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Particle size

Particle size refers to the geometric or physical dimension of a discrete unit of matter, typically quantified as the diameter of an equivalent sphere that shares the same volume, mass, surface area, or settling velocity as the actual particle, which simplifies characterization of irregularly shaped particles across a wide range of materials and systems.^[1] This concept is fundamental in fields such as materials science, environmental monitoring, pharmaceuticals, and soil analysis, where particle dimensions typically span from nanometers (e.g., 10 nm) to millimeters (e.g., 1 mm), influencing key properties like reactivity, stability, and transport behavior.^[1] Various equivalent diameters are used depending on the measurement context, including the sieve diameter (smallest opening a particle can pass through), the Stokes diameter (based on settling velocity in a fluid), and the aerodynamic diameter (relevant for airborne particles).^[1]^[2] The distribution of particle sizes within a sample—often described by metrics such as the median diameter (d50) or percentiles (d10, d90)—is equally important, as most real-world powders and suspensions are polydisperse, exhibiting a range of sizes that affects overall material performance.^[1] Common characterization techniques include sieving for larger particles (20 µm to 125 mm), laser diffraction for broad ranges (0.04 µm to 8000 µm), dynamic light scattering for submicron particles, and microscopy for shape-inclusive analysis (0.01 µm to 1000 µm), each relying on principles like gravitational settling, light scattering, or direct imaging.^[1] Accurate measurement requires careful sampling and dispersion to avoid agglomeration, with standards like NIST Standard Reference Materials ensuring reproducibility.^[1] In practical applications, particle size profoundly impacts product quality and safety; for instance, in pharmaceuticals, smaller particles enhance drug dissolution and bioavailability but may alter stability and encapsulation efficiency in nanocarriers.^[3] In environmental science, fine particles like PM2.5 (less than 2.5 µm in diameter) pose significant health risks by penetrating deep into the lungs, while larger particles (PM10, up to 10 µm) affect air quality and ecosystem transport.^[4] Similarly, in ceramics and soils, optimal size distributions improve packing density, hydraulic properties, and nutrient retention, underscoring the need for tailored analysis in industrial processes.^[1]

Fundamentals

Definition

Particle size refers to the dimensions of individual solid particles (such as flecks), liquid particles (such as droplets), or gaseous particles (such as bubbles) within dispersed systems, including colloids, aerosols, and granular materials.^[5] This measure captures the characteristic length scales of these discrete entities, typically expressed as an equivalent diameter assuming a spherical shape for simplicity in non-spherical cases.^[5] Unlike bulk properties—such as the overall density, volume, or flow behavior of an aggregate material—particle size pertains exclusively to the attributes of single particles, independent of their collective arrangement or interactions in the system.^[5] The concept emerged in the 19th century through foundational work in colloidal chemistry, particularly Thomas Graham's 1861 investigations into diffusion rates, which distinguished colloidal substances by their particulate nature and slow diffusion compared to true solutions.^[6] Particle size is commonly quantified using metric units aligned with the International System of Units (SI), such as the micrometer (μm, where 1 μm = 10^{-6} m) for particles in the range of 1 to 1000 μm, the nanometer (nm, where 1 nm = 10^{-9} m) for nanoscale particles below 1 μm, and the millimeter (mm, where 1 mm = 10^{-3} m) for larger granules.^[7] While focused on individual particles, the term often relates to particle size distribution when characterizing populations in practical applications.

Importance

Particle size profoundly influences the physical properties of materials, primarily through the surface area-to-volume ratio, which increases dramatically as particles become smaller, enhancing reactivity and dissolution rates in processes such as catalysis and drug release. For instance, in catalytic applications, nanoparticles with high surface areas facilitate greater interaction with reactants, improving efficiency.^[8] Smaller particles also exhibit higher diffusion rates due to Brownian motion dominance over gravitational settling, while larger ones sediment more rapidly according to Stokes' law, affecting suspension stability and transport in fluids.^[9] In industrial contexts, particle size control is essential for optimizing product performance across sectors. In pharmaceuticals, nanoscale particles enhance drug solubility and bioavailability, enabling targeted delivery systems that improve therapeutic efficacy while minimizing side effects.^[10] In materials science, such as powder metallurgy, uniform particle sizes ensure better sintering and mechanical strength in final components.^[11] Similarly, in food processing, the size of emulsion droplets determines stability and texture, preventing separation and ensuring consistent quality in products like mayonnaise.^[12] Environmentally, particle size governs the fate and impact of pollutants, with fine aerosols under 10 μm (PM10) capable of penetrating deep into the respiratory system, causing inflammation, cardiovascular issues, and exacerbated asthma.^[13] In water treatment, controlling particle size through flocculation promotes aggregation of suspended solids for efficient removal, safeguarding water quality.^[14] Ecologically, finer soil particles facilitate nutrient cycling by increasing adsorption sites for minerals, supporting microbial activity and plant uptake essential for ecosystem health. Economically, precise particle size management in manufacturing reduces operational costs and waste; for example, in cement production, optimizing the size distribution lowers energy consumption during grinding by up to 20% and enhances product uniformity, boosting overall profitability.^[15] This control minimizes defects and rework, directly contributing to higher yields and compliance with quality standards across industries.^[16]

Measurement Techniques

Direct Methods

Direct methods for measuring particle size involve the visual observation and imaging of individual particles, providing direct quantification of their dimensions and shapes without relying on inferred physical properties. These techniques are particularly valuable for applications requiring detailed morphological information, such as in pharmaceuticals, materials science, and environmental monitoring.^[8] Optical microscopy serves as a foundational direct method suitable for particles larger than approximately 1 μm, leveraging visible light to capture high-contrast images for manual or automated sizing. The resolving power of optical microscopes limits effective characterization to particles around 1 μm and greater, as smaller sizes approach the diffraction limit of light, typically around 0.2 μm under optimal conditions.^[17] This technique allows for the assessment of particle morphology, including length, width, and aspect ratio, often using a calibrated eyepiece micrometer for precise measurements.^[18] For submicron particles below 1 μm, electron microscopy techniques such as scanning electron microscopy (SEM) and transmission electron microscopy (TEM) offer superior resolution, enabling imaging down to the nanometer scale. SEM provides surface topography details with resolutions as fine as 1 nm, making it ideal for dry, non-volatile samples where particle size and shape can be directly measured from two-dimensional projections.^[19] TEM, by contrast, transmits electrons through ultra-thin samples to reveal internal structures, achieving sub-nanometer resolution for particles as small as a few nanometers, though sample preparation often requires embedding or sectioning.^[20] Both methods typically involve image analysis software to quantify diameters, perimeters, and other geometric parameters from captured micrographs.^[21] Dynamic image analysis extends direct imaging to flowing or suspended particles, employing high-speed cameras to capture thousands of images per second for real-time size and shape evaluation. Systems like the CAMSIZER from Microtrac analyze dry powders dispersed by vibration, covering a size range from 30 μm to 5 mm, and provide metrics such as equivalent circle diameter and aspect ratio through automated processing of particle silhouettes.^[22] Similarly, the QICPIC from Sympatec uses pulsed illumination and high-resolution cameras to image particles in liquids or gases, measuring sizes from 2 μm to 30 mm while quantifying shape descriptors like convexity and elongation for irregular particles.^[23] These approaches enable rapid analysis of large sample volumes, often exceeding 100,000 particles per measurement, enhancing statistical reliability.^[24] Inline imaging systems facilitate continuous monitoring during industrial processes, integrating directly into pipelines or vessels for non-invasive particle sizing. The SOPAT photo-optical analyzer, for instance, employs high-speed cameras and LED illumination to capture images of particles in multiphase flows, determining sizes from 0.5 μm to 50,000 μm and concentrations in real time, which supports process optimization in crystallization, emulsification, and filtration.^[25] This method includes automated particle counting and shape classification, allowing for immediate feedback on distribution changes without interrupting production.^[26] Direct imaging methods excel in providing high-fidelity data on particle shape and individuality, offering advantages over ensemble techniques for heterogeneous samples, though they can be labor-intensive for manual interpretation and limited by sample preparation requirements.^[27] Modern integrations of machine learning algorithms address these limitations by automating feature extraction and classification from images, improving throughput and reducing operator bias in analyzing complex morphologies, as demonstrated in nanoparticle characterization workflows.^[28]

Indirect Methods

Indirect methods for particle size analysis infer dimensions from physical responses such as light scattering, sedimentation behavior, electrical impedance, or acoustic attenuation, rather than direct visualization. These techniques are particularly valuable for analyzing large numbers of particles in suspensions or powders, offering rapid ensemble measurements but often requiring assumptions like particle sphericity for accurate interpretation.^[1] Light-based methods, including laser diffraction and dynamic light scattering (DLS), rely on interactions between particles and electromagnetic waves to estimate size distributions. Laser diffraction measures the angular distribution of laser light scattered by particles, applying Mie theory to model scattering patterns for spherical particles and deriving size distributions typically in the 0.1 to 1000 μm range. This method assumes optical properties like refractive index are known, enabling volume-equivalent sphere calculations, though non-spherical particles may introduce biases.^[29] DLS, suited for sub-micron colloids down to about 1 nm, analyzes fluctuations in scattered light intensity caused by Brownian motion, from which the diffusion coefficient and hydrodynamic radius are computed using the Stokes-Einstein relation. It provides intensity-weighted average sizes but is sensitive to aggregation in concentrated samples.^[30] Sedimentation and centrifugation techniques determine particle size based on settling velocities under gravitational or centrifugal forces, governed by Stokes' law for spherical particles in a fluid medium. The Andreasen pipette method uses gravitational sedimentation in a suspension, where samples are withdrawn at timed intervals from a fixed depth to measure the mass of settled particles, yielding cumulative size distributions for particles from 1 to 100 μm. This incremental approach directly provides mass-based distributions but requires low concentrations to avoid particle interactions.^[31] Centrifugal methods, such as disc centrifugation, accelerate settling by spinning a hollow disc filled with a density gradient fluid, detecting particles via light transmission as they sediment outward based on size and density, effective for nanoparticles from 2 nm to 0.5 μm. These offer high resolution for polydisperse samples but demand precise density gradients to prevent band broadening.^[32] Electrical sensing zones, based on the Coulter principle, measure particle volume by detecting transient changes in electrical impedance as particles pass through a small aperture in an electrolyte. Each particle displaces a volume of conductive fluid equivalent to its own, producing a pulse whose height is proportional to the displaced volume, allowing size calibration for particles from 0.4 to 1600 μm with high counting rates up to 10,000 per second. This method excels for cellular and industrial particles but assumes non-conductive particles to avoid short-circuiting.^[33] Acoustic methods utilize ultrasound attenuation in dense suspensions to infer particle size distributions, where sound waves propagate through the medium and lose intensity due to scattering and absorption by particles. By fitting attenuation spectra to theoretical models like the Epstein-Carhart or Allegra-Hawley theories, sizes from 0.1 to 100 μm can be estimated, including the particle density as a variable for unknown cases, making it suitable for concentrated slurries up to 50% volume fraction. Limitations include sensitivity to polydispersity and the need for multiple frequencies to resolve broad distributions.^[34] Field methods, such as electrical field-flow fractionation (EFFF), apply an electric field perpendicular to a flowing suspension to separate charged particles by their electrophoretic mobility, which correlates with size for uniform charge density. Particles migrate to an accumulation wall and elute based on field-induced velocity differences, enabling fractionation of sub-micron to micron-sized charged colloids with resolution enhanced by channel geometry. This technique assumes spherical shape and constant zeta potential, potentially underestimating sizes for irregular or variably charged particles.

Equivalent Diameters

Volume- and Mass-Based Expressions

The volume-equivalent diameter, denoted as d_v, represents the diameter of a hypothetical sphere that has the same volume as the actual particle, providing a standardized measure for comparing irregular particles in terms of their bulk volume. This concept is particularly useful in particle size analysis where volumetric properties dominate, such as in processes involving displacement or settling. The formula for d_v is derived from the volume of a sphere, V = \frac{4}{3} \pi r^3, where r is the radius. Solving for the diameter d = 2r, substitute r = \frac{d}{2} to obtain V = \frac{\pi d^3}{6}, and rearranging yields d_v = \left( \frac{6V}{\pi} \right)^{1/3}, where V is the particle's volume. This derivation assumes the particle can be equated to a sphere for volumetric equivalence, facilitating calculations in techniques like sieve analysis, where particles are separated based on their ability to pass through apertures approximating volume-based sizes, and centrifugation, where settling rates depend on volume-displaced buoyancy.^[35]^[36] For instance, in studies of quartz particles commonly used as reference materials in aerosol research, a sample of high-purity quartz dust exhibited a volume-equivalent diameter of approximately 15.1 μm, with 90% of particles below 39 μm, highlighting how d_v quantifies fine mineral distributions relevant to environmental and industrial applications. Similarly, shape factor analyses of quartz aerosols have reported d_v values around 440 nm for submicron particles, demonstrating the metric's role in assessing non-spherical effects on optical and transport properties. These examples underscore the practical utility of d_v in deriving consistent size metrics from measured volumes without requiring direct shape reconstruction.^[37]^[38] The mass-equivalent diameter, d_m, extends this approach by defining the diameter of a sphere with the same mass as the particle, assuming a uniform density \rho. It is expressed as d_m = \left( \frac{6M}{\pi \rho} \right)^{1/3}, where M is the particle mass, directly linking to volume via M = \rho V and thus equating to d_v for non-porous particles of constant density. This equivalence makes d_m valuable in weight-based separations, such as gravitational or centrifugal sorting, where mass influences trajectory and collection efficiency. In weight-based methods, d_m approximates the effective size for irregular particles by normalizing for density variations, ensuring comparability across diverse materials.^[35]^[39] Closely related is the sieve diameter, which for irregular particles approximates the volume-equivalent size as the width of the minimum square aperture through which the particle can pass, effectively representing the diameter of a sphere of equivalent volume that navigates the sieve mesh. This definition aligns sieve analysis outcomes with volumetric metrics, as the aperture size correlates with the particle's projected volume in the plane of passage, though it may slightly overestimate for highly elongated shapes. In practice, for quasi-spherical particles like ground quartz, the sieve diameter closely matches d_v, enabling seamless integration of sieving data into broader volume-based characterizations without additional corrections.^[40]^[41]

Surface- and Aerodynamic Expressions

The surface-equivalent diameter, denoted as d_s, represents the diameter of a hypothetical sphere that possesses the same external surface area A as the irregular particle in question. This parameter is particularly useful in contexts where surface-dependent phenomena dominate, such as optical granulometry, where light scattering correlates with surface interactions, and adsorption studies, where the available surface for molecular attachment influences reaction rates. For a sphere, the surface area is given by A = \pi d^2, leading to the derivation d_s = \sqrt{\frac{A}{\pi}}.^[42]^[42] In imaging-based particle analysis, the projected area diameter d_{pa} is employed to characterize the size from two-dimensional projections, equivalent to the diameter of a circle with the same area A_{proj} as the particle's silhouette. This is derived from the circle area formula \pi (d_{pa}/2)^2 = A_{proj}, yielding d_{pa} = \sqrt{\frac{4 A_{proj}}{\pi}}. It finds application in microscopic and electro-microscopic evaluations, providing a measure of effective optical cross-section without requiring full 3D reconstruction.^[43]^[44] The aerodynamic diameter d_a defines the diameter of a spherical particle with unit density (\rho_0 = 1 g/cm³) that exhibits the same terminal settling velocity in air as the actual particle, making it essential for aerosol dynamics and inhalation toxicology. Its expression accounts for particle density \rho_p, the Cunningham slip correction factor C_c (which adjusts for non-continuum effects at small sizes), and the dynamic shape factor \chi (reflecting deviations from sphericity): d_a = d_p \sqrt{\frac{\rho_p C_c}{\rho_0 \chi}}, where d_p is the physical diameter. This derives from Stokes' law for gravitational settling, v_t = \frac{\rho_p g d_p^2 C_c}{18 \mu \chi}, equating velocities for the reference sphere (C_c = 1, \chi = 1) and the real particle, with \mu as air viscosity and g as gravity. For aerosols, d_a better predicts transport and deposition than geometric measures, especially in polydisperse systems where volume-based expressions might overestimate mobility for low-density particles.^[45] A key application of d_a is in assessing health risks from fine particulate matter, such as PM2.5, defined as particles with d_a \leq 2.5 μm that can penetrate deep into the respiratory tract. These particles, often from combustion sources, deposit preferentially in the alveolar region due to their settling behavior, contributing to inflammation and cardiovascular effects; for instance, they show significant deposition in the pulmonary region for d_a around 1-2.5 μm under typical breathing patterns.^[46]^[47]

Standards and Conventions

International Standards

The International Organization for Standardization (ISO) plays a central role in establishing global norms for particle size measurement and reporting, with the ISO/TC 24 committee on particle characterization overseeing relevant technical standards to promote consistency across industries such as materials science, pharmaceuticals, and environmental monitoring. These standards ensure that particle size data can be compared internationally by defining uniform methods for representation, calibration, and validation, thereby facilitating trade, research reproducibility, and regulatory compliance.^[48] The ISO 9276 series provides comprehensive guidelines for the representation of particle size distributions, applicable to solid particles, droplets, or gas bubbles across all size ranges. Specifically, ISO 9276-1 outlines rules for graphical representations, including histograms, density distributions, and cumulative distributions, to visualize particle size analysis results effectively. ISO 9276-2 details procedures for calculating moments, mean particle sizes (such as arithmetic, geometric, and harmonic means), standard deviations, and other statistical measures using coherent nomenclature and equations. For log-normal distributions, common in many particulate systems, ISO 9276-5 specifies methods to calculate parameters like the geometric mean (the antilogarithm of the arithmetic mean of the logarithms of individual particle sizes) and the median, enabling standardized characterization of skewed distributions. Additionally, ISO 26824:2022 provides definitions for terms used in particle characterization, including size, shape, and morphology, to support standardized terminology across applications.^[49]^[50] Sieving remains a foundational technique standardized by ISO, with ISO 3310-1 specifying requirements for test sieves made of metal wire cloth, covering aperture sizes from 125 mm down to 20 μm to accommodate a wide range of particle analyses.^[51] This standard ensures sieve construction meets tolerances for wire diameter and aperture uniformity, aligning with ISO 565, which defines nominal sizes of openings for metal wire cloth, perforated plates, and electroformed sheets used as sieving media.^[52] ISO 565 includes principal and supplementary aperture sizes in the ISO R20/ISO R40 series, providing a progressive scale for precise classification.^[52] Complementary standards from other bodies address specific applications; for instance, ASTM International's E11 standard specifies the design, construction, and tolerances for woven wire test sieve cloth and sieves, including average opening, maximum opening, and standard deviation limits to verify compliance.^[53] In the pharmaceutical sector, the United States Pharmacopeia (USP) <811> chapter on Powder Fineness classifies powders by sieve-based categories using cumulative oversize distributions (Q3 values; e.g., coarse where Q3(355 μm) < 0.50 [more than 50% of particles exceed 355 μm], moderately fine primarily 180–355 μm [Q3(180 μm) < 0.50 and Q3(355 μm) ≥ 0.50], fine primarily 125–180 μm [Q3(125 μm) < 0.50 and Q3(180 μm) ≥ 0.50]) and recommends methods like analytical sieving for particles above 75 μm, ensuring quality control in drug formulations.^[54] The evolution of these standards since the 1980s reflects advancements in measurement technologies, with initial focuses on sieving and microscopy giving way to integrations of optical methods; for example, ISO 3310-1 was revised in 2016 to refine tolerances, while the ISO 9276 series saw updates in the 1990s, 2000s, and 2025 (e.g., Part 1 revised in 2025) to incorporate statistical rigor for emerging techniques.^[51] In the 2010s, revisions to related standards like ISO 13320 (2009 and later, including 2020) explicitly addressed laser diffraction for particle size analysis, standardizing validation protocols to account for Mie and Fraunhofer scattering theories and ensuring measurement reproducibility across instruments. This progression underscores the rationale for standardization: to mitigate variability from diverse methodologies, enabling reliable inter-laboratory comparisons and supporting innovations in fields like nanotechnology, where colloidal contexts serve as specialized extensions of these ISO norms.^[55]^[56]

Colloidal and Specialized Contexts

In colloidal systems, particles typically range from 1 nm to 1 μm in diameter, though the upper limit is sometimes extended to 1000 nm depending on the context.^[57] This size regime enables distinctive behaviors such as Brownian motion, where random collisions with solvent molecules prevent sedimentation and maintain suspension, and the Tyndall effect, characterized by visible light scattering that distinguishes colloids from true solutions.^[57] Particle size in this domain is often quantified using the hydrodynamic radius, which accounts for the effective size including the solvation layer, as measured by dynamic light scattering (DLS) techniques that analyze diffusion coefficients via the Stokes-Einstein equation.^[58] Specialized conventions adapt particle size definitions to domain-specific behaviors and regulatory needs. In aerosols, PM₁₀ and PM_2.5 refer to particulate matter with aerodynamic diameters of less than 10 μm and 2.5 μm, respectively, emphasizing inhalability and atmospheric settling velocity rather than geometric size.^[13] Similarly, in soil science, the USDA texture classification delineates clay as particles smaller than 2 μm, silt from 2 to 50 μm, and sand from 50 to 2000 μm, influencing soil properties like water retention and fertility.^[59] For nanoparticles, defined as those below 100 nm, quantum size effects introduce unique challenges to size definition, as discrete energy levels alter optical, electronic, and magnetic properties, shifting from bulk continuum behaviors to quantized states that depend critically on precise dimensional control.^[60] This historical perspective traces to early 20th-century work by Wilhelm Ostwald, who in the 1910s helped establish the colloidal size domain as approximately 1–100 nm, laying foundational conventions for distinguishing colloids from molecular solutions and coarse dispersions.^[61] Unlike bulk particles, colloidal sizes profoundly impact stability and aggregation; smaller particles exhibit higher surface energy, promoting flocculation unless stabilized by electrostatic or steric barriers, as governed by DLVO theory, where aggregation rates increase inversely with size due to enhanced van der Waals attractions relative to Brownian diffusion.^[62] These dynamics underscore the need for context-specific size reporting in colloids, often aligning with international standards for consistent characterization across applications like pharmaceuticals and environmental monitoring.

Particle Size Distributions

Core Concepts

Particle size distribution (PSD) refers to the frequency or relative amount of particles within specified size ranges in a sample, providing a statistical description of the sizes present in a population rather than individual particles. This distribution captures the inherent variability in particle dimensions, often expressed through histograms, curves, or mathematical functions that aggregate measurements from numerous particles. In essence, PSD shifts focus from single-particle characterization—such as equivalent diameters—to the collective behavior of ensembles, enabling assessment of overall sample heterogeneity.^[63] A key aspect of PSD is polydispersity, which quantifies the degree of size variation: monodisperse systems consist of particles nearly identical in size, yielding narrow distributions, while polydisperse systems exhibit a broad range of sizes, common in natural and processed materials like soils, aerosols, and powders. Polydispersity arises from manufacturing processes, environmental factors, or natural formation, influencing properties such as flowability, reactivity, and dissolution rates. For instance, a monodisperse sample might have a polydispersity index near 0, indicating uniformity, whereas polydisperse samples show values approaching 1, reflecting multimodal or skewed size populations.^[63]^[64]^[3] PSDs are typically represented in two main forms: cumulative distributions, which plot the proportion of particles undersize (below a given size) or oversize (above it) as an S-shaped curve, and density functions that show the relative frequency per size interval. Density functions can be based on number (counting individual particles), volume (proportional to size cubed), or mass (assuming uniform density), each emphasizing different aspects—number-based for fine particles, volume- or mass-based for bulk properties. The log-normal distribution is prevalent in many natural systems, such as atmospheric aerosols or pulverized ores, due to multiplicative growth processes that favor logarithmic scaling of sizes.^[65]^[66]^[67] Central parameters of a PSD include various mean sizes, such as the arithmetic mean (simple average of diameters) for symmetric distributions and the geometric mean (logarithmic average) for skewed ones like log-normal; the mode, representing the most frequent size; and the span, a width metric calculated as (D90 - D10)/D50, where D values denote percentiles. These parameters provide concise summaries: for example, in powder metallurgy, a bimodal PSD—with two distinct modes—can enhance packing density by optimizing the combination of fine and coarse particles to improve sinterability. Equivalent diameters parameterize these distributions for irregular particles, ensuring consistent statistical aggregation across measurements.^[68]^[69]

Characterization and Analysis

Particle size distributions are characterized using statistical measures derived from cumulative undersize curves, where D10, D50, and D90 represent the particle diameters below which 10%, 50%, and 90% of the sample mass resides, respectively, providing key indicators of distribution span and central tendency.^[70] These percentiles are particularly useful for quality control, as they highlight the range from fines to coarse fractions without assuming a specific distributional form.^[68] For volume-based analyses, moment means such as the De Brouckere mean, denoted D[4,3], calculate the average diameter weighted by volume (fourth moment over third moment), emphasizing larger particles' contribution to bulk properties like sedimentation.^[70] Analysis techniques often involve fitting empirical data to parametric models to interpolate and extrapolate distributions, with the log-normal model assuming a symmetric distribution on a logarithmic scale and the Rosin-Rammler model capturing the cumulative oversize via an exponential decay suitable for grinding processes.^[71] These fits enable prediction of unmeasured fractions and assessment of process uniformity, though model selection depends on the material's generation mechanism.^[72] Software like the Mastersizer from Malvern Panalytical interprets laser diffraction data by applying Mie theory to compute these distributions, outputting fitted parameters and visualizations for routine interpretation in industrial settings.^[73] Modern advances incorporate machine learning to predict distributions in real-time during manufacturing, such as convolutional neural networks analyzing images for component-specific sizing in milling operations, reducing reliance on offline sampling.^[74] These models are validated against standards like ISO 9276-2, which specifies moment calculations and nomenclature to ensure reproducibility across techniques.^[70] In pharmaceutical processes, neural network approaches forecast distribution parameters from process variables, enhancing control of bioavailability.^[75] Challenges arise in handling multimodal distributions, where multiple peaks indicate mixed populations from blending or attrition, complicating fits to unimodal models and requiring decomposition techniques for accurate percentiles in quality control.^[76] Irregular shapes further bias equivalent diameter calculations, as non-spherical particles in pharmaceuticals like needle-like crystals affect flow and dissolution, necessitating shape-corrected analyses to meet uniformity criteria.^[77]

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None
### Summary of Particle Characterization Guide (Malvern Instruments, 2015)
[37]
Diameter of a Sphere Formula Using Volume - Cuemath
The diameter of a sphere formula using volume is d = (6V/π)^(1/3). Learn this formula along with solved examples and practice questions.
[38]
Measurement of size-dependent dynamic shape factors of quartz ...
May 17, 2016 · For example, quartz particles with a volume equiva- lent diameter of 440 nm have measured values of xv between 1.45 and 1.7, and xt, between ...
[39]
Equivalent Diameter - an overview | ScienceDirect Topics
Equivalent diameter is defined as the diameter of a circle that has an equal aggregate sectional area, calculated using the formula d = 2 Area / π.
[40]
[PDF] The Basic Principles of Sieve Analysis - Retsch
Equivalent diameter. In sieving the equivalent diameter of a particle is the diameter of a sphere with the same mass or volume. Undersize. Mass fraction below ...
[41]
Particle Size Measurements - Fundamentals, Practice, Quality
... volume equivalent diameter has the advantage that it relates to the volume ... sieve analysis [11]. The number of apertures (or wires) per linear inch ...
[42]
[PDF] A Compact Introduction to the Numerical Modeling of Multiphase ...
the diameter of a sphere that has the same volume Vp as the particle deq ≡. 6Vp π. 1. 3. Similarly, a surface equivalent diameter can be defined ds ≡. Ap π. 1.
[43]
Basic Principles of Particle Size Analysis-1 - Malvern Panalytical
Jan 10, 2012 · The formula for the sphere's surface area is 4πr2. Therefore, we must first find the square of the diameters divided by the number of particles ...
[44]
Particle Shape Analysis - Sympatec
Diameter of a circle of equal projection area / EQPC. This is the diameter of a circle that has the same area as the projection area of the particle. It is ...
[45]
Particle size analysis - Basicmedical Key
Jun 2, 2016 · The projected area diameter is based on a circle of equivalent area to that of the projected image of a particle; the perimeter diameter is ...
[46]
Aerodynamic Diameter - an overview | ScienceDirect Topics
An Introduction to Aerosols The aerodynamic diameter of a particle, dae, is defined as the diameter of a spherical particle having a standard density, ρo = 1, ...
[47]
[PDF] Overview of Particulate Matter (PM) Air Quality in the United States
Feb 7, 2023 · The EPA defines PM2. 5, also referred to as fine particles, as particles with aerodynamic diameters generally less than or equal to 2.5 µm.
[48]
The Impact of PM2.5 on the Host Defense of Respiratory ... - Frontiers
Mar 3, 2020 · PM less than 2.5 μm in diameter poses the greatest problems, because it can get deep into the terminal bronchioles and alveoli, and some with < ...
[49]
Representation of results of particle size analysis — Part 1 - ISO
This document specifies guidelines and instructions for the graphical representation of particle size analysis data in histograms, distribution densities and ...
[50]
ISO 9276:2005 Representation of results of particle size analysis
Aug 5, 2005 · The main objective of ISO 9276-5:2005 is to provide the background for the representation of a cumulative particle size distribution which ...
[51]
ISO 3310-1:2016 - Test sieves of metal wire cloth
In stock 2–5 day deliveryIt applies to test sieves having aperture sizes from 125 mm down to 20 µm, in accordance with ISO 565.Missing: mesh | Show results with:mesh
[52]
ISO 565:1990 - Test sieves — Metal wire cloth, perforated metal ...
In stock 2–5 day deliveryApplies to metal wire cloth with square openings and perforated plates or sheets with square or circular openings. Tables include principal sizes and ...
[53]
E11 Standard Specification for Woven Wire Test Sieve Cloth ... - ASTM
Jun 3, 2024 · Standard E11 specifies 3 different tolerances for the sieve cloth openings; the average, the maximum, and the maximum standard deviation of the ...
[54]
[PDF] 〈811〉 POWDER FINENESS - US Pharmacopeia (USP)
May 1, 2012 · Sub-sieve Size Particles—The average particle size meas-. Coarse. >355. Q3(355) <0.50 ured is in the range of 0.2 to 50 µm. The test specimen ...
[55]
Setting New Standards for Laser Diffraction Particle Size Analysis
Jan 7, 2010 · As noted in the introduction to ISO13320:2009, laser diffraction is now the dominant method of particle size distribution analysis. Fast and ...
[56]
The central role of colloids to explain the crystallization dynamics of ...
Jul 3, 2024 · Colloidal particles (dispersed phase) typically fall within the size range of 1 nm to 1 μm. These particles experience the influence of the ...Overview Of Colloid Theory · Acknowledgments · Declaration Of Interests
[57]
Dynamic light scattering: a practical guide and applications in ...
Oct 6, 2016 · The diffusion coefficient, and hence the hydrodynamic radii calculated from it, depends on the size and shape of macromolecules. In this review, ...
[58]
[PDF] Soil Survey Manual 2017; Chapter 3
USDA textural triangle showing the percentages of clay, silt, and sand in the 12 basic texture classes. ... clayey family particle-size classes defined in Soil ...
[59]
Quantum Size Effect - an overview | ScienceDirect Topics
Quantum size effect refers to the variations in physical properties of metallic or semiconductor films as their dimensions decrease to the atomic scale, ...
[60]
Impact and repercussions of the Ostwald-de Izaguirre theory for ...
Ostwald set for the colloidal domain (1–100 nm). These coincide today with the limits of nanoscience, to which some authors [22] consider not as a new science, ...
[61]
Experimental and Theoretical Studies of the Colloidal Stability of ...
May 5, 2011 · We study ZnO in ethanol for which the influence of particle size and reactant ratio as well as surface coverage on colloidal stability in dependence of the ...Stability Of Zno Quantum... · Dimensional Analysis · Zno Quantum Dots...
[62]
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 ...Missing: colloids | Show results with:colloids
[63]
Polydispersity Index (PDI) - NanoTemper Technologies
The polydispersity index (PDI) indicates particle size heterogeneity. A small PDI means a monodisperse sample, while a large PDI means a polydisperse sample.
[64]
Particle size distribution | Anton Paar Wiki
The most common values used in several particle sizing techniques are the D-values. These are related to the cumulative distribution and are only meaningful ...<|control11|><|separator|>
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Particle Size Result Interpretation: Number vs. Volume Distributions
The particle size distribution can be calculated based on several models: most often as a number or volume/mass distribution.Missing: functions | Show results with:functions
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Particle Size Distribution Functions:
Particle size distributions can be obtained by hand counting TEM images, or through the use of differential mobility analysis (DMA) with fractionation.
[67]
Understanding & Interpreting Particle Size Distribution Calculations
The Di value for each channel is the geometric mean, the square root of upper x lower diameters. For the numerator take the geometric Di to the fourth power ...
[68]
Definition and Measurement Methods of Particle Size Distribution
Jun 28, 2018 · Methods for Measuring Particle Size Distribution · 1. Optical Bench · 2. Sample Dispersion Unit · 3. Measuring Device Software.Missing: micrometers | Show results with:micrometers
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Powdered Tungsten Particle Size Distribution
However, some tungsten powder applications require bimodal or even trimodal distributions. The Malvern Mastersizer plays an important part in helping us meet ...<|separator|>
[70]
ISO 9276-2:2014 - Representation of results of particle size analysis
CHF 155.00 In stockISO 9276-2:2014 provides relevant equations and coherent nomenclatures for the calculation of moments, mean particle sizes and standard deviations from a given ...Missing: validation | Show results with:validation
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Particle size distribution models, their characteristics and fitting ...
In this study all developed PSD models, their characteristics, behavior of their parameters, and their fitting capability to the UNSODA database soil samples ...
[72]
Comparison of six particle size distribution models on the goodness ...
The lognormal distribution model offered overall good approximations to measured PSD data, but was inferior to the gamma and Weibull distribution models when ...
[73]
Particle Size Analysis - An Explanation - Malvern Panalytical
Malvern Panalytical offers leading instrumentation for all types of particle size analysis and characterization from sub-nanometer to millimeters in particle ...
[74]
Real-time component-based particle size measurement and ...
Jun 1, 2025 · This work presents a system, in which machine vision combined with artificial intelligence-based image analysis was used to determine the component-based ...
[75]
An Innovative Artificial Intelligence Approach for Predicting Particle ...
Sep 5, 2025 · This study aims to develop and validate a neural network-based modeling approach to predict particle size distribution characteristics and ...
[76]
Challenges Of Particle Characterisation - Pharmaceutical Technology
Apr 1, 2011 · Particle characteristics can affect pharmaceutical formulations and products in a number of ways, and a variety of techniques are available ...Missing: irregular | Show results with:irregular
[77]
A Review of Particle Shape Effects on Material Properties for ... - MDPI
Today, many industries recognize the importance of measuring particle shape as well as particle size for a full understanding of products and processes, since ...