Introduction to Particle Sizing with Dynamic Light Scattering System

Introduction to Dynamic Light Scattering for Particle Size Analysis

Dynamic Light Scattering (DLS) is a highly suitable technique for characterizing the size distribution of small particles, typically ranging from nanometres to a few micrometers, dispersed in a liquid medium. In DLS, a coherent laser is used to illuminate the sample, and the instrument monitors time-dependent fluctuations in the intensity of the scattered light. These fluctuations arise from the Brownian motion of the particles, which is driven by thermal energy in the solvent.

By analyzing these intensity fluctuations through correlation functions, DLS provides access to the particle diffusion coefficient and, via the Stokes–Einstein equation, to the hydrodynamic radius and size distribution.

Theory

Theory of Dynamic Light Scattering for Particle Size Analysis

Field and intensity correlation functions

The intensity fluctuation of the scattered light can be converted to a correlation function to quantitatively analyze the Brownian motion of the particles using the light scattering technique.

The intensity fluctuations in terms of the field-time correlation function is shown in Equation (1).

(1)

g^{(1)}(\tau) = \frac{\langle E(t)\,E^{*}(t+\tau)\rangle} {\langle E(t)\,E^{*}(t)\rangle^{2}}

The above function describes the correlation between the scattering intensity at time $\tau = 0$ to time $t + \tau$ , where $\tau$ is the time interval, and $E$ is the scattered electric field. It is not possible to directly measure the scattered electric field; however, by monitoring the scattered intensity, it is possible to construct the intensity-time autocorrelation function, $g^{2}(\tau)$ , using the following equation:

(2)

g^{(2)}(\tau) = \frac{\langle I(t)\,I(t+\tau)\rangle}{\langle I(t)\rangle^{2}} \;

Where intensity $I(t) \propto |E(t)|^{2}$ . The Siegert relation can convert the intensity-time autocorrelation function $g^{2}(\tau)$ into the field-time autocorrelation function:

(3)

g^{(2)}(\tau) = 1 + B [g^{(1)}(\tau)]^2 \;

In Equation 3, $B$ is the factor that depends on the experimental geometry.

Exponential decay and diffusion coefficient

For monodisperse particles in solution, $g^{(1)}(\tau)$ decays exponentially:

(4)

g^{(1)}(\tau) = \exp(-\Gamma \tau) \;

Where the decay rate $\Gamma$ will be:

(5)

\Gamma = D_{\mathrm{app}}\, q^{2} \;

Where $q$ is the scattering vector and can be calculated by:

(6)

q = \frac{4\pi n}{\lambda_i}\,\sin\!\left(\frac{\Theta}{2}\right) \;

Where $n$ is the refractive index of the solvent, ${\lambda_i}$ is the incident laser wavelength, ${\Theta}$ is the scattering angle.

From diffusion coefficient to hydrodynamic radius

To determine the decay rate ( $\Gamma$ ) from Dynamic Light Scattering measurements, two common analysis methods are used depending on the sample’s size distribution. For monomodal distributions, the Cumulant method is used, while for broader or polymodal distributions, the CONTIN algorithm ¹ ² is employed. By plotting $\Gamma$ against the square of the scattering vector $q^{2}$ across multiple angles (Equation 5), the translational diffusion coefficient $D_T$ can be extracted from the slope ³ ⁴. This coefficient describes the rate of the particle motion due to random Brownian motion. The obtained $D_T$ is then used in the Stokes-Einstein equation to calculate the hydrodynamic radius $R_H$ , providing a more accurate and angle-independent particle size:

(7)

R_H = \frac{kT}{6\pi \eta D_T}

Where:

$k$ is Boltzmann’s constant
$T$ is the absolute temperature
$\eta$ is the dynamic viscosity of the solvent

Geometry

Multi-Angle DLS and Scattering Geometry

For larger $(> 60\,\text{nm})$ or weakly polydisperse particles, scattering becomes angle-dependent, meaning that light is scattered asymmetrically at different scattering angles.

Measuring at several angles allows one to probe different parts of the scattering pattern, quantify the angle dependence of the apparent size, and combine all angles to obtain a more robust diffusion coefficient.

Conclusion

Dynamic Light Scattering (DLS) measures the fluctuations in scattered light from particles over time due to Brownian motion, using this information to determine the rate of diffusion of the particles in the solvent. From this diffusion behavior, one can calculate an effective hydrodynamic size and, with suitable analysis methods, obtain a size distribution.

Cumulant analysis is typically used for single, narrow size populations, while CONTIN is better suited for broader or multimodal samples. Together with the possibility of using multiple scattering angles, this framework provides a quantitative and flexible approach for extracting particle size information from DLS measurements.

References

Frisken, Barbara J., 2001. Applied Optics. 40, 4087–4091.DOI: https://doi.org/10.1364/AO.40.004087 ↩︎
Provencher, S. W., 1982. Comput. Phys. Commun. 27, 213–227. DOI: https://doi.org/10.1016/0010-4655(82)90173-4 ↩︎
Lehner, D., Lindner, H., Glatter, O., 2000. Langmuir 16, 1689–1695. DOI: https://doi.org/10.1021/la9910273 ↩︎
Van Der Zande, B.M.I., Dhont, J.K.G., Böhmer, M.R., Philipse, A.P., 2000. Langmuir 16, 451–458. DOI: https://doi.org/10.1021/la9900425 ↩︎