KEVOS® MAKEOS · Quality
Particle and Polymer Coil Size from Dynamic Light Scattering (Decay Rate, Diffusion Coefficient, Stokes–Einstein)
From the decay rate of a dynamic light scattering correlation function, what are the diffusion coefficient and hydrodynamic size of the particles or polymer coils, and are they within my specification?
Dynamic light scattering (photon correlation) reduced by hand: the scattering vector from the laser wavelength, the refractive index of the liquid and the scattering angle, the field decay rate (half the decay rate of the intensity correlation), the translational diffusion coefficient D = Γ ÷ q², the hydrodynamic radius and diameter from the Stokes–Einstein equation at the liquid’s temperature and viscosity, the product qRh that shows whether internal motion of flexible coils can enter the decay, and the size against your specification limits.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Instrument | ||
| Laser wavelength in vacuum | nm | Wavelength of the instrument’s laser in vacuum, from its specification (for example a helium–neon laser). |
| Refractive index of the liquid | Refractive index of the liquid at the laser wavelength and sample temperature. | |
| Scattering angle | ° | Angle between the laser beam and the detector, from the instrument settings. |
| Measurement | ||
| Correlation function the decay rate was fitted to | choice | Which correlation function the decay rate comes from; most instruments fit the intensity correlation. |
| Measured decay rate | 1/s | Decay rate fitted to the correlation function: the first cumulant, or 1 ÷ the decay time. |
| Sample temperature | °C | Temperature of the sample cell during the measurement. |
| Viscosity of the liquid at the sample temperature | mPa·s | Viscosity of the liquid alone at the sample temperature, from tables or a viscometer. |
| Specification | ||
| Smallest acceptable hydrodynamic diameter | nm | Smallest hydrodynamic diameter the specification accepts. |
| Largest acceptable hydrodynamic diameter | nm | Largest hydrodynamic diameter the specification accepts. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Instrument
What this means: Laser wavelength in vacuum
- What it is
- The wavelength of the instrument’s laser measured in vacuum; with the refractive index and angle it sets the length scale q at which the scattering is observed.
- Units and format
- Enter a number in nm: nm. The form accepts 200 nm to 2000 nm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 632.8 nm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- At the same decay rate a longer wavelength gives a smaller q and so a smaller calculated size.
- Get it right
- Enter the vacuum wavelength; the refractive index of the liquid is applied separately.
What this means: Refractive index of the liquid
- What it is
- The refractive index of the liquid the particles are dispersed in, at the laser wavelength and sample temperature; it shortens the wavelength inside the sample.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.332 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- q is proportional to the refractive index, so the size changes with its square.
- Get it right
- Use the liquid’s index, not the particles’; about 1.33 for water.
What this means: Scattering angle
- What it is
- The angle between the incident laser beam and the direction in which the detector collects scattered light.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 1° to 179°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger angle raises q; at the same decay rate the size scales with sin²(θ ÷ 2).
- Get it right
- Many instruments use 90° or a backscatter angle near 173°; read it from the instrument settings.
Measurement
What this means: Correlation function the decay rate was fitted…
- What it is
- Whether the decay rate was fitted to the intensity correlation g₂ − 1, which decays twice as fast as the field correlation g₁ that diffusion governs.
- Units and format
- Choose one of 2 options from the list: Intensity correlation g₂(τ) − 1; Field correlation g₁(τ).
- Example
- The form opens with “Intensity correlation g₂(τ) − 1” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Choosing the intensity correlation halves the decay rate used and doubles the size.
- Get it right
- Most instruments report the intensity correlation; check the software’s definition before choosing.
What this means: Measured decay rate
- What it is
- The decay rate fitted to the correlation function, the first cumulant or one divided by the decay time, which is proportional to how fast the particles diffuse.
- Units and format
- Enter a number in 1/s: 1/s. The form accepts 0.01 1/s or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6000 1/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The size is inversely proportional to the decay rate.
- Get it right
- A decay time in microseconds must be inverted and converted: 1 ÷ (100 µs) = 10,000 s⁻¹.
What this means: Sample temperature
- What it is
- The temperature of the sample during the measurement, which drives Brownian motion in the Stokes–Einstein equation and sets the liquid’s viscosity.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -20 °C to 150 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- At the same decay rate and viscosity the size is proportional to absolute temperature.
- Get it right
- Let the cell reach temperature before measuring; enter the viscosity at this same temperature.
What this means: Viscosity of the liquid at the sample temperatu…
- What it is
- The dynamic viscosity of the liquid alone, without particles, at the sample temperature; it resists the particles’ motion in the Stokes–Einstein equation.
- Units and format
- Enter a number in mPa·s: mPa·s. The form accepts 0.01 mPa·s to 10000 mPa·s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.89 mPa·s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The size is inversely proportional to the viscosity.
- Get it right
- Water is about 0.89 mPa·s at 25 °C and changes about 2 % per kelvin; use a measured value for other liquids.
Specification
What this means: Smallest acceptable hydrodynamic diameter
- What it is
- The smallest hydrodynamic diameter the product specification accepts.
- Units and format
- Enter a number in nm: nm. The form accepts 0 nm to 1000000 nm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 nm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the lower line of the specification.
- Get it right
- Compare only with limits set for DLS z-average sizes, not microscope or sieve sizes.
What this means: Largest acceptable hydrodynamic diameter
- What it is
- The largest hydrodynamic diameter the product specification accepts.
- Units and format
- Enter a number in nm: nm. The form accepts 0.1 nm to 1000000 nm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 nm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the upper line of the specification.
- Get it right
- Growth above the limit can show aggregation, ripening or swelling of the particles.
Method and formulas
- q = 4πn sin(θ ÷ 2) ÷ λ₀; the field correlation decays as exp(−Γt) with Γ = D q² (Klenin Eqs. 3.3-57 to 3.3-60); the intensity correlation g₂ − 1 = |g₁|² decays at 2Γ.
- Stokes–Einstein: Rh = kB T ÷ (6πηD), diameter d = 2Rh.
- For flexible coils the decay is purely translational when qR ≪ 1 (Klenin Eq. 3.3-67); otherwise measure at several angles and extrapolate Γ ÷ q² to q = 0.
- Example: 632.8 nm, n = 1.332, 90°: q = 18.70 µm⁻¹; an intensity decay rate of 6,000 s⁻¹ is a field rate of 3,000 s⁻¹, so D = 8.58 × 10⁻¹² m²/s and, at 25 °C and 0.89 mPa·s, Rh = 28.6 nm and d = 57.2 nm: in range, with qRh = 0.54.
Limitations
- One decay rate gives the intensity-weighted (z-average) size; it cannot describe a broad or multimodal distribution.
- Dilute, non-interacting particles in a Newtonian liquid are assumed; concentrated samples need extrapolation to zero concentration.
- The hydrodynamic size includes any solvation layer or adsorbed stabiliser and differs from microscope or sieve sizes.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a DLS instrument’s reported size from its decay rate
- Recalculate a size for a different liquid viscosity or temperature
- Accept or reject an emulsion, latex, dispersion or nanoparticle batch on size
- Judge whether a polymer measurement needs extrapolation to zero angle
Who it is for
- Quality and laboratory staff
- Formulation and process engineers
- Colloid and polymer scientists
What you get
- Scattering vector and field decay rate
- Diffusion coefficient
- Hydrodynamic radius and diameter
- qRh check for internal motion
- In or out of specification
Interpreting the result
- A larger decay rate means faster diffusion and smaller particles; at fixed decay rate the size is proportional to temperature ÷ viscosity.
- If the size changes with scattering angle, the sample is polydisperse or the particles are large or flexible compared with 1 ÷ q.
Before relying on the result
- Measure a size standard of known diameter on the same instrument and settings.
References
- Klenin V. J., Thermodynamics of Systems Containing Flexible-Chain Polymers (Elsevier, 1999; ISBN 0-444-82373-5), §3.3.2 (Brownian motion of macromolecules in solution and inelastic light scattering: Eqs. 3.3-57 to 3.3-61, the condition qR ≪ 1 of Eq. 3.3-67 and the method of cumulants). Method reference; no tables or figures are reproduced.
- Berne B. J. and Pecora R., Dynamic Light Scattering (Wiley, 1976; Dover, 2000).
- Koppel D. E., J. Chem. Phys. 57, 4814 (1972): the method of cumulants.
- ISO 22412, Particle size analysis: dynamic light scattering (DLS). Consult the current edition.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Brownian diffusion: Einstein's relation at work
In Fine Particles by the Numbers
What it covers: Calculate the diffusion coefficient and random displacement of LX-3's fume with Einstein's relation and slip, find where diffusion overtakes settling, and see why sample lines lose the smallest fume.
Method explained here · KEVOS® original book
Measuring chains and building an incoming check
In Polymer Chains by the Numbers
What it covers: Size dissolved coils by dynamic light scattering, compare the methods that measure chains, and combine them into a practical check of resin lots, regrind, cement batches, cleaners and containers.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.