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

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 2D fluid (Stokes paradox).

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

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

$$\mathbf{b}\_{\Gamma} = \frac{1}{\mathbf{f}} = \frac{1}{4\pi (\eta\_1 + \eta\_2)\mathbf{R}\Lambda(\varepsilon)}\tag{10}$$

Where Λ(ε) is non-linear function of ε, 1 2 L R h η η ε η <sup>+</sup> <sup>=</sup> . Λ(ε) cannot be expressed analytically except for two limit cases,

$$\Lambda(\varepsilon) = \left[\varepsilon \left(\ln\left(\frac{2}{\varepsilon}\right) - \gamma + \frac{4}{\pi}\varepsilon - \frac{1}{2}\varepsilon^2 \ln\left(\frac{2}{\varepsilon}\right) + \mathcal{O}(\varepsilon^2)\right)\right]^{-1} \qquad \text{(Highly viscous members, e.p1)}$$
  $\Lambda(\varepsilon) = \frac{2}{\pi}$ 

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

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

π

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

Microrheology of Complex Fluids 159

theoretical point of view, the results presented by Danov are valid only in the limit E >>1, and for arbitrary values of the contact angle. Sickert & Rondelez (2003) were the first to applied Danov's ideas to obtain the surface shear viscosity by particle tracking using spherical microparticles trapped at the air-water interface, which was covered with Langmuir films. They have measured the surface viscosity of three monolayers formed by pentadecanoic acid (PDA), L-a-dipalmitoylphosphatidylcholine (DPPC) and N-palmitoyl-6 n-penicillanic acid (PPA) respectively. The values of the shear viscosities for PDA, DPPC and PPA reported were in the range of 1 to 11.10-10 N· s· m-1 in the liquid expanded region of the monolayer. These values are beyond the range that can be reached by macroscopic

Fischer considered that a monolayer cannot be considered as compressible. Due to the presence of a surfactant, Marangoni forces (forces due to surface tension gradients) strongly suppress any motion at the surface that compress or expands the interface. Any gradient in the surface pressure is almost instantly compensated by the fast movement of the surfactant at the interface given a constant surface pressure, behaving thus as a incompressible monolayer (Fischer assumed that the velocity of the 2D surfactant diffusion is faster than the movement of the beads). The fact that the drag on a disk in a monolayer is that of an incompressible surface has been verified experimentally by Fischer (2004). In the case of Langmuir films of polymers, the monolayer could be considered as compressible or incompressible depending on the rate of the polymer dynamics at the interface compared to the velocity of the beads probes. Bonales et al. (2007) have calculated the shear viscosity of two polymer Langmuir films using Danov's theory, and compared these values with those obtained by canal viscosimetry. Video Particle tracking and Danov's theory were used by Maestro et al. (2011.a) to show the glass transition in Langmuir films. Figure 8 shows the results obtained for a monolayer of poly(4-hydroxystyrene) onto water. For all the monolayers reported by Bonales et al. (2007) and Maestro et al. (2011.b) the surface shear viscosity calculated from Danov's theory was lower than that measured with the macroscopic canal surface viscometer. Similar qualitative conclusions were reached at by Sickert et al. (2007) for monolayers of fatty acids and phospholipids in the liquid expanded

mechanical methods, that usually have a lower limit in the range of 10-7 N· s· m-1.

**3.1.3 Fischer's theory for a sphere in a incompressible surfactant layer** 

s 12 ( ) ( )·a , a being the radius of the spherical particle:

Fischer et al. (2006) have numerically solved the problem of a sphere trapped at an interface with a contact angle θ moving in an *incompressible* surface. They showed that contributions due to Marangoni forces account for a significant part of the total drag. This effect becomes most pronounced in the limit of vanishing surface compressibility. In this limit the Marangoni effects are simply incorporated to the model by approximating the surface as incompressible. They solved the fluid dynamics equations for a 3D object moving in a monolayer of surface shear viscosity, ηs between two infinite viscous phases. The monolayer surface is assumed to be flat (no electrocapillary effects). Then the translational drag coefficient, kT,, was expressed as a series expansion of the Boussinesq number,

For B=0, and for an air-water interface (η1, η2=0), the numerical results for kT are fitted with

01 2 k k Bk O(B ) TT T =+ + (11)

region.

B = + η ηη

an accuracy of 3% by the formula,

fluid phases separating the interface. Danov et al. (1995) and Fischer et al. (2006) have made numerical calculations of the drag coefficient of spherical microparticles trapped at fluidfluid interfaces. While Danov considered the interface as compressible, Fischer assumed that the interface is incompressible, both authors predicted the dynamics of the particles adsorbed on bare fluid interfaces, i.e. with no surfactant monolayers (the so-called the limit of cero surface viscosity). The predictions of their theories are different, and will be discussed in detail below. More recently, Reynaert et al. (2007) and Madivala et al. (2009) have studied the dynamics of spherical, weakly aggregated, and of non-spherical particles at interfaces, though using macroscopic rheometers.

Danov et al. (1995) have calculated the hydrodynamic drag force and the torque acting on a micro spherical particle trapped at the air-liquid interface (they consider the viscosity of air to be zero) interface, and moving parallel to it. This model was later extended by Dimova et al. (2000) and by Danov et al. (2000) to particles adsorbed to flat or curved (spherical) interfaces separating two fluids of non vanishing viscosity. The interface was modeled as a *compressible*, 2D fluid characterized by two dimensionless parameters K and E defined as E a =ηsh ( ) η and K a =η<sup>d</sup> ( ) η , being ηsh and ηd the surface shear and dilational viscosity respectively (Note that E is the inverse of ε used by Hughes). Danov et al. made the following assumptions: 1) The movement implies a low Reynolds number, thus they ignored any inertial term; 2) the moving particle is not affected by capillarity or electrodipping; 3) the contact line does not move to respect to the particle surface, and 4) they considered E=K, i.e. the interface is compressible. With these assumptions they solved numerically the Navier-Stokes equation to obtain the values of the drag coefficient f as a function the contact angle and of E (or K). They presented their results in graphical form, and their results are reproduced in Figure 7.

Fig. 7. Left: Effect of contact angle on the diffusion coefficient of a particle trapped at a fluid interface according to Danov's theory. Ds0 is the diffusion coefficient for the bare interface. The different lines correspond to the following values of E (=K): 1) 0; 2) 1; 3) 5. Right: Effect of the surface to bulk shear viscosity on the diffusion coefficient. The different lines correspond to the following values of E (=K): 1) 0; 2) 1; 3) 5; 4) 10. Figures reproduced from Dimova et al. (2000).

These curves can be used to obtain the shear viscosity of compressible surfactant layer once one has obtained the diffusion coefficient from particle tracking experiments, D0, for a free interface and in the presence of a surfactant layer. It must be stressed that, from a strict

fluid phases separating the interface. Danov et al. (1995) and Fischer et al. (2006) have made numerical calculations of the drag coefficient of spherical microparticles trapped at fluidfluid interfaces. While Danov considered the interface as compressible, Fischer assumed that the interface is incompressible, both authors predicted the dynamics of the particles adsorbed on bare fluid interfaces, i.e. with no surfactant monolayers (the so-called the limit of cero surface viscosity). The predictions of their theories are different, and will be discussed in detail below. More recently, Reynaert et al. (2007) and Madivala et al. (2009) have studied the dynamics of spherical, weakly aggregated, and of non-spherical particles at

Danov et al. (1995) have calculated the hydrodynamic drag force and the torque acting on a micro spherical particle trapped at the air-liquid interface (they consider the viscosity of air to be zero) interface, and moving parallel to it. This model was later extended by Dimova et al. (2000) and by Danov et al. (2000) to particles adsorbed to flat or curved (spherical) interfaces separating two fluids of non vanishing viscosity. The interface was modeled as a *compressible*, 2D fluid characterized by two dimensionless parameters K and E defined as

respectively (Note that E is the inverse of ε used by Hughes). Danov et al. made the following assumptions: 1) The movement implies a low Reynolds number, thus they ignored any inertial term; 2) the moving particle is not affected by capillarity or electrodipping; 3) the contact line does not move to respect to the particle surface, and 4) they considered E=K, i.e. the interface is compressible. With these assumptions they solved numerically the Navier-Stokes equation to obtain the values of the drag coefficient f as a function the contact angle and of E (or K). They presented their results in graphical form,

Fig. 7. Left: Effect of contact angle on the diffusion coefficient of a particle trapped at a fluid interface according to Danov's theory. Ds0 is the diffusion coefficient for the bare interface. The different lines correspond to the following values of E (=K): 1) 0; 2) 1; 3) 5. Right: Effect

correspond to the following values of E (=K): 1) 0; 2) 1; 3) 5; 4) 10. Figures reproduced from

These curves can be used to obtain the shear viscosity of compressible surfactant layer once one has obtained the diffusion coefficient from particle tracking experiments, D0, for a free interface and in the presence of a surfactant layer. It must be stressed that, from a strict

of the surface to bulk shear viscosity on the diffusion coefficient. The different lines

, being ηsh and ηd the surface shear and dilational viscosity

interfaces, though using macroscopic rheometers.

 and K a =η<sup>d</sup> ( ) η

and their results are reproduced in Figure 7.

E a =ηsh ( ) η

Dimova et al. (2000).

theoretical point of view, the results presented by Danov are valid only in the limit E >>1, and for arbitrary values of the contact angle. Sickert & Rondelez (2003) were the first to applied Danov's ideas to obtain the surface shear viscosity by particle tracking using spherical microparticles trapped at the air-water interface, which was covered with Langmuir films. They have measured the surface viscosity of three monolayers formed by pentadecanoic acid (PDA), L-a-dipalmitoylphosphatidylcholine (DPPC) and N-palmitoyl-6 n-penicillanic acid (PPA) respectively. The values of the shear viscosities for PDA, DPPC and PPA reported were in the range of 1 to 11.10-10 N· s· m-1 in the liquid expanded region of the monolayer. These values are beyond the range that can be reached by macroscopic mechanical methods, that usually have a lower limit in the range of 10-7 N· s· m-1.

Fischer considered that a monolayer cannot be considered as compressible. Due to the presence of a surfactant, Marangoni forces (forces due to surface tension gradients) strongly suppress any motion at the surface that compress or expands the interface. Any gradient in the surface pressure is almost instantly compensated by the fast movement of the surfactant at the interface given a constant surface pressure, behaving thus as a incompressible monolayer (Fischer assumed that the velocity of the 2D surfactant diffusion is faster than the movement of the beads). The fact that the drag on a disk in a monolayer is that of an incompressible surface has been verified experimentally by Fischer (2004). In the case of Langmuir films of polymers, the monolayer could be considered as compressible or incompressible depending on the rate of the polymer dynamics at the interface compared to the velocity of the beads probes. Bonales et al. (2007) have calculated the shear viscosity of two polymer Langmuir films using Danov's theory, and compared these values with those obtained by canal viscosimetry. Video Particle tracking and Danov's theory were used by Maestro et al. (2011.a) to show the glass transition in Langmuir films. Figure 8 shows the results obtained for a monolayer of poly(4-hydroxystyrene) onto water. For all the monolayers reported by Bonales et al. (2007) and Maestro et al. (2011.b) the surface shear viscosity calculated from Danov's theory was lower than that measured with the macroscopic canal surface viscometer. Similar qualitative conclusions were reached at by Sickert et al. (2007) for monolayers of fatty acids and phospholipids in the liquid expanded region.

#### **3.1.3 Fischer's theory for a sphere in a incompressible surfactant layer**

Fischer et al. (2006) have numerically solved the problem of a sphere trapped at an interface with a contact angle θ moving in an *incompressible* surface. They showed that contributions due to Marangoni forces account for a significant part of the total drag. This effect becomes most pronounced in the limit of vanishing surface compressibility. In this limit the Marangoni effects are simply incorporated to the model by approximating the surface as incompressible. They solved the fluid dynamics equations for a 3D object moving in a monolayer of surface shear viscosity, ηs between two infinite viscous phases. The monolayer surface is assumed to be flat (no electrocapillary effects). Then the translational drag coefficient, kT,, was expressed as a series expansion of the Boussinesq number, B = + η ηηs 12 ( ) ( )·a , a being the radius of the spherical particle:

$$\mathbf{k}\_{\rm T} = \mathbf{k}\_{\rm T}^{0} + \mathbf{B} \mathbf{k}\_{\rm T}^{1} + \mathbf{O}(\mathbf{B}^{2}) \tag{11}$$

For B=0, and for an air-water interface (η1, η2=0), the numerical results for kT are fitted with an accuracy of 3% by the formula,

Microrheology of Complex Fluids 161

0 0 0

 θξ

 θ

ξ

ξ

attributed to specific interactions between the particles and the monolayer.

→

conditions which confirms the observation of Barentin et al. (2000).

was found for the water-n-octane interface.

**0**

**5**

**10**

**15**

**f**(θ)

**20**

**25**

**30**

(Danov) (Danov)

= = <sup>+</sup>

D0 being the diffusion coefficient of the beads at a free surface (compressible), and D→0 is the value of an incompressible monolayer which surface concentration is tending to zero. They found that this relation is not equal to 1 but to 0.84 for their systems and experimental

Figure 9 shows the friction coefficient for latex particles at the water-air interface obtained from particle tracking for polystyrene latex particles. It also shows the values calculated from Danov's and from Fischer's theories (notice that for the bare interface E = B =0). The figure clearly shows that both theories underestimate the experimental values over the whole θ range. An empirical factor of η(θ)exp/η(θ)Fisher = 1.8±0.2 brings the calculated values in good agreement with the experiments at all the contact angle values. A similar situation

The values of the shear viscosities calculated by Sickert & Rondelez (2003) by using the modified-Fisher theory are 2 or 3 times higher than the previous values. Sickert et al. (2007) also refers to a model developed by Stone which would be valid over the whole range of E, although only for a contact angle of 90º. Figure 10 shows clearly the large difference found between micro- and macrorheology for monolayers of poly(t-butyl acrylate) at the so-called Γ\*\* surface concentration (Muñoz et al., 2000). The macrorheology results have been obtained using two different oscillatory rheometers. The huge difference cannot be

In effect, Figure 11 shows that the values obtained are the same for particles of rather different surface characteristics. Moreover, the values calculated from the modified-Fisher's theory or by direct application of the GSE equation lead to almost indistinguishable surface shear viscosities. It must be stressed that in all the cases the contact angle used is the experimentally measured using the gel-trapping technique described by Paunov et al.

**0 20 40 60 80 100 120 140 160 180**

θ (**º**)

Fig. 9. Friction coefficients calculated from the experimental diffusion coefficients measured by particle tracking experiments (symbols), by Danov's theory (dotted line), by Fischer's

theory (dashed line), and by the corrected Fischer's theory (continuous line).

**<sup>g</sup>(**θ**)** <sup>3</sup><sup>π</sup> **experimental**

**g(**θ**) Fischer g(**θ**) Danov g'(**θ**)**

6π

**Sickert & Rondelez**

(Fisher) 0 1 0 T T D () () D ( ) k ( ) Ek ( )

θ

 θ

> θ

(14)

Fig. 8. Temperature dependence of the surface shear viscosity of a monolayer of poly(4 hydroxystyrene) at the air-water interface obtained by particle tracking (the insets show the corresponding values measured with a macroscopic canal viscometer. Left: experiments done at Π=8 mN·m-1. Right: triangles correspond to Π=3 mN·m-1 and circles to Π=2 mN·m-1. Notice that the results obtained by particle tracking are much smaller than those obtained with the canal viscometer. Data taken from Hilles et al. (2009).

$$\mathbf{k}\_{\rm T}^{0} = 6\pi \sqrt{\tanh\left(32\left(\frac{\rm d}{\rm R} + 2\right) \Big/ \left(9\pi^{2}\right)\right)}\tag{12}$$

where d is the distance from the apex of the bead to the plane of the interface (which defines the contact angle). Note that if d goes to infinity, <sup>0</sup> k 6 <sup>T</sup> = π , which is the correct theoretical value for a sphere in bulk (Stokes law). The translational drag in a half immersed sphere in a non viscous monolayer is <sup>0</sup> k 11 <sup>T</sup> ≈ which is about 25% higher than the drag on a sphere trapped at a free surface, k 3 <sup>T</sup> = π . This means that even in the absence of any appreciable surface viscosity the drag coefficient of an incompressible monolayer is higher than that of a free interface, and the data cannot be used to extract the surface shear viscosity using Danov's theory especially in the limit of low surface viscosities.

The numerical results for kT(1) are fitted within an accuracy of 3% to,

$$\mathbf{k}\_{\Gamma}^{(1)} = \begin{cases} -4\ln\left(\frac{2}{\pi}\arctan\left(\frac{2}{3}\right)\right) \left(\frac{\mathbf{a}^{3/2}}{\left(\mathbf{d}+3\mathbf{a}\right)^{3/2}}\right) & \text{(d / a > 0)}\\ -4\ln\left(\frac{2}{\pi}\arctan\left(\frac{\mathbf{d}+2\mathbf{a}}{3\mathbf{a}}\right)\right) & \text{(d / a < 0)} \end{cases} \tag{13}$$

Sickert & Rondelez (2003) have introduced in an ad-hoc way the incompressibility effect in Danov's theory by renormalizing his master curve (Figure 7 above) by the empirical value of 1.2, and they have later reanalyzed their data by combining the Danov's and Fischer's theories (Sickert et al., 2007). First they used the value determined by Danov et al. (2000) for the resistance coefficient of a sphere at a clean, compressible surface and at the contact angle of their experiments (50º). Afterwards, they used the predictions of Fischer et al. (2006) for a sphere in a surfactant monolayer (incompressible) with the contact angle corrected by the change in the surface tension, and in the case of E <<<1 (notice that this is the opposite Elimit than for the original Danov's theory),

**10-11**

**10-10**

**10-9**

η**s (N s m-1**

Fig. 8. Temperature dependence of the surface shear viscosity of a monolayer of poly(4 hydroxystyrene) at the air-water interface obtained by particle tracking (the insets show the corresponding values measured with a macroscopic canal viscometer. Left: experiments done at Π=8 mN·m-1. Right: triangles correspond to Π=3 mN·m-1 and circles to Π=2 mN·m-1. Notice that the results obtained by particle tracking are much smaller than those obtained

( ) 0 2

where d is the distance from the apex of the bead to the plane of the interface (which defines

value for a sphere in bulk (Stokes law). The translational drag in a half immersed sphere in a non viscous monolayer is <sup>0</sup> k 11 <sup>T</sup> ≈ which is about 25% higher than the drag on a sphere

surface viscosity the drag coefficient of an incompressible monolayer is higher than that of a free interface, and the data cannot be used to extract the surface shear viscosity using

2 2a 4ln arctan (d /a 0) <sup>3</sup> d 3a <sup>k</sup>

− > <sup>+</sup> <sup>≈</sup>

<sup>+</sup> − <

Sickert & Rondelez (2003) have introduced in an ad-hoc way the incompressibility effect in Danov's theory by renormalizing his master curve (Figure 7 above) by the empirical value of 1.2, and they have later reanalyzed their data by combining the Danov's and Fischer's theories (Sickert et al., 2007). First they used the value determined by Danov et al. (2000) for the resistance coefficient of a sphere at a clean, compressible surface and at the contact angle of their experiments (50º). Afterwards, they used the predictions of Fischer et al. (2006) for a sphere in a surfactant monolayer (incompressible) with the contact angle corrected by the change in the surface tension, and in the case of E <<<1 (notice that this is the opposite E-

3/2 (1)

( )

2 d 2a 4ln arctan (d /a 0) 3a

3/2

R

 π

π

≈ + (12)

. This means that even in the absence of any appreciable

<sup>d</sup> k 6 tanh 32 2 9

**)**

**3.0x10-3 3.2x10-3 3.4x10-3**

1/T

**10-8**

**1E-6**

ηs /Nsm-1

**2E-6 3E-6 4E-6**

**10-7**

**10-6**

**3,2 3,3 3,4 3,5 3,6**

, which is the correct theoretical

(13)

b)

**10<sup>3</sup> / T (K)**

**3,1 3,2 3,3 3,4 3,5 3,6**

with the canal viscometer. Data taken from Hilles et al. (2009).

T

π

Danov's theory especially in the limit of low surface viscosities. The numerical results for kT(1) are fitted within an accuracy of 3% to,

π

π

the contact angle). Note that if d goes to infinity, <sup>0</sup> k 6 <sup>T</sup> =

π

a)

**10<sup>3</sup> / T (K)**

**2.9 3.0 3.1 3.2 3.3 3.4 1E-6**

103 / T(K)

trapped at a free surface, k 3 <sup>T</sup> =

T

limit than for the original Danov's theory),

**10-7**

**10-6**

η**s (N s m-1**

**)**

**1E-5 1E-4 1E-3**

ηs / N · s · m -1

**10-5**

$$\frac{\mathbf{D}\_0}{\mathbf{D}\_{\rightarrow 0}} = \frac{\xi\_0^{\text{(Danny)}}(\theta)}{\xi^{\text{(Fisher)}}(\theta)} = \frac{\xi\_0^{\text{(Darcy)}}(\theta)}{\mathbf{k}\_\mathcal{\Gamma}^0(\theta) + \mathbf{E}\mathbf{k}\_\mathcal{\Gamma}^1(\theta)}\tag{14}$$

D0 being the diffusion coefficient of the beads at a free surface (compressible), and D→0 is the value of an incompressible monolayer which surface concentration is tending to zero. They found that this relation is not equal to 1 but to 0.84 for their systems and experimental conditions which confirms the observation of Barentin et al. (2000).

Figure 9 shows the friction coefficient for latex particles at the water-air interface obtained from particle tracking for polystyrene latex particles. It also shows the values calculated from Danov's and from Fischer's theories (notice that for the bare interface E = B =0). The figure clearly shows that both theories underestimate the experimental values over the whole θ range. An empirical factor of η(θ)exp/η(θ)Fisher = 1.8±0.2 brings the calculated values in good agreement with the experiments at all the contact angle values. A similar situation was found for the water-n-octane interface.

The values of the shear viscosities calculated by Sickert & Rondelez (2003) by using the modified-Fisher theory are 2 or 3 times higher than the previous values. Sickert et al. (2007) also refers to a model developed by Stone which would be valid over the whole range of E, although only for a contact angle of 90º. Figure 10 shows clearly the large difference found between micro- and macrorheology for monolayers of poly(t-butyl acrylate) at the so-called Γ\*\* surface concentration (Muñoz et al., 2000). The macrorheology results have been obtained using two different oscillatory rheometers. The huge difference cannot be attributed to specific interactions between the particles and the monolayer.

In effect, Figure 11 shows that the values obtained are the same for particles of rather different surface characteristics. Moreover, the values calculated from the modified-Fisher's theory or by direct application of the GSE equation lead to almost indistinguishable surface shear viscosities. It must be stressed that in all the cases the contact angle used is the experimentally measured using the gel-trapping technique described by Paunov et al.

Fig. 9. Friction coefficients calculated from the experimental diffusion coefficients measured by particle tracking experiments (symbols), by Danov's theory (dotted line), by Fischer's theory (dashed line), and by the corrected Fischer's theory (continuous line).

Microrheology of Complex Fluids 163

The set of microrheological techniques offer the possibility of studying the rheology of very small samples, of systems which are heterogeneous, and facilitate to measure the shear modulus over a broad frequency range. Particle tracking techniques are especially well suited for the study of the diffusion of microparticles either in the bulk or at fluid interfaces. Different types of mean squared displacements, MSD, (one-particle, two-particle) allow one to detect spatial heterogeneities in the samples. Even though good agreement has been found between micro- and macrorheology (at least when two-particle MSD is used) in bulk systems, the situation is still not clear for the case of fluid interfaces, where the shear surface microviscosity is much smaller than the one measured with conventional surface

This work has been supported in part by MICIN under grant FIS2009-14008-C02-01, by ESA under grant FASES MAP-AO-00-052, and by U.E. under grant Marie-Curie-ITN "MULTIFLOW". L.J. Bonales and A. Maestro are grateful to MICINN for their Ph.D. fellowships. We are grateful to Th.M. Fisher, R. Miller and L. Liggieri for helpful

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**4. Conclusions** 

rheometers.

discussions.

**6. References** 

**5. Acknowledgments** 

Fig. 10. Surface shear viscosity for monolayers of poly(t-butyl acrylate) as a function of the molecular weight and for a surface pressure of 16 mN·m-1. The lower curve corresponds to data obtained from particle tracking. The upper curve was obtained from conventional oscillatory rheometers.

Fig. 11. Surface shear viscosity of a monolayer of poly(t-butyl acrylate) (molecular weight 4.6 kDa) measured by particle tracking. Different microparticles where used: poly(styrene) of 1.6 and 5.7 µm (stabilized by sulfonate groups); poly(methylmethacrylate) stabilized by Coulombic repulsions (PMMA1), or by steric repulsions (PMMA2); Silica particles stabilized by Coulombic repulsions. Empty symbols: the viscosities were calculated using Fischer theory. Full symbols: calculated by the GSE equation.

(2003). This discrepancy between micro- and macrorheology in the study of monolayers seems to be a rather general situation (Schmidt et al., 2000; Khair & Brady, 2005; Oppong & de Bruyn, 2010; Lee et al., 2010) and no clear theoretical answer has been found so far.
