Fact-checked by Grok 2 weeks ago

Stokes radius

The Stokes radius (also known as the Stokes-Einstein radius or hydrodynamic radius) is the radius of a hypothetical hard sphere that experiences the same frictional resistance to diffusion in a fluid as the actual solute particle or macromolecule, providing a measure of its effective size in solution that incorporates effects like hydration and shape.^[1] It is fundamentally derived from the Stokes-Einstein equation, which relates the translational diffusion coefficient D of a spherical particle to its radius r_s as D = \frac{k_B T}{6 \pi \eta r_s}, where k_B is Boltzmann's constant, T is the absolute temperature, and \eta is the solvent viscosity; this equation assumes a no-slip boundary condition at the particle surface and is widely used to estimate r_s from experimentally measured diffusion coefficients.^[2] In practice, the Stokes radius is determined through techniques such as dynamic light scattering, fluorescence correlation spectroscopy, or size-exclusion chromatography, where it serves as the radius of an equivalent sphere matching the solute's hydrodynamic behavior, often expressed in nanometers for biomolecules like proteins.^[3] This parameter is essential in biophysics, colloid science, and chemical engineering for characterizing macromolecules, as it reflects not just geometric size but also interactions with the surrounding solvent, enabling assessments of conformational changes, aggregation states, and solvation shells in proteins and polymers.^[4] For instance, in protein studies, variations in Stokes radius can indicate binding events or structural transitions, such as the reduction from 2.77 nm to 2.53 nm upon calcium binding to the Arabidopsis calmodulin AtCaM1, highlighting its utility in monitoring functional dynamics.^[3] Beyond diffusion, the concept extends to sedimentation and electrophoresis, where it informs frictional coefficients under applied forces, aiding in the design of drug delivery systems and purification processes.^[4]

Fundamentals

Definition

The Stokes radius, also known as the hydrodynamic radius, is defined as the radius of a hypothetical hard sphere that would exhibit the same frictional resistance to motion or the same diffusion coefficient as the actual particle or macromolecule in a dilute suspension through a viscous fluid.^[3] This effective radius provides a measure of the particle's size under hydrodynamic conditions, accounting for interactions with the surrounding solvent.^[4] The Stokes radius r_s (or r_h) can be calculated from the particle's diffusion coefficient D using the Stokes-Einstein equation:

r_s = \frac{k T}{6 \pi \eta D}

where k is Boltzmann's constant, T is the absolute temperature, and \eta is the viscosity of the solvent.^[4]^[5] Unlike the actual geometric radius, which describes the physical dimensions of the particle, the Stokes radius is a hydrodynamic parameter derived from experimental observables such as diffusion or sedimentation rates, and it may differ from the geometric radius due to factors like molecular shape, solvation shell, and partial unfolding.^[3] This distinction arises from the foundational Stokes' law, which quantifies the viscous drag on a spherical particle in low-Reynolds-number flow.^[4]

Historical Context

The concept of the Stokes radius emerged from George Gabriel Stokes' foundational work in 1851, where he derived the expression for the viscous drag force on a spherical particle moving through a fluid at low Reynolds numbers, as detailed in his paper on the internal friction of fluids affecting pendulum motion. This derivation, now known as Stokes' law, provided the hydrodynamic basis for relating a particle's terminal velocity in sedimentation to its size, initially applied to geophysical and fluid mechanical problems rather than biological or colloidal systems.^[6] In 1905, Albert Einstein utilized Stokes' law to derive the Stokes-Einstein equation, relating the translational diffusion coefficient of a particle to its hydrodynamic radius, thereby extending the concept to Brownian motion and diffusion processes in fluids.^[7] In the early 20th century, the Stokes radius gained prominence in the study of macromolecules through Theodor Svedberg's pioneering ultracentrifugation experiments during the 1920s and 1930s. Svedberg, who received the 1926 Nobel Prize in Chemistry for his colloid research, adapted Stokes' law to analyze sedimentation velocities in high-speed centrifuges, enabling the calculation of molecular weights and effective hydrodynamic radii for proteins and other large molecules, thus establishing the technique as a cornerstone for macromolecular characterization.^[8] Following World War II, the concept evolved significantly in colloid science with the advent of light scattering methods, particularly through Peter Debye's 1947 paper on molecular weight determination via light scattering, which linked scattering intensities to particle dimensions and complemented diffusion measurements for estimating the Stokes radius in dilute solutions. This integration allowed researchers to probe hydrodynamic properties noninvasively, expanding applications from sedimentation to optical techniques for colloidal dispersions.^[9]

Theoretical Basis

Derivation from Diffusion

The derivation of the Stokes radius begins with the Einstein relation, which connects the diffusion coefficient D of a particle undergoing Brownian motion to the frictional coefficient f experienced by that particle in the surrounding fluid: D = \frac{kT}{f}, where k is Boltzmann's constant, T is the absolute temperature, and this relation arises from the balance between thermal energy and dissipative friction in the overdamped limit of Brownian dynamics.^[10] For a spherical particle, the frictional coefficient is given by Stokes' law as f = 6\pi\eta r_s, where \eta is the dynamic viscosity of the fluid and r_s is the Stokes radius, representing the effective hydrodynamic radius of the sphere.^[11] Substituting the expression for f into the Einstein relation yields the Stokes-Einstein equation: r_s = \frac{[kT](/page/KT)}{6\pi\eta [D](/page/D*)}.^[12] This equation allows the Stokes radius to be directly computed from a measured diffusion coefficient, provided the temperature and solvent viscosity are known. The derivation assumes a dilute solution where particles experience no significant interactions with each other, such that Brownian motion is independent and unhindered; the particles are perfectly spherical to satisfy the symmetry required by Stokes' low-Reynolds-number hydrodynamics; and the fluid is a continuum with Newtonian viscosity, valid for particle sizes much larger than molecular scales but small enough for inertial effects to be negligible.^[13] In practice, the diffusion coefficient D is obtained through techniques that probe Brownian motion at the single-particle or ensemble level. Dynamic light scattering (DLS) measures fluctuations in scattered laser light intensity from a suspension of particles, from which the autocorrelation function is analyzed to extract D via the Stokes-Einstein relation, typically for sizes from 1 nm to 1 \mum in dilute solutions.^[14] Fluorescence correlation spectroscopy (FCS) complements DLS by monitoring fluorescence intensity fluctuations in a small focal volume, yielding D from the autocorrelation decay time of diffusing fluorescently labeled molecules, offering high sensitivity for concentrations down to nanomolar levels and sub-micrometer scales.^[15]

Derivation from Sedimentation

The Stokes radius can be derived from the sedimentation behavior of particles in a gravitational or centrifugal field, where the downward motion reaches a terminal velocity when the effective gravitational force balances the viscous drag force. The effective gravitational force on a particle is given by m g' (1 - \bar{v} \rho), where m is the particle mass, g' is the effective acceleration due to gravity (or \omega^2 r in centrifugation), \bar{v} is the partial specific volume, and \rho is the solvent density.^[16] The opposing drag force follows Stokes' law for low-Reynolds-number flow: $6 \pi \eta r_s v, where \eta is the solvent viscosity, r_s is the Stokes radius, and v is the sedimentation velocity.^[17] At terminal velocity, these forces balance, yielding v = \frac{m g' (1 - \bar{v} \rho)}{6 \pi \eta r_s}. The sedimentation coefficient s, defined as the velocity per unit effective acceleration s = v / g', simplifies to s = \frac{m (1 - \bar{v} \rho)}{6 \pi \eta r_s}. Substituting the particle mass m = M / N_A, where M is the molar mass and N_A is Avogadro's number, gives the key relation s = \frac{M (1 - \bar{v} \rho)}{N_A 6 \pi \eta r_s}. Rearranging for the Stokes radius produces r_s = \frac{M (1 - \bar{v} \rho)}{N_A 6 \pi \eta s}.^[16]^[17] This derivation assumes creeping flow conditions (low Reynolds number, Re ≪ 1), ensuring Stokes' law validity; spherical particle geometry; negligible wall effects or particle-solvent interactions; and dilute solutions to avoid non-ideality.^[16] In practice, deviations from sphericity are accounted for via the frictional ratio f / f_0, where f_0 = 6 \pi \eta r_{\min} and r_{\min} is the radius of a compact sphere of equivalent mass.^[17] The sedimentation coefficient s is experimentally determined using analytical ultracentrifugation, where macromolecules sediment in a centrifugal field (\omega up to 60,000 rpm) and boundaries are monitored via absorbance or interference optics to fit the Lamm equation describing the concentration profile.^[16] Values of s are corrected to standard conditions (water at 20°C, s_{20,w}) for comparability across studies. Diffusion measurements provide a complementary approach to refine hydrodynamic parameters when combined with s, as detailed elsewhere.^[17]

Extensions and Limitations

Non-Spherical Particles

For non-spherical particles, such as ellipsoids or rods, the Stokes radius concept is adapted by incorporating shape-dependent corrections to the frictional coefficient derived from the Stokes-Einstein relation, allowing estimation of the hydrodynamic radius r_h (the Stokes radius) equivalent to that of a sphere with the same diffusion or sedimentation behavior, as well as inferences about the underlying geometric size. The frictional coefficient f for these particles is expressed as f = f_0 \cdot P(\rho), where f_0 = 6\pi\eta r_0 is the friction for a sphere of equivalent volume (with radius r_0), \eta is the solvent viscosity, and P(\rho) is the Perrin shape factor greater than 1, which accounts for increased drag due to asymmetry; \rho denotes the aspect ratio (e.g., major/minor semi-axis for ellipsoids). This factor P(\rho) originates from analytical solutions for the hydrodynamic resistance of prolate or oblate ellipsoids under low Reynolds number flow, as derived by Perrin in his foundational work on Brownian motion of spheroids. The hydrodynamic Stokes radius r_h is obtained experimentally from diffusion (r_h = kT / (6\pi\eta D)) or sedimentation data. To infer the equivalent volume radius r_0, which estimates the geometric size, r_0 = r_h / P(\rho). For rod-like particles, such as cylindrical macromolecules, more specialized models extend these corrections beyond simple ellipsoids. Broersma's theoretical framework from the 1960s provides shape factors for the translational and rotational diffusion of long, thin rods, incorporating end effects and aspect ratios up to high elongations, with explicit expressions for parallel and perpendicular diffusion coefficients that enable averaging for isotropic solutions. These were refined by Tirado and García de la Torre in the 1980s using bead-shell hydrodynamic models for short rods (aspect ratios 2–30), yielding accurate Perrin-like corrections for friction, such as end-effect corrections \delta \approx 0.307 + 0.426/\rho - 0.689/\rho^2 + 0.426/\rho^3 incorporated into the overall shape factor for the average translational friction f / (3\pi \eta L) = [\ln(2\rho) - \gamma + \delta]/\rho, where \gamma \approx 0.307 and L is the rod length, validated against experimental data for cylinders.^[18] Examples include double-stranded DNA fragments, often modeled as worm-like chains or rigid rods with persistence length around 50 nm and diameter 2 nm, where these factors adjust the apparent Stokes radius to reveal lengths from 10–100 nm; similarly, rod-shaped viruses like tobacco mosaic virus (length ~300 nm, diameter ~18 nm) use Tirado-García de la Torre models to correlate hydrodynamic data with structural dimensions from electron microscopy. Despite these advances, adaptations for non-spherical particles introduce limitations, particularly from anisotropic diffusion where parallel and perpendicular components differ significantly, leading to errors up to 20% in effective radius estimates if orientation averaging is neglected (e.g., D = (D_\parallel + 2D_\perp)/3). Highly elongated shapes (\rho > 10) amplify sensitivity to boundary conditions (stick vs. slip) and minor perturbations like flexibility, requiring complementary techniques like small-angle X-ray scattering for validation, as the Perrin-Broersma-Tirado frameworks assume rigid, isolated particles in dilute solutions.

Polydisperse Systems

In polydisperse systems, where particles exhibit a range of sizes, the Stokes radius cannot be represented by a single value without accounting for the distribution, leading to the use of weighted averages. Dynamic light scattering (DLS) commonly yields the z-weighted average Stokes radius \langle r_s \rangle_z, which is intensity-weighted and thus biased toward larger particles due to their greater scattering contribution proportional to the sixth power of the radius. In contrast, the number-weighted average \langle r_s \rangle_N provides an unweighted mean but is less directly accessible from DLS data without additional analysis. These averages are derived from the apparent diffusion coefficient via the Stokes-Einstein relation, with the z-average often serving as the primary metric for polydisperse samples in colloidal and macromolecular studies.^[19]^[20] To recover the underlying size distribution and mitigate biases in average values, deconvolution techniques are essential. Cumulants analysis expands the autocorrelation function in a power series to estimate the z-average diffusion coefficient and polydispersity index (PDI), offering a rapid assessment suitable for moderately polydisperse systems with PDI < 0.3. For broader distributions, the CONTIN algorithm applies regularization to invert the Laplace transform of the DLS autocorrelation function, enabling robust recovery of the distribution of diffusion coefficients and corresponding Stokes radii without assuming a specific form. These methods improve accuracy in characterizing heterogeneous samples, such as emulsions or aggregates, by distinguishing multiple size populations.^[7]^[21] Particle interactions in polydisperse systems further complicate Stokes radius determination, particularly at higher concentrations where interparticle effects modify diffusion. The second virial coefficient A_2, reflecting thermodynamic non-ideality, influences the effective diffusion through excluded volume and hydrodynamic interactions, causing the apparent Stokes radius to deviate from dilute limits—typically increasing for repulsive interactions and decreasing for attractive ones. This concentration dependence requires corrections, often via virial expansions of the diffusion coefficient, to obtain intrinsic values.^[22]^[23] Case studies in polymer solutions and latex dispersions highlight the practical implications of polydispersity. In polystyrene solutions analyzed by DLS, neglecting size heterogeneity when assuming a monodisperse model can overestimate the apparent Stokes radius by up to 20%, as the z-weighting amplifies contributions from larger chains. Similarly, for latex dispersions like polystyrene spheres in aqueous media, ignoring polydispersity leads to comparable errors in hydrodynamic sizing, underscoring the need for distribution-aware analysis to ensure reliable characterization in industrial formulations.^[24]^[25]

Applications

Macromolecular Characterization

The Stokes radius plays a crucial role in protein folding studies by providing insights into conformational changes through measurements of hydrodynamic size variations. Techniques such as nuclear magnetic resonance (NMR) spectroscopy enable the monitoring of the Stokes radius (r_s) via diffusion coefficients to distinguish between native, folded, and unfolded states of proteins, while small-angle X-ray scattering (SAXS) provides complementary information on the radius of gyration. For instance, in bovine serum albumin (BSA), the native form exhibits a Stokes radius of approximately 3.7 nm, while the unfolded state in denaturing conditions like 40% dimethyl sulfoxide expands to about 7.0 nm, reflecting increased chain flexibility and solvent exposure.^[26] These changes in r_s are quantified via pulsed-field gradient NMR for diffusion coefficients or SAXS for radius of gyration, allowing researchers to map folding pathways and intermediate states in real time.^[27]^[28] In the analysis of protein complexes, the Stokes radius facilitates estimation of oligomeric states by comparing measured hydrodynamic sizes to expected ratios for monomers, dimers, or higher-order assemblies. For hemoglobin, dynamic light scattering and analytical ultracentrifugation reveal a Stokes radius of roughly 2.77 nm for the functional tetramer, compared to 2.15 nm for the dimer and 1.85 nm for the monomer, enabling differentiation of dissociation events under varying pH or ligand conditions.^[29] This ratio-based approach is particularly valuable for confirming quaternary structures in multi-subunit proteins, where deviations from spherical assumptions are minimal, as seen in hemoglobin's near-globular tetrameric form.^[30] Integration of the Stokes radius with other biophysical metrics enhances macromolecular characterization. The partial specific volume (\bar{v}) for proteins is typically around 0.73 mL/g, which can be refined using sedimentation data and molecular weight estimates.^[31] Stokes radius measurements are used for calibrating size-exclusion chromatography (SEC) columns. SEC separates proteins based on hydrodynamic volume, with calibration curves plotted against known Stokes radii of standards to estimate r_s for unknowns, decoupling size from molecular weight for non-globular or denatured species.^[32] This facilitated analysis of protein mixtures, enabling precise oligomeric state assessments and purity checks without assuming globular shapes, as demonstrated in SEC-MALS (multi-angle light scattering) hybrids.^[33]

Environmental and Industrial Uses

In environmental monitoring, particularly water treatment and soil remediation, the Stokes radius (r_s) is employed to estimate the mobility of nanoparticles in porous media such as soils, derived from column filtration experiments that simulate transport under saturated conditions. For instance, engineered silver nanoparticles exhibit varying degrees of retention and breakthrough in soil columns, influencing their role in contaminant transport and potential leaching into groundwater.^[34]^[35] This assessment helps predict the environmental fate of nanomaterials released from industrial or agricultural sources, guiding remediation strategies to mitigate risks from persistent pollutants. In industrial processes, the Stokes radius characterizes the hydrodynamic size of latex particles in paints and emulsions, enabling optimization of formulation rheology through viscometric analysis. Capillary viscometry measures the flow behavior of these suspensions, where r_s informs particle interactions and viscosity profiles critical for application properties like spreadability and stability in water-based coatings. For example, associative thickeners in latex emulsions adjust effective r_s to achieve desired shear-thinning characteristics, enhancing performance in high-solids paints without excessive thickening agents. In aerosol science, the Stokes radius is calculated from mobility analyzers to determine settling rates of atmospheric pollutants, aiding in air quality modeling and exposure assessments. Differential mobility analyzers classify particles by electrical mobility, yielding r_s values that, when input into Stokes' law with Cunningham corrections, predict gravitational deposition velocities for submicron pollutants like particulate matter.^[36]^[37] This approach is essential for evaluating the persistence and dry deposition of urban aerosols, informing regulatory limits on emissions. Developments in nanotoxicology have extended Stokes radius applications, where hydrodynamic size serves as a key predictor of nanoparticle bioavailability and toxicity in biological systems. Studies have demonstrated that smaller r_s (e.g., <50 nm) correlates with higher cellular uptake and clearance rates, influencing risk assessments for engineered nanomaterials in consumer products.^[38]^[39] These insights, often derived from dynamic light scattering measurements, have shaped guidelines for safer nanomaterial design by linking size-dependent bioavailability to reduced environmental and health hazards.

References

[1]
Glycocalyx volume measurements: a critical review of tracer dilution ...
The Stokes Einstein radius of a molecule is the radius of a sphere that faces the same friction resistance to diffusion through an aqueous solution as the ...Missing: definition | Show results with:definition
[2]
Stokes–Einstein Relation in Different Models of Water - PMC - NIH
Nov 26, 2024 · D = k B T 6 π η R . (2) The numerical coefficient in Equation (1) corresponds to the “stick” boundary condition at the sphere surface.
[3]
Determination of Hydrodynamic Radius of Proteins by Size ... - PMC
Apr 20, 2017 · In the case of proteins, the hydrodynamic value that can be experimentally derived is the Stokes radius (Rs), which is the radius of a sphere ...
[4]
Hydrodynamic Radius - an overview | ScienceDirect Topics
Hydrodynamic radius (R_h) is defined as the hydrodynamically active dimension of a macromolecule, derived from the translational diffusion coefficient using ...
[5]
Measurement of the Translational Diffusion Coefficient and ...
Oct 20, 2021 · ... diffusion coefficient and application of the Stokes-Einstein equation . ... Stokes radius), can be calculated using the Stokes-Einstein equation.<|control11|><|separator|>
[6]
[PDF] Stokes's Fundamental Contributions to Fluid Dynamics
Jul 12, 2019 · George Gabriel Stokes was one of the giants of hydrodynamics in the nine- teenth century. He made fundamental mathematical contributions to ...
[7]
Stokes' law, viscometry, and the Stokes falling sphere clock - Journals
Aug 3, 2020 · ... ultracentrifuge, which could be used to determine the molecular weights and size—the Stokes radius—of colloids with Stokes' law. Lastly ...
[8]
Molecular-weight Determination by Light Scattering.
Article January 1, 1947. Molecular-weight Determination by Light Scattering. Click to copy article linkArticle link copied! P. Debye. ACS Legacy Archive. Open ...
[9]
[PDF] the brownian movement
Brownian motion, but also the order of magnitude of the paths described by the particles correspond completely with the results the theory. I will not ...
[10]
[PDF] on the Motion of Pendulums. By G. G. Stokes, M.A., Fellow of ...
On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. By G. G. Stokes, M.A., Fellow of Pembroke College, and Lucasian Pro- fessor of ...Missing: George | Show results with:George
[11]
https://www3.nd.edu/~powers/ame.60635/stokes1851.pdf
[12]
[PDF] Stokes-Einstein relation
Oct 21, 2022 · We can use this rule of thumb to estimate diffusion coefficients of objects in cells. For a protein with radius 3nm, for example, we obtain .
[13]
Measurement of the Translational Diffusion Coefficient ... - PubMed
Oct 20, 2021 · Dynamic light scattering (DLS) is the most common method to measure the diffusion coefficient of macromolecules.Missing: Debye 1947
[14]
Precise Measurement of Diffusion Coefficients using Scanning ...
Fluorescence correlation spectroscopy (FCS) was introduced as a method for the measurement of diffusion coefficients and concentrations of fluorescent molecules ...
[15]
Modern analytical ultracentrifugation in protein science: A tutorial ...
Determination of sedimentation coefficients allows for the modeling of the hydrodynamic shape of proteins and protein complexes. The computational treatment of ...<|control11|><|separator|>
[16]
[PDF] Sedimentation velocity : experiment and analysis
Apr 17, 2025 · The sedimentation velocity (SV) allows observing the movement of macromolecules in solution, submitting to centrifuge force, ...
[17]
What is the Z-Average Size Determined by DLS - HORIBA
Dynamic light scattering (DLS) results are often expressed in terms of the Z-average. The Z-average arises when DLS data is analyzed by the use of the technique ...
[18]
[PDF] Dynamic Light Scattering Distributions by Any Means
Abstract Dynamic light scattering (DLS) is an essential technique for nanoparticle size analysis and has been employed extensively for decades, ...<|separator|>
[19]
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, ...
[20]
The CONTIN algorithm and its application to determine the size ...
Jun 17, 2015 · We review a powerful regularization method, known as CONTIN, for obtaining the size distribution of colloidal suspensions from dynamic light scattering data.INTRODUCTION · II. THEORY · IV. RESULTS AND DISCUSSION
[21]
First and Second Concentration-Dependent Coefficients of ...
Calculating the hydrodynamic radius, RH, for each sample from the Stokes−Einstein relation RH = kBT/6πηsD0 (RH representation of eq 17 is RH = 1.215 × 10-9Mw ...
[22]
[PDF] Assessing Nonspecific Interactions with Light Scattering
The first-order correction, known as the second virial coefficient A2 (with units of mol·mL/g²), is the primary parameter of interest and provides a correction ...
[23]
Light scattering characterization of polystyrene latex with and ...
Using the Stokes–Einstein equation, the hydrodynamic radius can be inferred from the measured diffusion coefficient. Adsorbing polymer onto the surface of ...
[24]
Measurement of Particle Size Distribution in Turbid Solutions ... - NIH
Jan 9, 2017 · A protocol for measuring polydispersity of concentrated polymer solutions using dynamic light scattering is described.
[25]
Structural transformation of bovine serum albumin induced by ...
Aug 5, 2013 · The FCS study reveals a change of the hydrodynamic radius of BSA from 3.7 nm in the native state to 7.0 nm in the presence of 40% DMSO.Missing: Stokes | Show results with:Stokes
[26]
Quantitative, Diffusion NMR Based Analytical Tool To Distinguish ...
Feb 11, 2019 · The key questions in folding studies are the protein dimensions and the degree of folding. These properties are best characterized by the ...
[27]
[PDF] Distribution of Molecular Size within an Unfolded State Ensemble ...
Here, we have used SAXS and PFG-NMR to measure the radius of gyration and hydrodynamic radius of the unfolded state ensemble of the isolated N-terminal SH3 ...
[28]
A novel process for transcellular hemoglobin transport from ...
Nov 27, 2024 · The median value obtained corresponded to the tetramer radius obtained in vitro, suggesting that a tetramer is a predominant form of Hb in the ...
[29]
PMC
In support of this approximation, we note that the Stokes radius of tetrameric hemoglobin is in reasonably good agreement with the known dimensions of ...
[30]
The molecular basis for hydrodynamic properties of PEGylated ...
AUC measures the sedimentation coefficient s0, relative shape f/f0, and total effective hydration VS/ v ¯ of macromolecules that give rise to a Stokes radius (R ...
[31]
Size and Shape of Protein Molecules at the Nanometer Level ... - PMC
It is recalled that a gel filtration column fractionates proteins on the basis of their Stokes radius, not molecular weight. The molecular weight can be ...
[32]
Size Exclusion Chromatography | Analytical Chemistry
More protein laboratories are now combining Stokes radius measurements ... calibration for determination of protein molecular weight or hydrated radius.
[33]
Size-exclusion chromatography—a review of calibration ...
Recent developments are reviewed in size exclusion chromatographic calibration methodologies, including direct calibration by using narrow and broad polymer ...Missing: Stokes proteomics
[34]
Retention and Dissolution of Engineered Silver Nanoparticles in ...
May 1, 2012 · Engineered nanomaterials added to soil can be taken up by plants or invertebrates leading to toxicity (1; 60), but how the bioavailability and ...<|control11|><|separator|>
[35]
Transport of engineered silver (Ag) nanoparticles through partially ...
The hydrodynamic radius is approximately 50 nm (Klein et al., 2011, Neukum et al., 2014). The silver particles are stabilized by polyoxyethylene glycerol ...
[36]
Synthesis and Rheological Characterization of Latexes Stabilized by ...
Aug 26, 2019 · The hydrodynamic radius (Rh) of 0.02% HASE polymer system detd. from dynamic light scattering indicates that the latex particle swells with ...Missing: viscometry | Show results with:viscometry
[37]
Differential Mobility Analysis of Aerosols: A Tutorial - J-Stage
The aerosol then enters the differential mobility analyzer that transmits particles within a narrow range of mobilities across a sheath flow to a classified ...Missing: settling | Show results with:settling
[38]
Laboratory study of aerosol settling velocities using Laser Doppler ...
Both the Stokes and Epstein models allow simple predictive expressions relating settling velocity to particle size with only a single empirical parameter.1. Introduction · 2. Materials And Methods · 3. ResultsMissing: pollutants | Show results with:pollutants<|separator|>
[39]
Size-dependent cytotoxicity of silver nanoparticles in human lung cells
Feb 17, 2014 · The hydrodynamic size distribution was performed in cell medium (BEGM) using PCCS. Data is presented as density distribution by volume ...
[40]
In Vitro Methods for Assessing Nanoparticle Toxicity - PMC - NIH
The diffusion coefficient is used in conjunction with the Stokes-Einstein equation to estimate the hydrodynamic radius of the particles assuming that the ...