Microrheology measurement using dynamic light scattering

Microrheology measurements are based on the use of flow tracers (generally nano- or microparticles) in the liquid system to test. A suitable method is used to record the trajectory of the random movement of the particles in the liquid medium. This method is used to investigate the rheological properties of protein solutions or gel structures, for example.

Measurement devices

BeNano Series Microrheology measurement DLSµR

The dynamic light scattering microrheology method (DLSµR), as implemented in the devices of the BeNano series, is a passive variant of the microrheology analysis method: The measurement is carried out without the application of an external force only due to the displacement of the tracer particles with a known average particle size caused by their thermal energy (Brownian motion). Dynamic light scattering is used as the detection technique; the entire measurement principle is shown in the following figure:

The sample with the tracer particles dispersed in it is placed in a transparent cuvette and is continuously illuminated with a monochromatic light source (laser). The light is elastically scattered by the particles and continuously detected at a certain angle – in the BeNano series either 90° or 173° relative to the incident radiation. Since the tracer particles move continuously, the intensity of the detected scattering intensity I(t) fluctuates as a function of time due to interference of the scattered radiation caused by the permanent change of location of the scattering centers, i.e. an analysis of the function I(t) provides information about the mobility of the scattering particles.
A so-called correlation analysis is carried out for the evaluation in the BeNano series: In principle, starting from the intensity I(t) at an arbitrarily chosen time t, the intensities I(t + τ), I(t + 2τ),… are determined and autocorrelated with the “original intensity” I(t), i.e. multiplied with each other and added up, and normalized. The resulting function G⁽²⁾(τ) is called the correlation function, τ is the so-called correlation time. Due to the fluctuation in the sign of the intensity, all correlation functions decrease continuously as a function of τ until they reach zero. Particles that move more slowly show a slower decay behavior than those that move quickly. It should be mentioned that, for practical reasons, the correlation function G⁽²⁾(τ) of the scattered light intensity is shown as a function of a dimensionless field autocorrelation function g⁽¹⁾(τ), which also drops to 0 (for details, see DIN ISO 22412:2018-09).
The actual relevant parameter of the DLSµR is the so-called mean square displacement (MSD, Δr²(τ)), which results mathematically from the dimensionless field autocorrelation function g⁽¹⁾(τ) via

Where q = 4πn⋅sin(θ/2)/λ is the scattering vector.

The MSD curve Δr²(τ) is a measure of the volume that particles performing a random movement traverse on average in a given time interval (decorrelation time in this case). To understand this measured parameter, the following figure shows the Brownian motion on a flat surface of several particles (trajectories of different colors) that “start” at the same place (center of the circles) at the same time:

If all particle positions are now averaged after an observation period τ, the particles sweep over an area that corresponds to the area of the concentric circles in the figure (for different observation periods). The resulting MSD curve then shows the increase in area (y-coordinate in nm²) as a function of the observation period (x-coordinate τ in µS).

The random movement of the tracer particles depends largely on the rheological properties of the surrounding medium under investigation: the so-called frequency-dependent, complex shear modulus G^*(ω) is relevant for assessing these viscous and elastic components of the medium. This basically describes the elastic component (storage modulus G‘(ω) = deformation energy stored in the system) and viscous component (loss modulus G‘‘(ω) = loss component of the energy due to internal friction, converted into heat) of the energy in an oscillating shear load of the frequency ω of a system:

When considering the DLSµR experiment (tracer particle in the medium under investigation, without the effect of an external force), the speed of the particle’s own movement reflects the viscosity, the particle displacement the elasticity: the faster the particle diffuses, the lower the viscosity, the greater the displacement of the particle, the lower the elasticity of the system. The MSD curve Δr²(τ) describes this inherent movement of the particles and is directly related to the frequency-dependent complex modulus G^*(ω) via the generalized Stokes-Einstein equation:

Here k_bT is the thermal energy and R the particle radius, which is assumed to be known.

The rheological variables G‘(ω) and G‘‘(ω) as well as other rheological parameters can therefore be determined directly from the MSD curve.

Literature and Norms

/1/ ISO 22412:2017 – Particle size analysis — Dynamic light scattering (DLS)
/2/ Particle World 25/2024: “BeNano series light scattering in life sciences – Now with DLS microrheology and DLS flow mode option for enhanced protein analysis”
/3/ Particle World 24/2023: “BeNano series for particle analysis: New Autotitrator and DLS” Microrheology Option”
/4/ Cai,P.C. et al. “Dynamic Light Scattering Microrheology for Soft and Living Materials”; Soft Matter. 2021, 17(7); 1929-1939