Stokes-Einstein Particle Size from Diffusion (DLS)

Understanding the Stokes-Einstein particle size calculator — concept and principle

Dynamic light scattering (DLS) instruments do not measure size directly. They measure how fast scattered light fluctuates, which gives a diffusion coefficient D, and convert D to size with the Stokes-Einstein equation. This calculator does that conversion, so you can check an instrument report, convert literature data or see how temperature and viscosity affect the size of nanoparticles, proteins or liposomes.

The dashed ring is the solvent shell that moves with the particle — part of the hydrodynamic size.

How it is calculated

d_H = k_B·T / (3π·η·D)

k_B = 1.380649 × 10⁻²³ J/K, the Boltzmann constant.
T = absolute temperature: °C + 273.15.
η = solvent viscosity; mPa·s ÷ 1000 gives Pa·s (water at 25 °C ≈ 0.89 mPa·s).
D = diffusion coefficient in m²/s.

The result in metres is multiplied by 10⁹ to give nanometres.

The principle

Particles in a liquid are kicked randomly by solvent molecules (Brownian motion). Einstein showed that D equals thermal energy divided by the friction coefficient, D = k_B·T / f. For a sphere moving slowly through a viscous liquid, Stokes' law gives f = 6π·η·r = 3π·η·d. Combining them gives the formula. Higher temperature means faster diffusion; higher viscosity or larger particles mean slower diffusion. The result is the diameter of a sphere that diffuses at the same rate, including any solvation layer, hence "hydrodynamic" diameter.

Worked example

Inputs: D = 4.4 × 10⁻¹² m²/s, T = 25 °C, η = 0.89 mPa·s.
T = 298.15 K, so k_B·T = 4.1164 × 10⁻²¹ J.
Denominator: 3π × 0.00089 × 4.4 × 10⁻¹² = 3.6907 × 10⁻¹⁴.
d_H = 4.1164 × 10⁻²¹ / 3.6907 × 10⁻¹⁴ = 1.1153 × 10⁻⁷ m = 111.53 nm.

Each point is the calculator re-run with only “Diffusion coefficient (m²/s)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Diffusion coefficient (m²/s)	Hydrodynamic diameter (nm)
1.1e-12	446.1329264
2.2e-12	223.0664632
3.3e-12	148.7109755
4.4e-12	111.5332316
5.5e-12	89.22658529
6.6e-12	74.35548774
8.8e-12	55.76661581

How to read the result

Size is inversely proportional to D: halve D and the diameter doubles. Small proteins have D near 10⁻¹⁰ m²/s and sizes of a few nanometres; 100 nm particles sit near 4–5 × 10⁻¹² m²/s. DLS reports an intensity-weighted average (z-average), so a few aggregates can raise it; check the polydispersity index too.

Common mistakes

Using water's viscosity for buffers with glycerol, sugar or high salt.
Using °C instead of kelvin in a hand calculation.
Using 6π instead of 3π, which gives a radius.

Keep in mind: The equation assumes dilute, non-interacting spheres in a Newtonian solvent. Concentrated, charged, rod-like or aggregated samples need corrections.

Frequently asked questions

Why is the DLS size larger than the TEM size?

DLS includes the solvation shell, ligands and loose aggregates and weights large particles heavily; TEM sees only the dried core.

Does measurement temperature change the size?

The particle usually does not change, but T and η do. Water viscosity changes about 2% per kelvin near room temperature, so use the right η.

Does it work below 1 nm?

Stokes' law loses accuracy when the solute is not much larger than solvent molecules, so results are approximate.

Sources: A. Einstein, Annalen der Physik, 1905 (Brownian motion paper) · ISO 22412, Particle size analysis — Dynamic light scattering (DLS)

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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