# Microrheology

Microrheology is a rheology method that uses colloidal tracer particles, dispersed within a sample, as probes. Tracer particles (with diameters ranging from 0.3 to 2.0 \(\mu\)m) may be naturally present in the system, as in suspensions and emulsions, or added to the medium under interest. The motion of the tracer particles reflects the rheological properties of their local environment. In purely viscous samples, tracer particles freely diffuse through the whole sample (Fig. 1a), which results in a particle mean square displacement \(\langle \Delta r^2(\tau) \rangle\) that is linear with time (red line in Fig. 2a).

\begin{gather*} \langle \Delta r^2(\tau) \rangle= 6 D \tau \end{gather*}

where *D* is the particle diffusion coefficient as expressed by the (Standard) Stokes-Einstein equation: \begin{gather*} D= \frac{k_B T}{6 \pi \eta R}. \end{gather*}

Fitting the measured \( \langle \Delta r^2(\tau)\rangle\) with these equations yields the viscosity \(\eta \) of a Newtonian solvent containing tracer particles with known radius \(R\). However, in materials that also contain elastic components, the \(\langle\Delta r^2(\tau)\rangle\) shows a more complex time-dependency, which makes the above equations not generally applicable.

*Fig 1: a) Particles freely diffuse in *

*purely viscous liquids and can*

*explore the whole sample.*

* Fig 1: b) Particles trapped in a *

*gel network.*

This can be illustrated with the example of a gelatin solution containing polystyrene tracer particles. At elevated temperatures (e.g. 50°C), the gelatin solution behaves purely liquid and the tracer can diffuse freely (Fig. 1a). However, at low temperature (15°C), the gelatin forms a gel and the tracer particles are trapped within the network (Fig. 1b). The thermal energy (Brownian motion) only allows local deformation with amplitudes that depend on the stiffness of the local environment. The restricted motion of the tracer particles is reflected in a \(\langle\Delta r^2(\tau)\rangle\) that shows a plateau (blue line in Fig. 2a) with amplitude corresponding for the maximal displacement. This is characteristic of the strength of the gelatin network.

Many materials are complex fluids that exhibit both viscous and elastic behaviors, and are called, for this reason, viscoelastic materials. Their response typically depends on the length and time scale probed in the measurements. A natural way to incorporate viscoelastic behavior is to generalize the Stokes-Einstein relation [1]

\begin{gather*} G^*(\omega)= \frac{k_B T}{\pi\, R\, i\, \omega\, \langle \Delta r^2(i\,\omega) \rangle} =G'(\omega)+i\,G''(\omega). \end{gather*}

This equation allows calculation of the frequency-dependent storage \(G’(\omega)\) and loss \(G''(\omega)\) moduli from the measured \(\langle \Delta r^2(\tau) \rangle \). In our example of the gelatin solution, the values obtained for \(G'(\omega)\) and \(G''(\omega)\) from microrheology are shown in Fig. 2b. At high temperatures (red line) \(G''(\omega)\) is proportional to the frequency \(\omega\), indicating a pure liquid, whereas \(G'(\omega)\) is very small and out of plotting range. However, at low temperatures (blue lines), \(G'(\omega)\) dominates \(G''(\omega)\) over an extended frequency range; only at very high frequency a cross-over to a domain where \(G''(\omega)\) is larger than \(G'(\omega)\) is observed. Such a behavior may be approximately described by the Kelvin-Voigt model (Fig. 2c), which consists of a spring and dashpot connected in parallel. The spring stands for elasticity of the gelatin network, whereas the dashpot represents a viscous damper that describes the dissipative effect of water around the gelatin network.

*Fig. 2: a) Mean square displacement \(\langle \Delta r^2(\tau) \rangle \) from freely diffusing (red) and trapped (blue) particles. *

*b) Storage \(G'(\omega)\) and and loss \(G''(\omega)\) moduli from freely diffusing (red) and trapped particles (blue). *

*c) Kelvin-Voigt model consists of a spring and a dashpot connected in parallel.*

Most microrheology methods are said to be "passive", i.e., they exclusively rely on thermal energy (i.e. Brownian motion) to displace the tracer particles within the sample. Only a few specialized methods (e.g. optical tweezers, magnetic microrheology) are "active", i.e., external (optical, magnetic) forces move tracer particles with energies that are stronger than thermal energy \(k_B T\). The advantage of active methods is that the amplitude of the particle displacements can be controlled, which allows either linear or non-linear rheology to be performed. On the other hand, passive microrheology is ideal for measurements in the linear viscoelastic region (LVR) because the weak thermal energy, \(k_B T\), ensures small amplitudes in the displacement of the tracer particles.

Microrheology can be further differentiated by the method used to measure the \(\langle \Delta r^2(\tau) \rangle \) of the tracer particles. The most common techniques are [2]:

## Particle tracking microrheology

Sequences of images are recorded with a video camera mounted on a microscope, and software is used to track the motion of the particles. From this, the \(\langle \Delta r^2(\tau) \rangle \) of the particles and, subsequently, the medium's rheological properties are computed. This technique yields additional information on inhomogeneous samples where particles probe different local environments. The disadvantages of this technique are that it requires laborious tuning of tracking parameters and data treatment. Moreover, the limited spatial resolution of optical microscopy restricts tracking to samples with low viscosity, i.e., where tracer particles can displace over significant distances.

**DLS microrheology **

The \(\langle \Delta r^2(\tau) \rangle \) of the tracer particles is extracted using **Dynamic Light Scattering (DLS)**. The spatial resolution is about as low as for microscopy-based microrheology. As a consequence, DLS microrheology can only be used for samples with low viscosity where tracer particles can displace over significant distances. The main advantage compared to microscopy-based microrheology is that the measurements and data treatment are straight forward and fast.

## DWS microrheology

The \(\langle \Delta r^2(\tau) \rangle \) of the tracer particles is extracted using **Diffusing Wave Spectroscopy (DWS)**. This technique probes much larger sample volumes than DLS- and microscopy-based microrheology, which yields enhanced statistics at measuring times on the order of one minute. In addition, DWS has a greatly improved spatial resolution with respect to DLS and microscopy. Therefore, it allows for measurements on highly viscous and even arrested (non-ergodic) samples where motion of tracer particles is strongly restricted. Finally, the accessibility to frequencies as high as 10^{7} rad/s is one of the most important features of DWS microrheology. For comparison, mechanical rheometers are typically limited to frequencies up to 10^{2} rad/s.

The** DWS RheoLab** from LS Instruments is a compact and versatile instrument, which takes advantage of modern light scattering technology. In contrast to traditional mechanical rheology, samples are sealed in glass cuvettes and, therefore, can be studied over extended time ranges to assess their stability or aging. Moreover, typical mechanical rheometers are limited to frequencies up to ~100 rad/s, and may take several hours to complete a frequency sweep. By harnessing our patented DWS Echo-technique, the DWS RheoLab can perform accurate and reliable measurements of \(G'(\omega)\) and \(G''(\omega)\) over an extended frequency range from 10^{-1} to 10^{6} rad/s in a matter of minutes.

*Fig. 3: Storage \(G'(\omega)\) and loss \(G''(\omega)\) moduli of an aqueous solution with 0.55% wt/v xanthan, measured by a mechanical rheometer and DWS RheoLab. For microrheology, polystyrene particles (980 nm diameter) were added at 1%wt/v. Note that DWS extends the measurements of \(G'(\omega)\) and \(G''(\omega)\) to considerably higher frequencies.*

[1] T.G. Mason and D. A. Weitz:

Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids,

Physical Review Letters 74, 1250-1253 (1995).