Diffusion Coefficient Estimate Calculator

1. Select the estimation method that matches your experiment.
2. Enter values and choose units beside each input.
3. Press Calculate to display results above the form.
4. Use Download CSV for spreadsheets and logging.
5. Use Download PDF for a shareable summary report.

Article: Diffusion Coefficient Estimation in Practice

1) Why the diffusion coefficient matters

The diffusion coefficient (D) quantifies how quickly particles, ions, or molecules spread due to random thermal motion. Its SI unit is m²/s, while cm²/s is common in electrochemistry and materials work. Typical magnitudes range from about 10⁻⁹ m²/s for small molecules in water to 10⁻¹⁶ m²/s or lower in dense solids.

2) Choosing an estimation route

Experiments rarely measure D directly; they measure motion, concentration change, or transport response. This calculator includes five routes: mean square displacement (MSD), Stokes–Einstein, Nernst–Einstein, Arrhenius temperature dependence, and a concentration profile method based on the complementary error function.

3) MSD: linking tracked motion to D

For random walks, the relationship ⟨r²⟩ = 2nDt connects the squared displacement to time and dimension n. In 2D tracking, doubling the observation time halves the estimated D for the same observed displacement. MSD is powerful for microscopy tracking, diffusion Monte Carlo checks, and short-time regimes.

4) Stokes–Einstein: fluid viscosity and particle size

Stokes–Einstein uses D = k_BT/(6π η r) and is most appropriate for spherical particles in Newtonian fluids. It predicts that increasing viscosity reduces D linearly, while doubling particle radius halves D. At 298 K and η ≈ 0.89 mPa·s, a 50 nm particle yields D on the order of 10⁻¹² m²/s.

5) Nernst–Einstein: mobility to diffusion

When electrical mobility μ is known, D can be estimated using D = μ k_BT/(|z|e). This is common for ions and charge carriers, and it highlights how temperature increases D. Because charge magnitude appears in the denominator, multivalent ions typically give smaller D for the same μ.

6) Arrhenius modeling across temperature

Many solids follow D(T) = D₀ exp(−E_a/(RT)). Plotting ln(D) versus 1/T often produces a near-linear trend whose slope relates to E_a. For E_a = 50 kJ/mol, raising temperature from 300 K to 350 K can increase D by several times, depending on D₀.

7) Concentration profiles and depth-time data

For semi-infinite diffusion with a constant surface concentration, profiles follow an erfc form. With measured C(x,t), C₀, and C_s, you can solve for D from a single depth-time point. This approach is common in carburizing, doping, and permeation studies when the boundary condition is controlled.

8) Interpreting results and reducing uncertainty

Always check that units match the physics: viscosity must be dynamic viscosity, radius should be hydrodynamic, and temperature should be in Kelvin for high accuracy. Use multiple methods when possible and compare orders of magnitude. If methods disagree, the assumptions may be violated, such as non-spherical particles, slip effects, concentration-dependent viscosity, or finite-domain boundaries.

FAQs

1) What is a realistic range for diffusion coefficients?

Liquids often fall near 10⁻¹¹ to 10⁻⁹ m²/s for small species. Polymers and glasses can be 10⁻¹⁴ to 10⁻¹⁸ m²/s. Metals and ceramics vary widely with temperature and defect density.

2) Should I use 1D, 2D, or 3D in the MSD method?

Use 1D for confined motion along a channel, 2D for surface diffusion or planar tracking, and 3D for bulk motion. Choosing the wrong dimension systematically biases D by the factor n.

3) When does Stokes–Einstein fail?

It can fail for non-spherical particles, non-Newtonian fluids, strong interactions, or when the particle is comparable to solvent structure. In such cases, the effective hydrodynamic radius or viscosity may differ from tabulated values.

4) How do I keep units consistent for the profile method?

C, C0, and Cs can be any concentration unit, but they must match. Depth and time are converted to meters and seconds internally. If Cs equals C0, normalization becomes undefined and D cannot be solved.

5) Why does temperature increase D in most models?

Thermal energy increases random motion and helps overcome barriers. Stokes–Einstein scales linearly with T, Nernst–Einstein also increases with T, and Arrhenius behavior can increase rapidly because the exponential factor becomes less negative.

6) What is the difference between D in m²/s and cm²/s?

They describe the same quantity in different units. 1 cm²/s equals 10⁻⁴ m²/s. This calculator shows both to match common reporting styles across physics, chemistry, and materials science.

7) How can I improve accuracy from noisy measurements?

Average repeated measurements, use longer time windows when appropriate, and fit multiple points rather than relying on one. In profiles, use several depths at the same time and regress D to reduce sensitivity to any single data point.