Biophysics, Colloid & Nanoparticle Science & Dynamic Light Scattering
Stokes-Einstein Nanoparticle Diffusion & RMS Walk Formula
Calculate Stokes-Einstein Brownian Translational Diffusion Coefficient D (µm²/s) from Temperature, Dynamic Viscosity, and Hydrodynamic Radius (nm), plus 3D and 1D Root-Mean-Square (RMS) Displacement (µm) and Characteristic Diffusive Transit Time.
Scale nanoparticle Brownian diffusion and Dynamic Light Scattering (DLS) sizing without SI exponent slips!
In lipid-nanoparticle (LNP) drug delivery, microfluidics, and Dynamic Light Scattering (DLS), spherical particles of hydrodynamic radius r_H (nm) suspended in a liquid of dynamic viscosity η (mPa·s) at absolute temperature T (K) undergo thermal Brownian motion with Stokes-Einstein diffusivity D = (k_B * T) / (6 * π * η * r_H)! Combining Boltzmann’s constant k_B = 1.38065 × 10^-23 J/K with 1 mPa·s = 10^-3 Pa·s, 1 nm = 10^-9 m, and 1 m² = 10^12 µm² collapses all powers of ten into one clean engineering formula: D (µm²/s) = (13.8065 * T_K) / (6 * π * η_mPas * r_nm)! Crucially, Brownian displacement grows with the SQUARE ROOT of time—3D RMS displacement is SQRT(6 * D * t) whereas 1D channel projection is SQRT(2 * D * t), so crossing a gap L takes L² / (6 * D) seconds!
Verify on a 100 nm Diameter (50 nm Radius) Lipid Nanoparticle in Water—Temp_T = 300 K (B2), Water_Viscosity = 1.00 mPa·s (B3), Hydrodynamic_Radius = 50.0 nm (B4 -> Diffusivity D = (13.8065 * 300) / (6 * π * 1 * 50) = 4.3947 µm²/s in E2), and Observation_Window = 10.0 s (B5): 3D RMS Displacement is SQRT(6 * 4.394745 * 10) = 16.24 µm (E3), 1D Channel RMS Projection is SQRT(2 * 4.394745 * 10) = 9.38 µm (E4), and Characteristic 3D Time to Diffuse Across a 10 µm Synaptic/Microfluidic Gap is 100 / (6 * 4.394745) = 3.79 s (E5)!
| Stokes-Einstein Metric (300 K, η = 1.0 mPa·s, r_H = 50 nm, t = 10 s) | Formula | Verified Output |
|---|---|---|
| Diffusion Coefficient D (µm²/s) | (13.8065 * T_K) / (6 * PI() * η_mPas * r_nm) | 4.3947 µm²/s (2.1974 @ 100 nm) |
| 3D RMS Displacement (µm) | SQRT(6 * D * t_s) | 16.24 µm |
| 1D Projected RMS Walk (µm) | SQRT(2 * D * t_s) | 9.38 µm |
| 3D Diffusion Time for 10 µm (s) | (10^2) / (6 * D) | 3.79 s (7.58 s @ 100 nm) |
Stokes-Einstein Diffusion Coefficient Formula (µm²/s)
Validate T_K > 0, Viscosity_mPas > 0, Hydrodynamic_Radius_nm > 0, and Observation_Time_s > 0!
=LET(tk_v, B2, et_m, B3, rh_n, B4, tm_s, B5, IF(OR(tk_v <= 0, et_m <= 0, rh_n <= 0, tm_s <= 0), "Invalid Brownian Input", ROUND((13.8065 * tk_v) / (6 * PI() * et_m * rh_n), 4)))Aggregated Nanoparticle Doubling Check (Set Hydrodynamic Radius = 100 nm)
Simulate DLS dimer/aggregation doubling r_H B4 from 50 nm to 100 nm: Diffusion Coefficient E2 halves to 2.1974 µm²/s and 10 µm Diffusive Transit Time E5 doubles from 3.79 s to 7.58 s!
=IF(ISTEXT(E2), E2, ROUND(100 / (6 * ((13.8065 * B2) / (6 * PI() * B3 * B4))), 2))Try the example
Use the original values or change them to test your understanding.