Diffusion Coefficient Calculator

Estimate a particle's diffusion coefficient

What this calculator does. It estimates the translational diffusion coefficient of a small particle moving by Brownian motion in a viscous fluid. The core relation is the Einstein relation, D = kBT/ξ, where kB is the exact Boltzmann constant, T is absolute temperature, and ξ is the particle's hydrodynamic friction coefficient. For standard shapes, the calculator derives ξ from viscosity and geometry; for measured or simulated drag, the custom mode accepts ξ directly. The result is an idealized dilute-solution estimate, not a complete model of concentrated suspensions, molecular-scale solvent structure, slip boundaries, particle interactions, or non-Newtonian fluids.

When to use it. Typical uses include checking the expected mobility of colloids or nanoparticles, comparing how orientation changes disk or ellipsoid drag, estimating a Stokes radius experiment, and preparing a first-pass transport calculation before detailed simulation or measurement.

1. Select Particle shape. Choose a sphere, one of three disk orientations, one of three ellipsoid orientations, or Custom friction coefficient.
2. Enter Absolute temperature (T) in kelvin. Values must be finite and greater than zero; 298.15 K is room temperature near 25 °C.
3. Enter Solvent viscosity (η) in mPa·s. Water near 25 °C is about 0.89 mPa·s. Higher viscosity increases drag and lowers D.
4. Enter Radius / major semiaxis (a) in nanometres. For ellipsoids, also enter Minor semiaxis (b); this implementation requires a > b so the logarithmic slender-body expressions remain meaningful.
5. Read Diffusion coefficient (D), Friction coefficient (ξ), and Thermal energy (kBT). The comparison table shows what the same size and solvent would imply under the other compatible shape models.
6. Use Download Excel to create a validated OOXML workbook from the current values, or Reset to restore the documented defaults.

Input guide. Particle shape is required and controls the drag formula. Absolute temperature accepts ordinary decimal or scientific notation in kelvin; zero, negative, blank, ambiguous comma-formatted, and nonnumeric entries are rejected. Solvent viscosity is required for every geometric shape and uses mPa·s, numerically equal to cP; it is converted internally to Pa·s. Radius / major semiaxis and Minor semiaxis are required positive lengths in nm. Custom friction coefficient is required only in custom mode and is entered in kg/s. A frequent mistake is mixing radius with diameter or entering viscosity in Pa·s while the field is labeled mPa·s.

Output guide. Diffusion coefficient is reported in m²/s and is an estimate from the selected hydrodynamic model. A larger value means faster random spreading. Friction coefficient is in kg/s; larger particles, more viscous fluids, and higher-drag orientations raise ξ. Thermal energy is kBT in joules and rises linearly with temperature. Relative to sphere is a dimensionless ratio Dshape/Dsphere: 1 means equal diffusion, values below 1 mean slower diffusion than the sphere under the same conditions.

Worked example. With a spherical particle of radius 2 nm in a fluid of viscosity 0.89 mPa·s at 298.15 K, ξ = 6πηa = 3.3552 × 10 ^{– 11} kg/s. Thermal energy is kBT = 4.1164 × 10 ^{– 21} J. Dividing gives D = 1.2269 × 10 ^{– 10} m²/s, matching the displayed result apart from the final presentation rounding.

Learn more. The NIST value of the Boltzmann constant supports the constant used here, while the Encyclopaedia Britannica overview of Brownian motion explains the random microscopic motion behind diffusion.