time counting and sizing of thousands of particles per second for high-throughput applications.^[78]^[79] Interferometric variants enhance precision by analyzing phase shifts in light waves interfered by particles, enabling sub-micrometer resolution for dilute samples, though they are less common for routine PSD due to complexity.^[80] Examples include systems like the FlowCam or QICPIC, which integrate such imaging for both size and shape metrics in suspensions or powders.^[81] These methods offer significant advantages, including rapid analysis times (often under a minute) and non-destructive testing, making them ideal for quality control in high-volume production. However, limitations arise in opaque or highly concentrated suspensions, where multiple scattering distorts patterns in laser diffraction, necessitating dilution that may alter fragile samples. Image-based techniques, while superior for shape, suffer from lower throughput for very fine particles below 1 μm and potential biases from overlapping particles in dense flows. For dilute samples, brief cross-references to electrical sensing zones can provide complementary counting data, but optical methods dominate for their versatility.^[82]^[83]^[84]

Electrical and Acoustic Methods

Electrical and acoustic methods for particle-size distribution (PSD) analysis rely on detecting changes in electrical resistance or sound wave propagation caused by particles in suspensions, offering advantages for concentrated or opaque samples where optical techniques may fail. These approaches enable non-invasive measurements in liquids, providing insights into particle volumes or dimensions through signal perturbations. The electroresistance method, based on the Coulter principle, measures particle size by detecting transient changes in electrical resistance as particles pass through a small aperture in an electrolyte solution. Developed by Wallace H. Coulter in 1953, the technique involves suspending particles in a conductive fluid and drawing the suspension through a narrow orifice between two electrodes; each particle displaces an equivalent volume of electrolyte, generating a voltage pulse whose height is proportional to the particle's volume.^[85] This pulse height distribution is then converted to a size distribution, typically covering particles from 0.4 to 1600 μm in diameter, depending on aperture size.^[86] The method excels in biotech applications, such as sizing blood cells and microorganisms, where it supports rapid, automated counting and volume analysis in clinical hematology.^[86] Acoustic spectroscopy determines PSD by analyzing the attenuation of ultrasound waves propagating through a particle suspension, where attenuation spectra encode information about particle size via scattering and absorption mechanisms. In this technique, broadband ultrasound (1–100 MHz) is transmitted through the sample, and the frequency-dependent attenuation is measured; for particles larger than 5–7 μm, scattering dominates, while visco-inertial effects prevail for smaller sizes.^[87] The spectra are fitted to theoretical models, such as the Faran model for elastic scatterers, to invert the data and obtain the size distribution, accommodating concentrations up to 60 vol.% without dilution.^[88] This method spans particle sizes from 1 nm to 1000 μm and is particularly useful for opaque or high-concentration dispersions in industries like pharmaceuticals and paints.^[87] Laser obscuration time (LOT), also known as time-of-transition (TOT), sizes particles by measuring the duration a focused, rotating laser beam is obscured as it scans across individual particles in a flowing suspension. In the TOT implementation, a helium-neon laser beam, rotated at constant angular velocity by a prism, interacts with particles; the time the photodiode detects beam interruption is proportional to the particle's chord length, from which equivalent spherical diameter is calculated assuming known flow velocity.^[89] This single-particle approach yields number- and volume-based distributions, effective for sizes from 0.5 to 150 μm, and is validated against optical methods for broader accuracy in suspensions.^[89] Calibration of these methods commonly employs monodisperse polystyrene spheres as size standards, traceable to NIST reference materials, to verify instrument response and establish size-to-signal correlations across the measurement range. For instance, NIST SRM 1964 provides certified spheres for validating PSD instruments, ensuring reproducibility in applications like cell sizing in biotechnology.^[90]

Specialized Techniques for Aerosols and Nanoscale

For measuring ultrafine aerosols, the Scanning Mobility Particle Sizer (SMPS) is a widely adopted instrument that classifies particles based on their electrical mobility using a differential mobility analyzer (DMA), typically covering a size range from 1 nm to 1 μm.^[91] The DMA separates charged particles in an electric field according to their mobility diameter, with singly charged particles selected and counted downstream by a condensation particle counter (CPC), enabling high-resolution size distribution profiles essential for atmospheric and combustion aerosol studies.^[92] This technique, originating from foundational work on aerosol classification, provides number concentration distributions with logarithmic binning across up to 64 sizes in minutes.^[93] Condensation particle counters complement SMPS systems by detecting and enumerating particles as small as 2.5 nm through vapor condensation onto the particles, growing them to optically detectable sizes for laser-based counting.^[94] In aerosol applications, CPCs quantify total particle number concentration, often integrated with mobility analyzers to resolve sub-10 nm ultrafine particles in real-time, which is critical for assessing nucleation processes in polluted air or engine exhausts.^[95] At the nanoscale, Dynamic Light Scattering (DLS) determines particle size distributions by analyzing fluctuations in scattered laser light caused by Brownian motion of suspended nanoparticles.^[96] The diffusion coefficient D relates to hydrodynamic diameter d via the Stokes-Einstein equation:

D = \frac{kT}{3 \pi \eta d}

where k is Boltzmann's constant, T is temperature, and \eta is solvent viscosity, allowing non-invasive sizing of polydisperse samples in liquids from 1 nm to 1 μm with intensity-weighted averages.^[97] For dry or surface-bound nanoscale particles, Atomic Force Microscopy (AFM) provides direct topographic imaging by raster-scanning a sharp tip over the sample, yielding height-based size distributions with sub-nanometer resolution for individual particles or aggregates.^[98] AFM excels in characterizing irregular shapes and surface features of nanoparticles, such as those in nanocomposites, by measuring vertical profiles on substrates.^[99] In emissions monitoring, real-time optical counters assess particulate matter (PM) size distributions in industrial stacks using light-scattering principles to detect and size particles in hot, dilute gas streams, aligning with U.S. EPA Performance Specification 11 (PS-11) for continuous emission monitoring systems (CEMS).^[100] These instruments, often employing nephelometry or forward-scattering optics, provide mass-based PM concentrations and coarse size fractions (e.g., PM10, PM2.5) for compliance, with dual-sensor designs enhancing accuracy in variable flow conditions.^[101] Recent advancements since 2020 integrate artificial intelligence (AI) into these instruments for enhanced data processing, such as automated inversion of raw mobility spectra in SMPS or noise reduction in DLS autocorrelation functions, improving resolution and reducing operator bias in complex aerosol distributions.^[102] AI-driven estimators, like neural networks, accelerate particle size distribution retrieval from optical counter data, achieving sub-minute analysis for dynamic processes such as pharmaceutical crystallization or environmental monitoring.^[103]

Mathematical Models

Probability Distribution Frameworks

Particle size distributions (PSDs) are often modeled using probability density functions (PDFs), which describe the relative likelihood of particles occurring within specific size intervals. The PDF, denoted as f(x), satisfies the normalization condition \int_{-\infty}^{\infty} f(x) \, dx = 1, ensuring the total probability across all sizes is unity. In PSD contexts, these functions can represent different physical quantities: number density, where f(x) gives the fraction of particles by count in size bin dx; surface density, proportional to x^2 f(x) for spherical particles; or volume density, proportional to x^3 f(x), which is particularly relevant for mass-based analyses in powders and suspensions.^[104] Common probabilistic frameworks for PSDs include the lognormal, gamma, and Weibull distributions, each suited to underlying formation mechanisms. The lognormal distribution arises naturally from multiplicative growth processes, such as successive fragmentations or condensations in aerosols and nanoparticles, where size variations accumulate logarithmically.^[105] The gamma distribution captures skewed distributions in systems like precipitation particles or milled powders, offering flexibility through shape and scale parameters to model multimodal behaviors. Similarly, the Weibull distribution models breakage-dominated processes in grinding and crushing, emphasizing tail behaviors for larger particles. Parameters for these distributions, such as the mean \mu and standard deviation \sigma in lognormal or gamma cases, are typically estimated using the method of moments or maximum likelihood estimation (MLE). The method of moments equates sample moments (e.g., mean and variance) to theoretical ones for quick, closed-form solutions, while MLE maximizes the likelihood of observed data under the model, providing asymptotically efficient estimates especially for grouped PSD data from sieving or optical measurements.^[106] These approaches assume particle sizes are independent and identically distributed, an idealization often violated in real aggregates where clustering introduces correlations and alters effective distributions.^[107]

Specific Models and Parameterization

The Rosin-Rammler distribution, also known as the Weibull distribution in particle size analysis, models the cumulative mass fraction undersize Q(x) as Q(x) = 1 - \exp\left(-\left(\frac{x}{X_c}\right)^n\right), where x is the particle size, X_c is the characteristic particle size (the size at which 63.2% of the mass is undersize), and n is the uniformity index representing the spread of the distribution.^[108] This model was originally developed for describing the fineness of powdered coal and has since become widely adopted in engineering for simulating particle size distributions in grinding and comminution processes, such as in cement production and mineral processing, due to its ability to capture the skewed nature of fragmented materials.^[109] The parameters X_c and n are typically estimated by linearizing the equation in logarithmic form, \ln(-\ln(1 - Q(x))) = n \ln(x) - n \ln(X_c), and fitting to sieve or sedimentation data, with higher n values indicating narrower distributions suitable for controlled milling operations.^[108] The Gates-Gaudin-Schuhmann model provides a power-law representation for the cumulative undersize fraction Q(x) = \left(\frac{x}{X_m}\right)^m, where X_m is the maximum particle size and m is the distribution modulus (typically between 0.5 and 2 for crushed materials), offering a simpler alternative for broad, coarse distributions.^[110] Developed for analyzing size distributions in ore crushing, this model excels in mineral engineering applications like beneficiation and heap leaching, where it approximates the straight-line behavior on log-log plots of cumulative percentage versus size, facilitating quick predictions of yield and energy requirements in breakage processes.^[111] Its parameters are determined by plotting \log Q(x) against \log x, yielding a slope of m and intercept related to X_m, making it particularly useful for heterogeneous feeds in industrial grinding circuits.^[112] Parameter estimation for these models commonly employs least-squares fitting to minimize the sum of squared residuals between observed cumulative fractions from experimental data (e.g., sieve analysis) and the model predictions, often after linear transformation to enable straightforward regression. Goodness-of-fit is assessed using the chi-square statistic \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}, where O_i and E_i are observed and expected frequencies in size bins, respectively; low \chi^2 values relative to degrees of freedom indicate reliable parameterization for engineering simulations.^[113] In modern engineering simulations, particularly for complex polydisperse systems like pharmaceutical powders or environmental aerosols, extensions to these models incorporate multimodal mixtures, combining multiple Rosin-Rammler or Gates-Gaudin-Schuhmann components weighted by mixing proportions to capture bimodal or trimodal peaks observed in advanced laser diffraction measurements.

References

[1]
[PDF] NIST recommended practice guide : particle size characterization
measurements can be conducted with some degree of reliability when using techniques such as microscopy-based analysis. In these cases, the particles.<|control11|><|separator|>
[2]
Particle and Size Distribution - PMC - NIH
Distribution of particle size also means the dispersity of particles, which represents the number or mass percentage of particles at different size within the ...
[3]
Particle Size Distribution - an overview | ScienceDirect Topics
Particle size distribution refers to the classification of particles based on their sizes, a critical parameter analyzed using techniques.
[4]
[PDF] A comprehensive review of particle size analysis techniques
Feb 7, 2024 · Abstract. In this review, we will discuss particle size analysis as it relates to specific pharmaceutical applications.
[5]
[PDF] A basic guide to particle characterization - Montana State University
particle size distribution, which means that the presence of oversized particles can dominate the particle size result. Instrumentation. A conventional ...
[6]
Incremental methods of particle size determination - SpringerLink
The particle size distribution of a fine powder may be determined by ... Andreasen, A.H.M. (1928), Kolloid Beith, 27, 405. Google Scholar. Andreasen ...
[7]
Particle Size Distribution Functions:
A log-normal distribution is natural to a system where particles are formed or broken up in accordance with some power of particle size such as the volume or ...
[8]
https://www.iso.org/obp/ui/#iso:std:iso:9276:-1:en
[9]
Understanding & Interpreting Particle Size Distribution Calculations
For particle size distributions the median is called the D50 (or x50 when following certain ISO guidelines). The D50 is the size in microns that splits the ...
[10]
[PDF] INTERNATIONAL STANDARD ISO 9276-2
May 15, 2014 · For both notation systems, a terminology of specific mean particle sizes is inserted in the corresponding clauses: Clause 4 and Clause 5, ...
[11]
Analysis of Particle Size Distribution - Microtrac
A particle size distribution indicates the percentage of particles of a certain size (or in a certain size interval). These intervals are also called size ...
[12]
D90, D50, D10, and span – for DLS? - Malvern Panalytical
Aug 25, 2025 · The span of a volume-based size distribution is defined as Span = (D90 – D10)/D50 and gives an indication of how far the 10 percent and 90 ...Missing: coefficient | Show results with:coefficient
[13]
Particle size distribution | Anton Paar Wiki
Particle size is one of the most important characteristics of particulate materials. It directly affects many material properties, from the accessibility of ...
[14]
Particle Size Distribution And Its Measurement | Agno Pharmaceuticals
The modal value is that size where the majority of particles are located. The median is value at which 50% (by whatever is being reported, mass for instance) ...
[15]
Understanding Sieve Analysis and Uniformity Coefficient (UC) in ...
Apr 23, 2025 · D10, D50, and D90 Values: D10: The particle size at which 10% of the material passes. Indicates the smallest particles that dominate filtration ...Missing: nomenclature span
[16]
The Language of Particle Size
Feb 1, 2011 · This discussion introduces the common values encountered on a typical particle size report and provides references for deeper mathematical interpretations.
[17]
Influence of particle size distribution skewness on dust explosibility
Skewness indicates the presence of a tail of fines or a tail of coarse particles. It is suggested that skewness should be reported to better describe PSD.
[18]
An Introduction to Particle Characterization - AZoM
Oct 7, 2019 · The number or arithmetic mean is purely the average value. It is frequently denoted as D1,0. Other means take into account volume weighting and ...Missing: nomenclature | Show results with:nomenclature
[19]
ISO 9276-1:1998 - Representation of results of particle size analysis
General information ; Status. : Withdrawn ; Publication date. : 1998-06 ; Stage. : Withdrawal of International Standard [95.99] ; Edition. : 2 ; Number of pages. : 9.Missing: history | Show results with:history
[20]
ISO 9276-1:2025 - Representation of results of particle size analysis
2–5 day deliveryThis document specifies guidelines and instructions for the graphical representation of particle size analysis data in histograms, distribution densities ...Missing: history | Show results with:history
[21]
The effects of particle size distribution on the rheological properties ...
Besides chemistry, one of the fundamental powder characteristics is the particle size distribution (PSD).
[22]
The effects of particle size distribution and surface area upon ...
Jan 10, 2009 · For example, Aiqin et al. [6] reported that allowing a wider PSD of the Portland cement can result in 35% greater packing density of the cement ...Missing: variation percentage
[23]
Innovative assessment of reactivity in fly ash: The role of particle size ...
Results revealed that particle size distribution had a more significant impact on reaction activity than amorphous phase content or average particle size alone.
[24]
Effect of particle size on solubility, dissolution rate, and oral ... - NIH
The particle size effect on dissolution was clearly influenced by the diffusion coefficients of the various dissolution media, and the dissolution velocity of ...
[25]
Particle size analysis reduces cement manufacturing costs
Reducing the particle size of the cement too much, within limits, tends to simply improve its quality, but an overly coarse product will fail to meet the ...Missing: pigments | Show results with:pigments
[26]
Optimization of cement quantity through the engineering of particle ...
This paper presents a novel approach to achieve enhanced efficiency through the engineering of particle size distribution.Missing: inefficiencies pigments
[27]
Effect of Particle Size Distribution on the Rheology of Dispersed ...
The degree of shear thinning depends on the breadth of particle size distribution, with the narrowest distribution suspension exhibiting the highest degree of ...
[28]
The influence of particle(s) size, shape and distribution on cake ...
Jun 11, 2021 · The review reveals that the particle characteristics can significantly impact the filterability, which subsequently affects the filtration rate.
[29]
The Importance of Particle Size Distribution on the Performance of ...
This appendix discusses and demonstrates the importance of suspended sediment particle size distributions on the removal effectiveness of sedimentation ...
[30]
Effect of Particle Size Distribution on Powder Packing and Sintering ...
The use of bimodal powder mixtures improves the powder's packing density (8.2%) and flowability (10.5%), and increases the sintered density (4.0%) while also ...
[31]
Particulate Matter (PM) Basics | US EPA
May 30, 2025 · Of these, particles less than 2.5 micrometers in diameter, also known as fine particles or PM2.5, pose the greatest risk to health. Fine ...
[32]
What are the WHO Air quality guidelines?
Sep 22, 2021 · Exposure to PM2.5 can cause diseases both to our cardiovascular and respiratory system, provoking, for example stroke, lung cancer and chronic ...Missing: PM1 | Show results with:PM1
[33]
Air pollution is responsible for 6.7 million premature deaths every year
The health risks associated with particulate matter of less than 10 and 2.5 microns in diameter (PM10 and PM2.5) are especially well documented. PM is capable ...Health impacts · Exposure & health impacts of... · Climate impacts of air pollutionMissing: PM1 | Show results with:PM1
[34]
[PDF] Chapter 3 Section 6 - Electrostatic Precipitators
The size distribution describes how many particles of a given size there are, which is important because ESP efficiency varies with particle size. In.
[35]
What is a HEPA filter? | US EPA
Aug 15, 2025 · Using the worst case particle size results in the worst case efficiency rating (i.e. 99.97% or better for all particle sizes). All air cleaners ...Missing: distribution | Show results with:distribution<|control11|><|separator|>
[36]
Deposition efficiency of inhaled particles (15-5000 nm) related to ...
Apr 8, 2017 · Individuals with high deposition of a certain particle size generally had high deposition for all particles <3500 nm. The individual variability ...Statistical Analysis · Airway Anatomy (pc 2) · Abbreviations
[37]
[PDF] Clouds and Aerosols - Intergovernmental Panel on Climate Change
Luo, 2009: Simulation of particle size distribution with a global aerosol model: Contribution of nucleation to aerosol and CCN number concentrations. Atmos ...
[38]
[PDF] Efficiency of cloud condensation nuclei formation from ultrafine ...
Feb 27, 2007 · The likelihood that an ultrafine particle will grow to CCN size depends on the competition between the rates of growth and removal processes.
[39]
Managing Air Quality - Ambient Air Monitoring | US EPA
Jun 5, 2025 · AirNow provides easy access to real-time and forecast air quality information using the Air Quality Index (AQI), which helps you understand ...Missing: 2020s PSD
[40]
https://www.theenergylawblog.com/2025/03/articles/industrial-project-development/epa-to-reconsider-previous-administrations-pm-2-5-naaqs-continuing-its-deregulatory-push/
[41]
[PDF] Particle Size Analysis in Pharmaceutics and Other Industries
MODEL DISTRIBUTIONS. Particle size distributions may take many forms, but there are a small number of model distributions which are of particular interest ...
[42]
Measurement of the Size Distribution of Multimodal Colloidal ...
May 25, 2021 · In this study, we assessed the accuracy of LD for the measurement of the modal diameter of both single and mixed populations of polystyrene particles.<|control11|><|separator|>
[43]
Multimodal Distribution - an overview | ScienceDirect Topics
Multimodal distributions refer to probability distributions that have multiple peaks, indicating that there are several values at which the probability function ...
[44]
Particle Size Distribution of Various Soil Materials Measured ... - MDPI
Apr 28, 2021 · Particle size distribution is an important soil parameter—therefore precise measurement of this characteristic is essential.
[45]
Statistical comparisons of sediment particle size distributions
Oct 1, 2021 · This is a logarithmic scale because natural particle size distributions are often highly positively skewed. All computing used the ...
[46]
Characterization of Fluvial Sediment (Appendix B) - River Dynamics
Apr 30, 2020 · If the distribution has two or more peaks, it is multimodal (e.g., bimodal or trimodal) (Figure B.1). The median particle size divides the ...
[47]
[PDF] Particle Size Distributions: Theory and Application to Aerosols ...
Nov 3, 2002 · This document describes mathematical and computational considerations pertaining to size distributions. The application of statistical ...
[48]
Influence of dry and wet grinding conditions on fineness and shape ...
Aug 7, 2025 · The particle size distribution width was lowered by using smaller grinding balls in wet condition and larger grinding balls in dry condition.<|control11|><|separator|>
[49]
Prevention of Crystal Agglomeration: Mechanisms, Factors ... - MDPI
The formation of agglomerates usually lowers product purity and generates a broad particle size distribution. This review focuses on preventing agglomeration in ...
[50]
Factors Affecting the Performance of Air Classifier Mill
Nov 5, 2024 · The rotational speed of classifying wheel is a critical factor that directly affects the particle size distribution. Higher speed can increase ...
[51]
[PDF] Understanding the components of the fundamental sampling error
The approach to sampling taken by Pierre. Gy provides an excellent inferential relationship between the mass of material to be sampled, the size of the ...
[52]
[PDF] Correct Sampling Using the Theories of Pierre Gy
Pierre Gy's theory addresses seven types of sampling error and offers proven techniques for their minimization. The seven major categories of sampling error ...
[53]
[PDF] Design of Optimum Sampling Plans: 1- Dry Powders
There are several sampling tools and methods that may be used to extract the primary sample. The choice of particular method depends on many factors including ...
[54]
[PDF] Sampling for Particle Size Analysis - HORIBA
This approach is only recommended for samples with particles size < 20-40 μm and without a large variation in density, size, or shape. A soil sample was sent ...
[55]
[PDF] A Personal Cascade Impactor: Design, Evaluation and Calibration
Jun 4, 2010 · Evaluations of the impactor with monodisperse particles in the size range of. 0.48 to 23 µm have shown that the impactor has sharp cut-off ...
[56]
Development of a Sampling Probe for ... - AMS Journals
Apr 1, 2018 · Comparison of aspiration ratio between single-shrouded probe and double-shrouded probe according to particle size and freestream velocity.
[57]
Particle Size Distribution | METTLER TOLEDO Technology
Particle size distribution information is critical for the analysis, control and optimization of a variety of processes, such as crystallization (and ...
[58]
D4057 Standard Practice for Manual Sampling of Petroleum and ...
Jun 16, 2022 · ASTM D4057 covers manual sampling of liquid, semi-liquid, and solid petroleum products, including free water, from various sources into primary ...
[59]
[PDF] The Basic Principles of Sieve Analysis - Retsch
The particle size distribution is defined via the mass or volume. Sieve analysis is used to divide the granular material into size fractions and then to ...
[60]
D6913/D6913M Standard Test Methods for Particle-Size Distribution ...
This test method separates soil particles by size using sieves to determine particle-size distribution (gradation) between 3-in and No. 200 sieves.
[61]
[PDF] 〈786〉 Particle Size Distribution Estimation by Analytical Sieving ...
Nov 22, 2016 · Sieving is a method for classifying powders by particle size, suitable for particles larger than 75 μm, and is a two-dimensional estimate.
[62]
Micromeritics SediGraph - Particle Size Analyzer - Malvern Panalytical
The SediGraph particle size distribution analyzer offers inorganic particle size distribution analysis by x-ray sedimentation. Find out more here.
[63]
Elutriation - an overview | ScienceDirect Topics
Strictly, an elutriator is a classifier rather than a particle sizer. It divides the group of particles into those above and those below a given cut size. In ...
[64]
127 Determination of Particle Size Distribution in Flour by the ...
Mar 14, 2018 · The Andreasen method measures particle size in cereal flours by assuming particles sediment based on size, using Stoke's Law, and measuring ...
[65]
[PDF] cyclosizer instruction manual - AusIMM
For particle size analysis, elutriation and sedimentation techniques are well- known procedures designed to give a sizing distribution on the basis of.
[66]
Impacts of particle shape on sedimentation of particles - ScienceDirect
Terminal settling velocity decreases with increased aspect ratio. Rectangular particles settle slower than ellipses, and circular particles differ ...
[67]
GRAIN SIZE OF WHITEWARE CLAYS AS DETERMINED BY THE ...
Sci., 12,. 306-21 (1922). ... The Andreasen sedimentation apparatus con- stitutes a refinement of the ordinary pipette method of particle-size determination.<|control11|><|separator|>
[68]
SediGraph - Micromeritics
The Micromeritics SediGraph continues to be a prefered standard instrument for particle size analysis in many laboratories throughout the world.
[69]
Using laser diffraction to measure particle size and distribution
... particle size and particle size distribution. It exploits the Mie theory of light, which relates the scattering pattern produced as light passes through a ...Missing: seminal papers
[70]
[PDF] Mie theory the first 100 years - ATA Scientific
The resulting Mie theory has various uses, one of the most important industrial applications being laser diffraction particle size analysis. Laser diffraction ...
[71]
Mastersizer | Laser Diffraction Particle Size Analyzers | Malvern ...
Mastersizer 3000+ Pro · Particle size range from 0.1 to 2500 µm · Automated dispersion · Advanced method development · Intelligent data quality assessment.Mastersizer 3000+ Ultra · Mastersizer 3000+ Lab · Mastersizer 3000+ Pro
[72]
Image Analysis of Particles - HORIBA
The major steps for image analysis of particles. First, high quality images are acquired, then the particles (or second phase for an emulsion) are identified.
[73]
Dynamic Image Analysis - How it Works - Vision Analytical
Aug 13, 2025 · Learn about the principles of operation behind Dynamic Image Analysis. Discover our particle shape and particle size analyzers.
[74]
Particle sizing using laser interferometry - Optica Publishing Group
It has been shown that particle size information is inherent in the scattered light when a single particle passes through the probe volume of a laser ...
[75]
Dynamic Image Analysis - Sympatec
Dynamic Image Analysis. With laser diffraction the particle size distribution is determined by the characteristic diffraction pattern of a particle collective.
[76]
What are the advantages and disadvantages of different particle ...
Jul 13, 2023 · Laser diffraction, Advantages: simple to use, fast analysis, wide measurement range, good repeatability and accuracy, optional sampling ...
[77]
Key Advantages and Challenges in Laser Diffraction for Particle ...
Mar 15, 2019 · Laser diffraction advantages include wide range and rapid results. Challenges include sampling errors and the need for visual confirmation.
[78]
Particle size analysis methods: Laser Diffraction vs. Dynamic Image ...
This review compares Laser Diffraction (LD) and Dynamic Image Analysis (DIA), two commonly used particle size analysis techniques.
[79]
Means for counting particles suspended in a fluid - Google Patents
A means for minimizing the effect of detector sensitivity on the number or size of particles detected is to provide means for sharply confining the electric ...
[80]
Coulter Principle, Counting and Sizing Particles
Wallace H. Coulter developed a technology for counting and sizing particles using impedance measurements.Missing: seminal paper<|separator|>
[81]
Acoustic / Particle size | 3P Instruments
The particle size is determined in proportion to mass by measuring the attenuation of ultrasonic waves in concentrated dispersions.
[82]
Polydisperse particle size characterization by ultrasonic attenuation ...
Ultrasonic attenuation spectroscopy offers advantages over other spectrometric methods for the determination of suspension and emulsion size distributions.
[83]
(PDF) Variations in Grain Size Analysis with a Time‐of‐Transition ...
Aug 6, 2025 · This study deals with grain size analysis with a Laser Sizer Galai CIS-50. This device utilizes the time-of-transition method and is ...Missing: TOT | Show results with:TOT
[84]
[PDF] Standard Reference Material 1964 Polystyrene Spheres
Apr 6, 2022 · CERTIFICATE OF ANALYSIS. Purpose: This Standard Reference Material (SRM) is intended for the calibration/validation of particle sizing.
[85]
[PDF] Scanning Mobility Particle Sizer (SMPS) Instrument Handbook
Scanning Mobility Particle Sizer (SMPS). Instrument Handbook. U.S. Department of Energy, Atmospheric. Radiation Measurement user facility, Richland, Washington.
[86]
Scanning Mobility Particle Sizer (SMPS)
Principle of Operation. The Scanning Mobility Particle Sizer is based on the principal of the mobility of a charged particle in an electric field.Missing: seminal paper
[87]
Condensation Particle Counters - TSI Incorporated
This Condensation Particle Counter (CPC) is the long-standing reference for counting ultrafine particles down to 2.5 nm. TSI CPCs are the most referenced ...
[88]
[PDF] fundamentals of condensation - particle counters (cpc) and
Oct 30, 2014 · Condensation Particle Counters. (CPCs) are used to measure total number concentration of an aerosol, while Scanning Mobility. Particle Sizer ( ...
[89]
Understanding Dynamic Light Scattering Theory - Wyatt Technology
Learn the theory behind how dynamic light scattering (DLS) measures the Brownian motion of molecules and particles to determine size and size distributions.
[90]
An Introduction to Dynamic Light Scattering (DLS)
Nov 4, 2010 · Brownian Motion. DLS measures Brownian motion and relates this to the size of the particles. Brownian motion is the random movement of particles ...
[91]
Size Measurement of Nanoparticles Using Atomic Force Microscopy
This assay protocol outlines the procedures for sample preparation of gold and the determination of nanoparticle size using atomic force microscopy (AFM).
[92]
Atomic force microscopy analysis of nanoparticles in non-ideal ...
Aug 30, 2011 · If the isolated nanoparticles of spherical shape are deposited on an ideally flat substrate, their size can be determined easily from the AFM ...
[93]
Performance Specification 11 for Particulate Matter | US EPA
Apr 21, 2025 · Performance Specification 11 is for particulate matter continuous emission monitoring systems at stationary sources, and includes ...
[94]
[PDF] Thermo Scientific PM CEMS: A dual-sensing technology
The dual-method PM CEMS uses real-time light-scattering (nephelometry) and the semi-continuous TEOM monitor, combining fast response with direct mass ...
[95]
Harnessing AI and Automation for Advanced Particle Size Analysis
Aug 12, 2024 · To address this, we have integrated AI and automation solutions to enable all users to obtain high-quality data without needing extensive ...
[96]
Accelerating particle size distribution estimation | MIT News
Sep 23, 2024 · MIT researchers have dramatically accelerated particle size distribution estimation, improving on a novel AI-based estimator for medication ...
[97]
Properties of the Log-Normal Particle Size Distribution
The log-normal distribution is a function distributing a dependent variable in a nor- mal or Gaussian fashion on a logarithmic.Missing: seminal | Show results with:seminal
[98]
Lognormal Size Distributions in Particle Growth Processes without ...
Mar 16, 1998 · A new model is proposed to explain lognormal particle size distributions found in vapor growth processes such as gas evaporation, ...Missing: seminal | Show results with:seminal
[99]
Particle size analysis utilizing grouped data and the log-normal ...
A maximum likelihood method for fitting and testing the fit of a log-normal function to grouped particle size data is described.Missing: seminal paper
[100]
On the relation between number-size distribution and the fractal ...
Aug 5, 2025 · If the probability of fragmentation is independent of fragment diameter, then the exponent may be identified with the boundary dimension, DB, of ...
[101]
Mean Particle Diameters. Part VII. The Rosin‐Rammler Size ...
Mar 5, 2013 · The Rosin-Rammler particle size distribution is used for a broad range of applications. Physical and statistical properties of the Rosin-Rammler ...
[102]
Empirical energy and size distribution model for predicting single ...
Sep 1, 2021 · Rosin and Rammler, 1933. P. Rosin, E. Rammler. Laws governing the fineness of powdered coal. J. Inst. Fuel, 7 (1933), pp. 29-36. Google Scholar.
[103]
Principles of Comminution, I-Size Distribution and Surface ...
Schuhmann Principles of Comminution, I-Size Distribution and Surface Calculations. The American Institute of Mining, Metallurgical, and Petroleum Engineers, ...Missing: particle | Show results with:particle
[104]
Particle Size Distribution Models for Metallurgical Coke Grinding ...
Six different particle size distribution (Gates–Gaudin–Schuhmann (GGS), Rosin–Rammler (RR), Lognormal, Normal, Gamma, and Swebrec) models were compared
[105]
AMIT 135: Lesson 3 Particle Size Distribution
The mean particle size of a distribution is typically measured by either the arithmetic or geometric mean of the maximum (dmax) and minimum (dmin) particle ...Missing: frequency | Show results with:frequency
[106]
[PDF] HORIBA - Technical Note
A lower value for either Chi square or R parameter indicates a better fit of the raw data to the calculated particle size distribution. The Chi square ...
[107]
Performance Comparison of Different Particle Size Distribution ...
Nov 17, 2022 · This paper compared the PSD prediction ability of three types of mathematical model. We selected nine models that have been proven to accurately predict sample ...