**2.2.2 Fluorescence correlation spectroscopy (FCS)**

It is usually combined with optical microscopy, in particular confocal or two-photon microscopy. In these techniques light is focused on a sample and the fluorescence intensity fluctuations (due to diffusion, physical or chemical reactions, aggregations, etc.) can be measured in the form of a temporal correlation function. Similarly to what has been discussed in the light scattering technique, it is possible to obtain the MSD from the correlation function. In most experiments, Brownian motion drives the fluctuation of fluorescent-labeled molecules (or particles) within a well-defined element of the measurement cell. The samples have to be quite dilute, so that only few probes are within the focal spot (usually 1 – 100 molecules in one fL). Because of the tiny size of the confocal volume (approx. 0.2 fL), the measurements can be carried out in living cells or on cell membranes. In case that the interactions between two molecules wish to be studied, two options are available depending on their relative size. If their size is quite different, only one of them has to be labeled with a fluorescent dye (autocorrelation). If the diffusion coefficients of both molecules are similar, both have to be labeled with different dies (crosscorrelation). A detailed description of FCS techniques and of the data analysis has been recently given by Riegler & Elson (2001). Recent problems to which FCS has been applied include: dynamics of rafts in membranes and vesicles, dynamics of supramolecular

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

 g1(t) = exp [-q2 <Δr2(t)>/6] (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"

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;

It is usually combined with optical microscopy, in particular confocal or two-photon microscopy. In these techniques light is focused on a sample and the fluorescence intensity fluctuations (due to diffusion, physical or chemical reactions, aggregations, etc.) can be measured in the form of a temporal correlation function. Similarly to what has been discussed in the light scattering technique, it is possible to obtain the MSD from the correlation function. In most experiments, Brownian motion drives the fluctuation of fluorescent-labeled molecules (or particles) within a well-defined element of the measurement cell. The samples have to be quite dilute, so that only few probes are within the focal spot (usually 1 – 100 molecules in one fL). Because of the tiny size of the confocal volume (approx. 0.2 fL), the measurements can be carried out in living cells or on cell membranes. In case that the interactions between two molecules wish to be studied, two options are available depending on their relative size. If their size is quite different, only one of them has to be labeled with a fluorescent dye (autocorrelation). If the diffusion coefficients of both molecules are similar, both have to be labeled with different dies (crosscorrelation). A detailed description of FCS techniques and of the data analysis has been recently given by Riegler & Elson (2001). Recent problems to which FCS has been applied include: dynamics of rafts in membranes and vesicles, dynamics of supramolecular

mean squared displacement of the particles, MSD, through

**2.2.2 Fluorescence correlation spectroscopy (FCS)** 

(Oppong & de Bruyn, 2010).

Marcus et al., 1996).

**2.2.1 Diffusion wave spectroscopy** 

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).

#### **2.2.3 Particle tracking techniques**

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, in which the translational motion takes place by

$$
\left\langle \Delta \mathbf{r}^2 \left( \mathbf{t} \right) \right\rangle = \left\langle \left[ \mathbf{\bar{r}} \left( \mathbf{t}\_0 - \mathbf{t} \right) - \mathbf{\bar{r}} \left( \mathbf{t}\_0 \right) \right]^2 \right\rangle = 2 \,\text{d}\mathbf{D}\mathbf{t}^\alpha \tag{2}
$$

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

$$\mathbf{D} = \frac{\mathbf{k}\_\mathrm{B} \mathbf{T}}{\mathbf{f}} \tag{3}$$

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 discussed in more detail in the next section.

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

Microrheology of Complex Fluids 153

the heterogeneities of the sample. Figure 5 shows a comparison of the MSD obtained by single particle and two-point tracking for a solution of entangled F-actin solutions at different length scales from 1 to 100 μm (Liu et al., 2006). Both methods agree when the particle size is of the same order than the scale of the inhomogeneities of the system when the particle probes the average structure. Otherwise, the two methods lead to different results. In general, quite good agreement is found between two-point tracking experiments

**012345**

**t (s)**

MSDabs

MSDrel

\* 0.00012 0.001 = ±

b)

ρ

**0.0 0.2 0.4 0.6 0.8 1.0 0.0**

Fig. 4. a) Typical results of mean square displacement for a 3D gel made out of a

polysaccharide in water [44]. Filled points are from DWS experiments, and open symbols are from particle tracking. The continuous line is an eye guide. b) Mean square displacement (MSDabs), circles, and relative square displacement (MSDrel), triangles, for latex particles at the water/n-octane interface. Experimental details: set of 300 latex particles of 1 μm of diameter, surface charge density: -5.8 mC·cm-2, and reduced surface density, ρ\*=1.2·10-3 (ρ\*=ρa2), 25 ºC. Figure 4.a is reproduced from Vincent et al. (2007). Inset corresponds to a

**t (s)**

**4 D 8 D** 

and macroscopic rheology experiments.

**MSD (**

smaller time interval.

μ

**m2**

**)**

**0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0**

**MSD** (μ**m2**

)

One of the experimental problems frequently found in particle tracking experiments is that the linear behavior of the MSD vs. t is relatively short. This may be due to poor statistics in calculating the average in Eq.(2), or to the existence of interactions between particles. As it will be discussed below, this may be a problem in calculating the shear modulus from the MSD. An additional experimental problem may be found when the interaction of the particles with the fluid surrounding them is very strong, which may lead to changes in its viscoelastic modulus, or when the samples are heterogeneous at the scale of particle size, a situation rather frequent in biological systems, e.g. cells (Konopka & Weisshaar, 2004), or gels (Alexander & Dalgleish, 2007), or solutions of rod-like polymers (Hasnain & Donald, 2006). In this case the so-called "two-point" correlation method is recommended (Chen et al., 2003). In this method the fluctuations of pairs of particles at a distance Rij are measured for all the possible values of Rij within the system. Vector displacements of individual particles are calculated as a function of lag time, t, and initial time, t0.

Fig. 3. Typical particle tracking setup for 2D microrheology experiments: 1: Langmuir trough; 2: illumination; 3: microscope objective; 4: CCD camera; 5: computer; 6: thermostat; 7: electronics for measuring the temperature and the surface pressure.

Then the ensemble averaged tensor product of the vector displacements is calculated (Chen et al., 2003):

$$D\_{\alpha\beta}(\mathbf{r},\mathbf{r}) = \left\langle \Delta \mathbf{r}\_{\alpha}^{\mathrm{i}}(\mathbf{r},\mathbf{t}) \Delta \mathbf{r}\_{\beta}^{\mathrm{j}}(\mathbf{r},\mathbf{t}) \delta \left[\mathbf{r} - \mathcal{R}\_{\mathrm{ij}}\left(\mathbf{t}\_{0}\right)\right] \right\rangle\_{\mathbf{i}\neq\mathbf{j},\mathbf{t}} \tag{4}$$

where a and b are coordinate axes. The average corresponding to i = j represents the oneparticle mean-squared displacement.

Two-point microrheology probes dynamics at different length scales larger than the particle radius, although it can be extrapolated to the particle's size thus giving the MSD (Liu et al., 2006). In fact it has been found that for Rij close to the particle radius, the two-point MSD matches the tendency of the MSD obtained by tracking single particles. However, both sets of results are different for Rij's much larger than the particle diameter. This is a consequence of the fact that single particle tracking reflects both bulk and local rheologies, and therefore

One of the experimental problems frequently found in particle tracking experiments is that the linear behavior of the MSD vs. t is relatively short. This may be due to poor statistics in calculating the average in Eq.(2), or to the existence of interactions between particles. As it will be discussed below, this may be a problem in calculating the shear modulus from the MSD. An additional experimental problem may be found when the interaction of the particles with the fluid surrounding them is very strong, which may lead to changes in its viscoelastic modulus, or when the samples are heterogeneous at the scale of particle size, a situation rather frequent in biological systems, e.g. cells (Konopka & Weisshaar, 2004), or gels (Alexander & Dalgleish, 2007), or solutions of rod-like polymers (Hasnain & Donald, 2006). In this case the so-called "two-point" correlation method is recommended (Chen et al., 2003). In this method the fluctuations of pairs of particles at a distance Rij are measured for all the possible values of Rij within the system. Vector displacements of individual

particles are calculated as a function of lag time, t, and initial time, t0.

Fig. 3. Typical particle tracking setup for 2D microrheology experiments: 1: Langmuir trough; 2: illumination; 3: microscope objective; 4: CCD camera; 5: computer; 6: thermostat;

Then the ensemble averaged tensor product of the vector displacements is calculated (Chen

β

where a and b are coordinate axes. The average corresponding to i = j represents the one-

Two-point microrheology probes dynamics at different length scales larger than the particle radius, although it can be extrapolated to the particle's size thus giving the MSD (Liu et al., 2006). In fact it has been found that for Rij close to the particle radius, the two-point MSD matches the tendency of the MSD obtained by tracking single particles. However, both sets of results are different for Rij's much larger than the particle diameter. This is a consequence of the fact that single particle tracking reflects both bulk and local rheologies, and therefore

( ) <sup>i</sup> <sup>j</sup> ij 0 <sup>i</sup> j,t D (r, ) r (r,t) r (r,t) r R t

 δ <sup>≠</sup> =Δ Δ −

(4)

7: electronics for measuring the temperature and the surface pressure.

α

αβ

particle mean-squared displacement.

τ

et al., 2003):

the heterogeneities of the sample. Figure 5 shows a comparison of the MSD obtained by single particle and two-point tracking for a solution of entangled F-actin solutions at different length scales from 1 to 100 μm (Liu et al., 2006). Both methods agree when the particle size is of the same order than the scale of the inhomogeneities of the system when the particle probes the average structure. Otherwise, the two methods lead to different results. In general, quite good agreement is found between two-point tracking experiments and macroscopic rheology experiments.

Fig. 4. a) Typical results of mean square displacement for a 3D gel made out of a polysaccharide in water [44]. Filled points are from DWS experiments, and open symbols are from particle tracking. The continuous line is an eye guide. b) Mean square displacement (MSDabs), circles, and relative square displacement (MSDrel), triangles, for latex particles at the water/n-octane interface. Experimental details: set of 300 latex particles of 1 μm of diameter, surface charge density: -5.8 mC·cm-2, and reduced surface density, ρ\*=1.2·10-3 (ρ\*=ρa2), 25 ºC. Figure 4.a is reproduced from Vincent et al. (2007). Inset corresponds to a smaller time interval.

Microrheology of Complex Fluids 155

material, (b) no slip boundary conditions, (c) the fluid surrounding the sphere is

The application of the GSE is limited to a frequency range limited in the high frequency range by the appearance of inertial effects. The high frequency limit is imposed by the fact that the viscous penetration depth of the shear waves propagated by the particle motion must be larger than the particle size. The penetration depth is proportional to 2 1/2 (G \* / )

where ρ is the density of the fluid surrounding the particles, and for micron-sized particles in water is of the order of 1 MHz. On the other hand, the lower limit is set by the time at which compressional modes become significant compared to the shear modes excited by the

2

≥ (7)

= Δ (8)

G' a ξ

η

ξ being the characteristic length scale of the elastic network in which the particles move. Again, for the same conditions mentioned above, the low-frequency limit is in the range of 0.1 to 10 Hz. Figure 6.a shows the frequency dependence of the shear modulus for a 3D gel using two passive techniques: DWS and particle tracking. As it can be observed the agreement is very good. It must be stressed that, in order to obtain reliable Laplace or Fourier transforms of the MSD, it is necessary to measure the particle trajectories over long t periods (minutes), which makes absolutely necessary to eliminate any collective drift in the system. Very recently Felderhof (2009) has presented an alternative method for calculating the shear complex modulus from the velocity autocorrelation function, VAF, that can be calculated from the particle trajectories. An experimental difficulty associated to this method is that the VAF decays very rapidly, and therefore it is difficult to obtain many experimental

Under the same conditions assumed for the GSE equation, the creep compliance is directly

( ) ( ) <sup>2</sup> B <sup>a</sup> Jt r t k T π

Even though the GSE method has been applied to different bulk systems, few applications have been done for studying the complex shear modulus of interfaces and thin films (Wu &

The two-point correlation method also provides information about the viscoelastic moduli of the fluid in which the particles are embedded. In effect, the ensemble averaged tensor

> <sup>B</sup> rr rr k T <sup>1</sup> D (r,s) ; D D D 2 rsG(s) <sup>2</sup>

where D r,s rr ( ) is the Laplace transform of Drr(r,t) and the off-diagonal terms vanish. Figure 6.b compares the G' and G" results calculated for a solution of F-actin (MSD data shown in Figure 4) using one- and two-particle tracking methods. The results agree with those obtained by single-particle methods as far as the scale of the inhomogeneities is similar to the particle size, otherwise the single particle method is affected both by local and global

π

θθ

 φφ

<sup>=</sup> = = (9)

particle motion. An approximate value for the low frequency limit is given by

L

ω

ρω,

incompressible, and (d) no inertial effects.

data in the decay region.

Dai, 2006; Prasad & Weeks, 2009; Maestro et al., 2011).

product, Eq.(4), leads to (Chen et al., 2003)

related to the MSD by

Fig. 5. Comparison of one-particle (open symbols) and two-particle (closed symbols) MSD for a solution of F-actin using particles of radius 0.42 μm. Different average actin filaments are used: a) 0.5 μm, b) 2 μm, c) 5 μm, d) 17 μm. Notice that when the scale of the inhomogeneities of the solution is similar to the particle size both methods lead to the same results. The figure is reproduced from Liu et al. (2006).

For the case in which the particles are embedded in a viscoelastic fluid, particle tracking experiments allow one to obtain the viscoelastic moduli of the fluids. Manson & Weitz (1995) first in an ad-hoc way, and later Levine & Lubensky (2000) in a more rigorous way, proposed a generalization of the Stokes-Einstein (GSE) equation:

$$
\left< \Delta \tilde{\mathbf{r}}^2 \left( \mathbf{s} \right) \right> = \frac{2 \mathbf{k}\_B \mathbf{T}}{3 \pi \mathbf{a} \mathbf{s} \tilde{\mathbf{G}} \left( \mathbf{s} \right)}\tag{5}
$$

where G(s) is the Laplace transform of the stress relaxation modulus, s is the Laplace frequency, and a is the radius of the particles. An alternative expression for the GSE equation can be written in the Fourier domain as:

$$\mathbf{G}^\*(\boldsymbol{\omega}) = \frac{\mathbf{k}\_\mathrm{B} \mathbf{T}}{\pi \mathbf{a} \mathbf{i} a \mathfrak{S} \left< \Delta \mathbf{r}^2(\mathbf{t}) \right>} \tag{6}$$

where ℑ represents a unilateral Fourier transform, which is effectively a Laplace transform generalized for a complex frequency iω. Different methods have been devised to obtain G(s) from the experimental MSD including direct Laplace or Fourier transformations (Dasgupta et al., 2002; Evans et al., 2009), or analytical approximations (Mason, 2000; Wu & Dai, 2006). It must be stressed that the GSE equation is valid under the following approximations: (a) the medium around the sphere may be treated as a continuum material, which requires that the size of the particle be larger than any structural length scale of the

Fig. 5. Comparison of one-particle (open symbols) and two-particle (closed symbols) MSD for a solution of F-actin using particles of radius 0.42 μm. Different average actin filaments

inhomogeneities of the solution is similar to the particle size both methods lead to the same

For the case in which the particles are embedded in a viscoelastic fluid, particle tracking experiments allow one to obtain the viscoelastic moduli of the fluids. Manson & Weitz (1995) first in an ad-hoc way, and later Levine & Lubensky (2000) in a more rigorous way,

> ( ) ( ) <sup>2</sup> <sup>B</sup> 2k T

frequency, and a is the radius of the particles. An alternative expression for the GSE

( ) ( ) B 2

where ℑ represents a unilateral Fourier transform, which is effectively a Laplace transform generalized for a complex frequency iω. Different methods have been devised to obtain

 from the experimental MSD including direct Laplace or Fourier transformations (Dasgupta et al., 2002; Evans et al., 2009), or analytical approximations (Mason, 2000; Wu & Dai, 2006). It must be stressed that the GSE equation is valid under the following approximations: (a) the medium around the sphere may be treated as a continuum material, which requires that the size of the particle be larger than any structural length scale of the

k T G \*

π ω

ω

3 asG s π

is the Laplace transform of the stress relaxation modulus, s is the Laplace

ai r t

Δ = (5)

<sup>=</sup> ℑ Δ (6)

are used: a) 0.5 μm, b) 2 μm, c) 5 μm, d) 17 μm. Notice that when the scale of the

r s

results. The figure is reproduced from Liu et al. (2006).

equation can be written in the Fourier domain as:

where G(s)

G(s)

proposed a generalization of the Stokes-Einstein (GSE) equation:

material, (b) no slip boundary conditions, (c) the fluid surrounding the sphere is incompressible, and (d) no inertial effects.

The application of the GSE is limited to a frequency range limited in the high frequency range by the appearance of inertial effects. The high frequency limit is imposed by the fact that the viscous penetration depth of the shear waves propagated by the particle motion must be larger than the particle size. The penetration depth is proportional to 2 1/2 (G \* / ) ρω , where ρ is the density of the fluid surrounding the particles, and for micron-sized particles in water is of the order of 1 MHz. On the other hand, the lower limit is set by the time at which compressional modes become significant compared to the shear modes excited by the particle motion. An approximate value for the low frequency limit is given by

$$
\rho \eta\_{\perp} \geq \frac{G' \xi^2}{\eta \mathbf{a}} \tag{7}
$$

ξ being the characteristic length scale of the elastic network in which the particles move. Again, for the same conditions mentioned above, the low-frequency limit is in the range of 0.1 to 10 Hz. Figure 6.a shows the frequency dependence of the shear modulus for a 3D gel using two passive techniques: DWS and particle tracking. As it can be observed the agreement is very good. It must be stressed that, in order to obtain reliable Laplace or Fourier transforms of the MSD, it is necessary to measure the particle trajectories over long t periods (minutes), which makes absolutely necessary to eliminate any collective drift in the system. Very recently Felderhof (2009) has presented an alternative method for calculating the shear complex modulus from the velocity autocorrelation function, VAF, that can be calculated from the particle trajectories. An experimental difficulty associated to this method is that the VAF decays very rapidly, and therefore it is difficult to obtain many experimental data in the decay region.

Under the same conditions assumed for the GSE equation, the creep compliance is directly related to the MSD by

$$\mathbf{J(t)} = \frac{\pi \mathbf{a}}{\mathbf{k\_B T}} \left< \Delta \mathbf{r}^2 \left( \mathbf{t} \right) \right> \tag{8}$$

Even though the GSE method has been applied to different bulk systems, few applications have been done for studying the complex shear modulus of interfaces and thin films (Wu & Dai, 2006; Prasad & Weeks, 2009; Maestro et al., 2011).

The two-point correlation method also provides information about the viscoelastic moduli of the fluid in which the particles are embedded. In effect, the ensemble averaged tensor product, Eq.(4), leads to (Chen et al., 2003)

$$\tilde{\mathbf{D}}\_{\text{rr}}(\mathbf{r}, \mathbf{s}) = \frac{\mathbf{k}\_{\text{B}} \mathbf{T}}{2\pi \text{rs} \tilde{\mathbf{G}}(\mathbf{s})}; \ \ \ \mathbf{D}\_{\theta\theta} = \mathbf{D}\_{\phi\phi} = \frac{1}{2} \mathbf{D}\_{\text{rr}} \tag{9}$$

where D r,s rr ( ) is the Laplace transform of Drr(r,t) and the off-diagonal terms vanish. Figure 6.b compares the G' and G" results calculated for a solution of F-actin (MSD data shown in Figure 4) using one- and two-particle tracking methods. The results agree with those obtained by single-particle methods as far as the scale of the inhomogeneities is similar to the particle size, otherwise the single particle method is affected both by local and global

Microrheology of Complex Fluids 157

For using particle tracking techniques to get insight of the interfacial microrheology it is first necessary to study the diffusion of particles in the bare interface. For an inviscid interface the drag comes entirely from the upper and lower fluid phases (in the usual air-water interface only from the water subphase). The MSD of particles trapped at fluid interfaces depends on the surface concentration, and for very low surface concentration it is linear with time, thus the diffusion coefficients, *D0*, can be easily obtained. However, for high surface concentrations, even below the threshold of aggregation or fluid-solid phase transitions (Bonales et al., 2011), the MSD is no longer linear with time, but shows a sub-

In the case of particles trapped at interfaces Einstein's equation, Eq.(3), is still valid. However, one cannot calculate the friction coefficient using Stokes equation and directly substituting the interfacial shear viscosity. Instead, f is a function of the viscosities of the phases (η's), the geometry of the particle (the radius "a" for spheres), the contact angle between the probe particle and the interface (θ), etc. For a pure 2D system there is no solution for the slow viscous flow equations for steady translational motion of a sphere in a

Saffman & Delbrück (1975) and Hughes et al. (1981) have solved the problem of the motion of a thin disk immersed in a membrane of arbitrary viscosity, ηL separating two phases of viscosities η1 and η2. The height of the disk is assumed to be equal to the membrane

**3.1.1 Motion of a disk in and incompressible membrane of arbitrary viscosity** 

thickness, h. They obtained the following expression for the translational mobility,

1 1 <sup>b</sup>

( ) <sup>T</sup>

1 2

 η

R h η η

1

 ε

L

η

<sup>+</sup> <sup>=</sup>

= = + Λ (10)

. Λ(ε) cannot be expressed

f 4 R() πη

ε

 ε

These works have been generalized by Stone & Adjari (1998) and by Barentin et al. (2000).

The above theories are limited to non protruding particles (or high membrane viscosities). In particle tracking experiments spherical particles are used that are partially immersed in both

<sup>−</sup> Λ = −+ − + (Highly viscous membranes, e<1)

Λ = (Low viscous membranes, ε>1)

 ε

**3.1.2 Danov's model for a sphere in a compressible surfactant layer** 

with α<1, hence D0 must be obtained from the time

**3. Dynamics of particles at interfaces** 

dependence of the MSD in the limit of short times.

**3.1 Shear micro-rheology of monolayers at fluid interfaces** 

Where Λ(ε) is non-linear function of ε, 1 2

2 41 2 2 2 ( ) ln ln O( ) <sup>2</sup>

 εε

analytically except for two limit cases,

 γ

επ

ε

<sup>2</sup> ( ) ε

 ε

π

diffusive behavior, MSD(t) ~ t<sup>α</sup>

2D fluid (Stokes paradox).

rheology. Notice that the results of the two-point technique agree with those obtained with conventional macroscopic rheometers.

Fig. 6. Real and imaginary components calculated from the MSD shown in: a) the Figure 4.a, and b) Figure 5. Notice the good agreement between the results calculated from DWS (closed symbols) and single particle tracking (open symbols) in Figure 5.a. The solid and dotted lines are guides for G'and G" results, respectively. In Figure 6.b the open symbols refer to G", and the full ones to G'. Triangles correspond to single particle tracking and squares to two-particle tracking. Circles correspond to conventional macro-rheology. Figure 6.a was taken from Vincent et al. (2007) and Figure 6.b from Cherdhirankorn et al. (2009).

rheology. Notice that the results of the two-point technique agree with those obtained with

Fig. 6. Real and imaginary components calculated from the MSD shown in: a) the Figure 4.a, and b) Figure 5. Notice the good agreement between the results calculated from DWS (closed symbols) and single particle tracking (open symbols) in Figure 5.a. The solid and dotted lines are guides for G'and G" results, respectively. In Figure 6.b the open symbols refer to G", and the full ones to G'. Triangles correspond to single particle tracking and squares to two-particle tracking. Circles correspond to conventional macro-rheology. Figure 6.a was taken from Vincent et al. (2007) and Figure 6.b from Cherdhirankorn et al.

(a)

(b)

conventional macroscopic rheometers.

(2009).
