Brownian motion is a fundamental physical process governing the movement of particles suspended in a fluid. In the context of nanoparticle characterization, particle size is inferred from this motion by quantifying translational diffusion. While Brownian motion occurs in three dimensions, Nanoparticle Tracking Analysis (NTA) captures particle trajectories within a two-dimensional imaging plane. The validity of using 2D Brownian motion to determine 3D diffusion behavior is therefore central to accurate particle sizing.
Brownian Motion, Diffusion, and Particle Size
The random motion of particles arises from continuous collisions with solvent molecules. The rate of this motion is described by the translational diffusion coefficient (Dₜ), which is inversely related to particle size. In NTA, particle size is calculated using the Stokes–Einstein relationship:
(1) dₕ = kᴮT / (3πηνDₜ)
where dₕ is the hydrodynamic diameter, kᴮ is the Boltzmann constant, T is the absolute temperature, η is the dynamic viscosity of the medium, and Dₜ is the translational diffusion coefficient. Smaller particles diffuse more rapidly, while larger particles exhibit slower Brownian motion.
Three-Dimensional Brownian Motion and Optical Tracking
Brownian motion occurs simultaneously along three orthogonal spatial axes (x, y, and z). Diffusion along each axis is statistically independent and equivalent, meaning that motion in three dimensions can be represented as the superposition of three independent one-dimensional diffusion processes.
NTA instruments track particles using optical microscopy, recording displacement within the focal plane of the camera. As a result, only x–y motion is directly observed, while movement along the optical (z) axis remains unmeasured. The absence of direct z-axis information does not invalidate the measurement, but it does influence how diffusion statistics are accumulated.
Mean Squared Displacement and the 3D Diffusion Model
The diffusion coefficient is obtained from the mean squared displacement (MSD) of a particle over a defined time interval t, corresponding to the frame-to-frame acquisition time of the camera. For ideal Brownian diffusion, MSD scales linearly with time and with the number of spatial dimensions considered:
(2) One dimension (1D): ⟨x²⟩ = 2Dₜt
(3) Two dimensions (2D): ⟨x² + y²⟩ = 4Dₜt
(4) Three dimensions (3D): ⟨x² + y² + z²⟩ = 6Dₜt
These expressions form the basis of the 3D diffusion model used to relate measured particle displacements to translational diffusion.
Determining the Diffusion Coefficient from 2D Brownian Motion
When particle trajectories are measured in two dimensions, the translational diffusion coefficient is calculated as:
(5) Dₜ = ⟨x² + y²⟩ / 4t
This diffusion coefficient is then converted to hydrodynamic diameter using the Stokes–Einstein equation. Particle size distributions reported by NTA are therefore derived entirely from 2D Brownian motion measurements.
Statistical Considerations in 2D Measurements
For a fixed number of displacement steps, higher-dimensional measurements yield greater statistical precision per step. Conversely, lower-dimensional measurements require a larger number of steps to achieve equivalent accuracy. In NTA, this requirement is addressed by tracking particles over many consecutive frames, allowing sufficient displacement statistics to accumulate.
Because diffusion proceeds identically and independently along each spatial axis, the dimensionality of observation affects only the rate of statistical convergence, not the underlying diffusion behavior. With adequate tracking duration, 2D Brownian motion provides an accurate representation of 3D diffusion.
Practical Implications for NTA Measurements
An accurate interpretation of 2D Brownian motion in the context of 3D diffusion is essential for reliable nanoparticle sizing. Measurement duration, frame rate, and particle count all influence the precision of diffusion estimates. When these parameters are appropriately selected, NTA delivers reproducible particle size distributions that reflect true 3D Brownian motion behavior.
This principle underlies the widespread use of NTA for nanoparticle, extracellular vesicle, and exosome characterization.
3D Brownian Motion with 2D Optical Projection
Figure: Particles diffuse randomly in three dimensions (x, y, z), while the NTA imaging system records only the projected displacement in the x–y plane. Because diffusion is statistically independent and equivalent along each spatial axis, the measured 2D trajectories provide an accurate basis for determining the three-dimensional translational diffusion coefficient.
Key Takeaway
Although Brownian motion is inherently three-dimensional, particle size determination in NTA relies on well-established diffusion theory that allows 3D Brownian motion to be quantified from 2D trajectories. With sufficient tracking statistics, 2D Brownian motion measurements provide a robust and accurate basis for nanoparticle sizing.