**2.1.1 Magnetic tweezers**

This is the oldest implementation of an active microrheology technique, and it has been recently reviewed by Conroy (Conroy, 2008). A modern design has been described by Keller et al. (2001). The method combines the use of strong magnets to manipulate embedded super-paramagnetic or ferromagnetic particles, with video microscopy to measure the displacement of the particles upon application of constant or time-dependent forces. Strong magnetic fields are required to induce a magnetic dipole in the beads and magnetic field gradients are applied to produce a force. The force exerted is typically in the range of 10 pN to 10 nN depending on the experimental details (Keller et al. 2001). The spatial resolution is typically in the range of 10-20 nm, and the frequency range is 0.01 – 1000 Hz. Three modes

Microrheology of Complex Fluids 149

Figure 2.b shows a detailed scheme of how an optical trap is created. Light carries a momentum, in the direction of propagation, that is proportional to its energy. Any change in the direction of light, by reflection or refraction, will result in a change of the momentum of the light. If an object bends the light, conservation momentum requires that the object must undergo an equal and opposite momentum change, which gives rise to a force acting on the subject. In a typical instrument the laser has a Gaussian intensity profile, thus the intensity at the center is higher than at the edges. When the light interacts with a bead, the sum of the forces acting on the particle can be split into two components: Fsc, the scattering force, pointing in the direction of the incident beam, and Fg, the gradient force, arising from the gradient of the Gaussian intensity profile and pointing in the plane perpendicular to the incident beam towards the center of the beam. Fg is a restoring force that pulls the bead into the center of the beam. If the contribution to Fsc of the refracted rays is larger than that of the reflected rays then a restoring force is also created along the beam direction and a stable trap exists. A detailed description of the theoretical basis and of modern experimental setups has been given in Refs. (Ou-Yang & Wei, 2010; Borsali & Pecora, 2008; Resnick, 2003) that also include a review of applications of optical and magnetic tweezers to problems of biophysical interest: ligand-receptor interactions, mechanical response of single chains of biopolymers, force spectroscopy of enzymes and membranes, molecular motors, and cell manipulation. A recent application of optical tweezers to study the non-linear mechanical response of redblood cells is given by Yoon et al. (2008). Finally, optical tweezers are also suitable for the

Fig. 2. a) Basic design of an optical tweezers instrument. b) Details of the physical principles

These techniques use the Brownian dynamics of embedded colloids to measure the rheology of the materials. Since passive methods use only the thermal energy of the beads, materials

study of interfacial rheology (Steffen et al., 2001).

leading to the optical trap.

**2.2 Passive techniques** 

of operation are possible: a viscosimetry measurement after applying a constant force, a creep response experiment after applying a pulse excitation, and the measurement of the frequency dependent viscoelastic moduli in response to an oscillatory stress (Riande et al., 2000). This technique has been extensively applied to characterize the bulk viscoelasticity of systems of biological relevance (Wilson & Poon, 2011; Gardel et al., 2005). Moreover, realtime measurements of the local dynamics have also been reported for systems which change in response to external stimuli (Bausch et al., 2001), and rotational diffusion of the beads has also been used to characterize the viscosity of the surrounding fluid and to apply mechanical stresses directly to the cell surfaces receptors using ligand coated magnetic colloidal particles deposited onto the cell membrane (Fabry et al., 2001). Finally, this technique is well suited for the study of anisotropic systems by mapping the strain-field, and for studying interfaces (Lee et al., 2009). In recent years (Reynaert et al., 2008) have described a magnetically driven macrorheometer for studying interfacial shear viscosities in which one of the dimensions of the probe (a magnetic needle) is in the μm range. This has allowed the authors to work at rather high values of the Boussinesq number, which is one of the typical characteristics of the microrheology techniques.

Fig. 1. Frequency and elasticity modulus range available to the different microrheological techniques. Continuous vertical represent the frequency range, and dashed arrows the range of shear moduli (G´, G´´) that are accessible to each technique. a)Video particle tracking. b) Optical Tweezers. c) Diffusing wave spectroscopy: upper line for transmission geometry, lower line for back geometry. d) Magnetic microrheology. e) Atomic Force Microscopy (AFM). Adapted from Waigh (2005).

#### **2.1.2 Optical tweezers**

This technique uses a highly focused laser beam to trap a colloidal particle, as a consequence of the momentum transfer associated with bending light. The most basic design of an optical tweezer is shown in Figure 2.a: A laser beam (usually in the IR range) is focused by a highquality microscope (high numerical aperture objective) to a spot in a plane in the fluid.

of operation are possible: a viscosimetry measurement after applying a constant force, a creep response experiment after applying a pulse excitation, and the measurement of the frequency dependent viscoelastic moduli in response to an oscillatory stress (Riande et al., 2000). This technique has been extensively applied to characterize the bulk viscoelasticity of systems of biological relevance (Wilson & Poon, 2011; Gardel et al., 2005). Moreover, realtime measurements of the local dynamics have also been reported for systems which change in response to external stimuli (Bausch et al., 2001), and rotational diffusion of the beads has also been used to characterize the viscosity of the surrounding fluid and to apply mechanical stresses directly to the cell surfaces receptors using ligand coated magnetic colloidal particles deposited onto the cell membrane (Fabry et al., 2001). Finally, this technique is well suited for the study of anisotropic systems by mapping the strain-field, and for studying interfaces (Lee et al., 2009). In recent years (Reynaert et al., 2008) have described a magnetically driven macrorheometer for studying interfacial shear viscosities in which one of the dimensions of the probe (a magnetic needle) is in the μm range. This has allowed the authors to work at rather high values of the Boussinesq number, which is one of

a b c d e

This technique uses a highly focused laser beam to trap a colloidal particle, as a consequence of the momentum transfer associated with bending light. The most basic design of an optical tweezer is shown in Figure 2.a: A laser beam (usually in the IR range) is focused by a highquality microscope (high numerical aperture objective) to a spot in a plane in the fluid.

Fig. 1. Frequency and elasticity modulus range available to the different microrheological techniques. Continuous vertical represent the frequency range, and dashed arrows the range of shear moduli (G´, G´´) that are accessible to each technique. a)Video particle tracking. b) Optical Tweezers. c) Diffusing wave spectroscopy: upper line for transmission geometry, lower line for back geometry. d) Magnetic microrheology. e) Atomic Force Microscopy

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104

Shear moduli (Pa)

the typical characteristics of the microrheology techniques.

10-2

**2.1.2 Optical tweezers** 

(AFM). Adapted from Waigh (2005).

10-1

100

101

102

Frequency range (Hz)

103

104

105

106

Figure 2.b shows a detailed scheme of how an optical trap is created. Light carries a momentum, in the direction of propagation, that is proportional to its energy. Any change in the direction of light, by reflection or refraction, will result in a change of the momentum of the light. If an object bends the light, conservation momentum requires that the object must undergo an equal and opposite momentum change, which gives rise to a force acting on the subject. In a typical instrument the laser has a Gaussian intensity profile, thus the intensity at the center is higher than at the edges. When the light interacts with a bead, the sum of the forces acting on the particle can be split into two components: Fsc, the scattering force, pointing in the direction of the incident beam, and Fg, the gradient force, arising from the gradient of the Gaussian intensity profile and pointing in the plane perpendicular to the incident beam towards the center of the beam. Fg is a restoring force that pulls the bead into the center of the beam. If the contribution to Fsc of the refracted rays is larger than that of the reflected rays then a restoring force is also created along the beam direction and a stable trap exists. A detailed description of the theoretical basis and of modern experimental setups has been given in Refs. (Ou-Yang & Wei, 2010; Borsali & Pecora, 2008; Resnick, 2003) that also include a review of applications of optical and magnetic tweezers to problems of biophysical interest: ligand-receptor interactions, mechanical response of single chains of biopolymers, force spectroscopy of enzymes and membranes, molecular motors, and cell manipulation. A recent application of optical tweezers to study the non-linear mechanical response of redblood cells is given by Yoon et al. (2008). Finally, optical tweezers are also suitable for the study of interfacial rheology (Steffen et al., 2001).

Fig. 2. a) Basic design of an optical tweezers instrument. b) Details of the physical principles leading to the optical trap.

### **2.2 Passive techniques**

These techniques use the Brownian dynamics of embedded colloids to measure the rheology of the materials. Since passive methods use only the thermal energy of the beads, materials

Microrheology of Complex Fluids 151

complexes, proteins, polymers, blends and micelles, electrically induced microflows, diffusion of polyelectrolytes onto polymer surfaces, normal and confined diffusion of molecules and polymers, quantum dots blinking, dynamics of polymer networks, enzyme kinetics and structural heterogeneities in ionic liquids (Winkler, 2007; Heuf et al., 2007; Ries & Schwille, 2008; Cherdhirankorn et al., 2009; Wöll et al., 2009; Guo et al., 2011). The use of microscopes makes FCS suitable for the study of the dynamics of particles at interfaces. Moreover, contrary to particle tracking techniques, it is not necessary to "see" the particles,

thus interfaces with nanometer sized particles can be studied (Riegler & Elson, 2001).

( ) ( ) <sup>2</sup> <sup>2</sup> 0 0 r t r t t r(t ) 2dDt

Δ = −− =

where the brackets indicate the average over all the particles tracked, and t0 the initial time. In case of diffusion in a purely viscous material or interface, α is equal to 1, and the usual linear relation is obtained between MSD and t. When the material or interface is viscoelastic, α becomes lower than 1 and this behavior is called sub-diffusive. It is worth noticing that sub-diffusivity can be found not only as a consequence of the elasticity of the material, but also due to particle interactions as concentration increases, an effect that is particularly important at interfaces. Anomalous diffusion is also found in many systems of biological interest where the Brownian motion of the particles is hindered by obstacles (Feder et al., 1996), or even constrained to defined regions (corralled motion) (Saxton & Jacobson, 1997). The diffusion coefficient is related to the friction coefficient, f, by the Einstein relation

<sup>D</sup> k TB

In 3D Stokes law, f=6πηa, applies and for pure viscous fluids the shear viscosity, η, can be directly obtained from the diffusion coefficient of the probe particle of radius a at infinite dilution. The situation is much more complex in the case of fluid interfaces, and it will be

Figure 3 shows a sketch of a typical setup for particle tracking experiments. A CCD camera (typically 30 fps) is connected to a microscope that permits to image either the interface prepared onto a Langmuir trough, or a plane into a bulk fluid. The series of images are transferred to a computer to be analyzed to extract the trajectories of a set of particles. Figure 4.a shows typical results of MSD obtained for a 3D gel, combining DWS and particle tracking techniques which shows a very good agreement between both techniques, and illustrates the broad frequency range that can be explored. Figure 4.b shows a typical set of results for the MSD of a system of latex particles (1 μm of diameter) spread at the water/n-octane interface. The analysis of MSD within the linear range in

The main idea in particle tracking is to introduce onto the interface a few spherical particles of micrometer size and follow their trajectories (Brownian motion) using videomicroscopy. The trajectories of the particles, either in bulk or on surfaces, allow one to calculate the mean square displacement, which is related to the diffusion coefficient, D, and the dimensions, d,

α

(2)

<sup>f</sup> <sup>=</sup> (3)

**2.2.3 Particle tracking techniques** 

in which the translational motion takes place by

discussed in more detail in the next section.

terms of Eq. (2) allows to obtain D.

must be sufficiently soft for the motion of the particles to be measure precisely. The resolution typically ranges from 0.1 to 10 nm and elastic modulus from 10 to 500 Pa can be measured with micron sized particles. Thermal fluctuations of particles in transparent bulk systems have traditionally been studied using light scattering techniques that allow one to measure the intensity correlation function from which the field correlation function g1(t) can be calculated, t being the time. For monodisperse particles g1(t) is directly related to the mean squared displacement of the particles, MSD, through

$$\mathbf{g}\_1(\mathbf{t}) = \exp\left[\mathbf{-}\mathbf{q}^2 \le \Delta \mathbf{r}^2(\mathbf{t}) \ge /\mathbf{6}\right] \tag{1}$$

q being the scattering wave vector (Borsali & Pecora, 2008). Once <Δr2(t)> is obtained, it is possible to calculate the real and imaginary components of the shear moduli, G' and G" (Oppong & de Bruyn, 2010).

## **2.2.1 Diffusion wave spectroscopy**

Diffusion wave spectroscopy, DWS, allows measurements of multiple scattering media, and therefore non-transparent samples can be studied. The output of the technique allows to calculate <Δr2(t)>, and because of the multiple scattering all q-dependent information is lost as photons average over all possible angles, thus resulting only in two possible scattering geometries: transmission and backscattering. The frequency range of both geometries is complementary (see Figure 1) spanning from 0.1 Hz to 1MHz. For bulk polymer solutions and gels excellent agreement of the G' and G" values obtained by DWS and those obtained with conventional rheology has been found (Dasgupta et al., 2002; Dasgupta & Weitz, 2005). Even though these light scattering techniques are quite powerful tools for bulk microrheology, they have been scarcely used to probe the rheology of interfaces; in fact, as far as we know, only in old papers of Rice's group a set-up was described to measure dynamic light scattering of polymer monolayers using evanescent waves (Lin et al., 1993; Marcus et al., 1996).
