**2.1.3 Irreversible aggregation**

4 Hydrodynamics

If both particles have identical magnetic properties and knowing that the dipole moment aligns with the field, we obtain the following two expressions for potential energy and force:

where *α* is the angle between the direction of the magnetic field *H*ˆ , and the direction set by *r*ˆ

From the above equations, it follows that the radial component of the magnetic force is

that the dipolar interaction defines an hourglass-shaped region of attraction-repulsion in the complementary region (see Fig.1). In addition, the angular component of the dipolar interaction always tends to align the particles in the direction of the applied magnetic field. Thus, the result of this interaction will be an aggregation of particles in linear structures

The situation depicted here is very simplified, especially from the viewpoint of magnetic interaction itself. In the above, we have omitted any deviations from this ideal behaviour, such as multipole interactions or local field (Martin & Anderson (1996)). Multipolar interactions can become important when *μp*/*μ<sup>s</sup>* � 1. The local field correction due to the magnetic particles themselves generate magnetic fields that act on other particles, increasing the magnetic interaction. For example, when the magnetic susceptibility is approximately *χ* ∼ 1, this interaction tends to increase the angle of the cone of attraction from 55◦ to about 58◦

One type of fluid, called electro-rheological (ER fluids) is the electrical analogue of MRF. This type of fluid is very common in the study of kinematics of aggregation. Basically, the ER fluids consist of suspensions of dielectric particles of sizes on the order of micrometers (up to hundreds of microns) in conductive liquids. This type of fluid has some substantial differences with MRF, especially in view of the ease of use. The development of devices using electric fields is more complicated, requiring high power voltage; in addition, ER fluids have many more problems with surface charges than MRF, which must be minimized as much as possible in aggregation studies. However, basic physics, described above, are very similar in both

Chains of magnetic particles, once formed, interact with other chains in the fluid and with single particles. In fact, the chains may laterally coalesce to form thicker strings (sometimes called columns). This interaction is very important, especially when the concentration of particles in suspension is high. The first works that studied the interaction between chains of particles come from the earliest studies of external field-induced aggregation

(<sup>1</sup> <sup>−</sup> 3 cos<sup>2</sup> *<sup>α</sup>*)*r*<sup>ˆ</sup> <sup>−</sup> sin(2*α*)*α*<sup>ˆ</sup>

*r*ˆ + (*m*� *<sup>j</sup>* · *r*ˆ)*m*� *<sup>i</sup>* + (*m*� *<sup>i</sup>* · *r*ˆ)*m*� *<sup>j</sup>*

<sup>4</sup>*πr*<sup>3</sup> (<sup>1</sup> <sup>−</sup> 3 cos<sup>2</sup> *<sup>α</sup>*) (4)

1

<sup>3</sup> � <sup>55</sup>◦, so

(3)

(5)

(*m*� *<sup>i</sup>* · *m*� *<sup>j</sup>*) − 5(*m*� *<sup>i</sup>* · *r*ˆ)(*m*� *<sup>j</sup>* · *r*ˆ)

attractive when *α* < *α<sup>c</sup>* and repulsive when *α* > *αc*, where *α<sup>c</sup>* = arccos √

and also the attractive radial force in a 25% and the azimuth in a 5% (Melle (2002)).

systems, due to similarities between the magnetic and electrical dipolar interaction.

Then, we can obtain the force generated by *m*� *<sup>i</sup>* under *m*� *<sup>j</sup>* as:

*Ud*

� *Fd*

*ij* <sup>=</sup> *<sup>μ</sup>*0*μsm*<sup>2</sup>

*ij* <sup>=</sup> <sup>3</sup>*μ*0*μsm*<sup>2</sup> 4*πr*<sup>4</sup>

� *Fd*

*ij* <sup>=</sup> <sup>3</sup>*μ*0*μ<sup>s</sup>* 4*πr*<sup>4</sup>

and where *α*ˆ is its unitary vector.

oriented in the direction of *H*ˆ .

**2.1.2 Magnetic interaction between chains**

(Fermigier & Gast (1992); Fraden et al. (1989))

The irreversible aggregation of colloidal particles is a phenomenon of fundamental importance in colloid science and its applications. Basically, there are two basic scenarios of irreversible colloidal aggregation. The first, exemplified by the model of Witten & Sander (1981), is often referred to as Diffusion-Limited Aggregation (DLA). In this model, the particles diffuse without interaction between them, so that aggregation occurs when they collide with the central cluster. The second scenario is when there is a potential barrier between the particles and the aggregate, so that aggregation is determined by the rate at which the particles manage to overcome this barrier. The second model is called Colloid Reaction-Limited Aggregation (RLCA). These two processes have been observed experimentally in colloidal science (Lin et al. (1989); Tirado-Miranda (2001)).

These aggregation processes are often referred as fractal growth (Vicsek (1992)) and the aggregates formed in each process are characterized by a concrete fractal dimension. For example, in DLA we have aggregates with fractal dimension *Df* ∼ 1.7, while RLCA provides *Df* ∼ 2.1. A very important property of these systems is precisely that its basic physics is independent of the chemical peculiarities of each system colloidal i.e., these systems have universal aggregation. Lin et al. (1989) showed the universality of the irreversible aggregation systems performing light scattering experiments with different types of colloidal particles and changing the electrostatic forces in order to study the RLCA and DLA regimes in a differentiated way. They obtained, for example, that the effective diffusion coefficient (Eq.28) did not depend on the type of particle or colloid, but whether the process aggregation was DLA or RLCA.

The DLA model was generalized independently by Meakin (1983) and Kolb et al. (1983), allowing not only the diffusion of particles, but also of the clusters. In this model, named Cluster-Cluster Aggregation (CCA), the clusters can be added by diffusion with other clusters or single particles. Within these systems, if the particles are linked in a first touch, we obtain the DCLA model. The theoretical way to study these systems is to use the theory of von Smoluchowski (von Smoluchowski (1917)) for cluster-cluster aggregation among Monte Carlo

**2.2 Brownian movement and microrheology**

aggregate) formed by a number of particles *N*:

the Stokes-Einstein expression.

*t*. This equation has as a solution:

by Eq.14 is given by:

of radius *a* immersed in a fluid of viscosity *η* at temperature *T*:

where *Rg* is the radius of gyration defined as *Rg*(*N*) =

particles moving in the fluid. The diffusion equation says that:

 (Δ*r*)<sup>2</sup> = 

particle for a fixed *δt*. In two dimensions, the equations 14 and 15 are:

New Philosophical Journal, Vol. 5, April to September, 1828, pp. 358-371)

*<sup>ρ</sup>*(Δ*r*, *<sup>t</sup>*) = <sup>1</sup>


Robert Brown1 (1773-1858) discovered the phenomena that was denoted with his name in 1827, when he studied the movement of pollen in water. The explanation of Albert Einstein in 1905 includes the named Stokes-Einstein relationship for the diffusion coefficient of a particle

<sup>325</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

in Water Suspension: Effects of Low-Confinement Conditions and Electrostatics Interactions

*<sup>D</sup>* <sup>=</sup> *kBT* 6*πaη*

where *kB* is the Boltzmann constant. This equation can be generalized for an object (an

*<sup>D</sup>* <sup>=</sup> *kBT* 6*πηRg*

distance between the *i* particle to the centre of mass of the cluster. If *Rg* = *a*, we recover

Let's see how to calculate the diffusion coefficient *D* from the observation of individual

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>D</sup>*∇2*<sup>ρ</sup>*

where *ρ* is here the probability density function of a particle that spreads a distance Δ*r* at time

(4*πDt*)

If the Brownian particle moves a distance Δ*r* in the medium on which is immersed after a time *δt*, then the mean square displacement (MSD) weighted with the probability function given

The diffusion coefficient can be obtained by 15 and observing the displacement Δ*r* of the

The equations 13 and 15 are the basis for the development of a experimental technique known as microrheology (Mason & Weitz (1995)). This technique consists of measuring viscosity and other mechanical quantities in a fluid by monitoring, using video-microscopy, the movement <sup>1</sup> Literally: *While examining the form of these particles immersed in water, I observed many of them very evidently in motion [..]. These motions were such as to satisfy me, after frequently repeated observation, that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself.* (Edinburgh


2 

3/2 *e*

2 

*∂ρ*

*<sup>ρ</sup>*(Δ*r*, *<sup>t</sup>*) = <sup>1</sup>

1/*N* ∑*<sup>N</sup>*

*<sup>i</sup>*=<sup>1</sup> *<sup>r</sup>*<sup>2</sup>

<sup>−</sup>Δ*r*2/4*Dt* (14)

= 6*Dt* (15)

(4*πDt*) *<sup>e</sup>*−Δ*r*2/4*Dt* (16)

= 4*Dt* (17)

*<sup>i</sup>* , where *ri* is the

(13)

simulations (Vicsek (1992)). This theory considers that the aggregation kinetics of a system of *N* particles, initially separated and identical, aggregate; and these clusters join themselves to form larger objects. This process is studied through the distribution of cluster sizes *ns*(*t*) which can be defined as the number of aggregates of size *s* per unit of volume in the system at a time *t*. Then, the temporal evolution is given by the following set of equations:

$$\frac{dn\_s(t)}{dt} = \frac{1}{2} \sum\_{i+j=s} K\_{ij} n\_i n\_j - n\_s \sum\_{j=1} K\_{sj} n\_j \tag{6}$$

where the kernel *Kij* represents the rate at which the clusters of size *i* and *j* are joined to form a cluster of size *s* = *i* + *j*. All details of the physical system are contained in the kernel *Kij*, so that, for example, in the DLA model, the kernel is proportional to the product of the cross-section of the cluster and the diffusion coefficient. Eq.6 has certain limitations because only allows binary aggregation processes, so it is just applied to processes with very low concentration of particles.

A scaling relationship for the cluster size distribution function in the DCLA model was introduced by Vicsek & Family (1984) to describe the results of Monte Carlo simulations. This scaling relationship can be written as:

$$m\_s \sim s^{-2} \lg\left(s/S(t)\right) \tag{7}$$

where *S*(*t*) is the average cluster size of the aggregates:

$$S(t) \equiv \frac{\sum\_{s} s^{2} n\_{s}(t)}{\sum\_{s} s n\_{s}(t)} \tag{8}$$

and where the function *g*(*x*) is in the form:

$$g(\mathfrak{x}) \begin{cases} \sim \mathfrak{x}^{\Delta} \text{ if } \mathfrak{x} \ll 1\\ \ll 1 \text{ if } \mathfrak{x} \gg 1 \end{cases}$$

One consequence of the scaling 7 is that a temporal power law for the average cluster size can be deduced:

$$S(t) \sim t^{z} \tag{9}$$

Calculating experimentally the average cluster size along time, we can obtain the kinetic exponent *z*. Similarly to *S*(*t*) is possible to define an average length in number of aggregates *l*(*t*):

$$d(t) \equiv \frac{\sum\_{s} s \ n\_{s}(t)}{\sum\_{s} n\_{s}(t)} = \frac{1}{N(t)} \sum\_{s} s \ n\_{s}(t) = \frac{N\_{p}}{N(t)} \tag{10}$$

where *N*(*t*) = ∑ *ns*(*t*) is the total number of cluster in the system at time *t* and *Np* = ∑ *s ns*(*t*) is the total number of particles. Then, it is expected that *N* had a power law form with exponent *z*� :

$$N(t) \sim t^{-z'} \tag{11}$$

$$l(t) \sim t^{z'} \tag{12}$$

#### **2.2 Brownian movement and microrheology**

6 Hydrodynamics

simulations (Vicsek (1992)). This theory considers that the aggregation kinetics of a system of *N* particles, initially separated and identical, aggregate; and these clusters join themselves to form larger objects. This process is studied through the distribution of cluster sizes *ns*(*t*) which can be defined as the number of aggregates of size *s* per unit of volume in the system

where the kernel *Kij* represents the rate at which the clusters of size *i* and *j* are joined to form a cluster of size *s* = *i* + *j*. All details of the physical system are contained in the kernel *Kij*, so that, for example, in the DLA model, the kernel is proportional to the product of the cross-section of the cluster and the diffusion coefficient. Eq.6 has certain limitations because only allows binary aggregation processes, so it is just applied to processes with very low

A scaling relationship for the cluster size distribution function in the DCLA model was introduced by Vicsek & Family (1984) to describe the results of Monte Carlo simulations. This

∑*s*

∑*s sns*(*t*)

 <sup>∼</sup> *<sup>x</sup>*<sup>Δ</sup> if *<sup>x</sup>* � <sup>1</sup> � 1 if *x* � 1

One consequence of the scaling 7 is that a temporal power law for the average cluster size can

*S*(*t*) ∼ *t*

Calculating experimentally the average cluster size along time, we can obtain the kinetic exponent *z*. Similarly to *S*(*t*) is possible to define an average length in number of aggregates

> <sup>=</sup> <sup>1</sup> *<sup>N</sup>*(*t*) <sup>∑</sup>*<sup>s</sup>*

where *N*(*t*) = ∑ *ns*(*t*) is the total number of cluster in the system at time *t* and *Np* = ∑ *s ns*(*t*) is the total number of particles. Then, it is expected that *N* had a power law form with

*N*(*t*) ∼ *t*

*l*(*t*) ∼ *t z*�

−*z*�

*s*2*ns*(*t*)

*Kijninj* − *ns* ∑

*j*=1

*Ksjnj*, (6)

(8)

(11)

(12)

<sup>−</sup>2*g* (*s*/*S*(*t*)) (7)

*<sup>z</sup>* (9)

*<sup>N</sup>*(*t*) (10)

*s ns*(*t*) = *Np*

at a time *t*. Then, the temporal evolution is given by the following set of equations:

<sup>2</sup> ∑ *i*+*j*=*s*

*ns* ∼ *s*

*S*(*t*) ≡

*g*(*x*)

*s ns*(*t*)

*dns*(*t*) *dt* <sup>=</sup> <sup>1</sup>

concentration of particles.

be deduced:

exponent *z*�

:

*l*(*t*):

scaling relationship can be written as:

and where the function *g*(*x*) is in the form:

where *S*(*t*) is the average cluster size of the aggregates:

*l*(*t*) ≡

∑*s*

∑*s ns*(*t*) Robert Brown1 (1773-1858) discovered the phenomena that was denoted with his name in 1827, when he studied the movement of pollen in water. The explanation of Albert Einstein in 1905 includes the named Stokes-Einstein relationship for the diffusion coefficient of a particle of radius *a* immersed in a fluid of viscosity *η* at temperature *T*:

$$D = \frac{k\_B T}{6\pi a\eta} \tag{13}$$

where *kB* is the Boltzmann constant. This equation can be generalized for an object (an aggregate) formed by a number of particles *N*:

$$D = \frac{k\_B T}{6\pi\eta R\_\mathcal{g}}$$

where *Rg* is the radius of gyration defined as *Rg*(*N*) = 1/*N* ∑*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *<sup>r</sup>*<sup>2</sup> *<sup>i</sup>* , where *ri* is the distance between the *i* particle to the centre of mass of the cluster. If *Rg* = *a*, we recover the Stokes-Einstein expression.

Let's see how to calculate the diffusion coefficient *D* from the observation of individual particles moving in the fluid. The diffusion equation says that:

$$\frac{\partial \rho}{\partial t} = D \nabla^2 \rho$$

where *ρ* is here the probability density function of a particle that spreads a distance Δ*r* at time *t*. This equation has as a solution:

$$\rho \left(\Delta r, t\right) = \frac{1}{\left(4\pi Dt\right)^{3/2}} e^{-\Delta r^2/4Dt} \tag{14}$$

If the Brownian particle moves a distance Δ*r* in the medium on which is immersed after a time *δt*, then the mean square displacement (MSD) weighted with the probability function given by Eq.14 is given by:

$$
\left\langle \left( \Delta r \right)^{2} \right\rangle = \left\langle \left| r(t + \delta t) - r(t) \right|^{2} \right\rangle = 6Dt \tag{15}
$$

The diffusion coefficient can be obtained by 15 and observing the displacement Δ*r* of the particle for a fixed *δt*. In two dimensions, the equations 14 and 15 are:

$$
\rho(\Delta r, t) = \frac{1}{(4\pi Dt)} \, e^{-\Delta r^2/4Dt} \tag{16}
$$

$$\left\langle \left| r(t + \delta t) - r(t) \right|^2 \right\rangle = 4Dt \tag{17}$$

The equations 13 and 15 are the basis for the development of a experimental technique known as microrheology (Mason & Weitz (1995)). This technique consists of measuring viscosity and other mechanical quantities in a fluid by monitoring, using video-microscopy, the movement

<sup>1</sup> Literally: *While examining the form of these particles immersed in water, I observed many of them very evidently in motion [..]. These motions were such as to satisfy me, after frequently repeated observation, that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself.* (Edinburgh New Philosophical Journal, Vol. 5, April to September, 1828, pp. 358-371)

Fig. 2. Comparative analysis between the relative diffusion coefficient for the Brenner equation (Eq.20) and the first order approximation (Eq.21), as a function of the distance to the wall *z* for a particle of diameter 1 (*z*-unit are in divided by the diameter of the particle). These

When a particle diffuses near a wall, thanks to the linearity of Stokes equations, the diffusion coefficient can be separated into two components, one parallel to the wall *D* and the other perpendicular *<sup>D</sup>*⊥. In the literature, several studies in this regard can be found (Crocker (1997); Lin et al. (2000); Russel et al. (1989)). One particularly important is the study of Faucheux & Libchaber (1994) where measurements of particles confined between two walls are reported. This work provides a table with the diffusion coefficients obtained (theoretical and experimental) for different samples (different radius and particles) and different distances from the wall, from 1 to 12 *μ*m. For example, for a particle diameter 2.5 *μ*m, a distance of 1.3 *μ* m from the wall and with a density 2.1 times that of water, a diffusion coefficient *D*/*D*<sup>0</sup> = 0.32

<sup>327</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

in Water Suspension: Effects of Low-Confinement Conditions and Electrostatics Interactions

There are no closed analytical solutions for this type of problem, with the exception of that obtained for a sphere moving near a flat wall in the direction perpendicular to it (Brenner

where *α* ≡ arccosh (*z*/*a*) and *a* is the radius particle and *z* is the distance between the centre

Theoretical calculations in this regard are generally based on the methods of reflections, which involves splitting the hydrodynamic interaction between the wall and the particle in a linear superposition of interactions of increasing order. Using this method, it is possible to obtain a iterative solution for this problem in power series of (*a*/*z*). In the case of the perpendicular

2 sinh[(2*n* + 1)*α*]+(2*n* + 1) sinh[2*α*] 4 sinh2[(*<sup>n</sup>* <sup>+</sup> 1/2)*α*] <sup>−</sup> (2*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)<sup>2</sup> sinh2[*α*]

− 1

−<sup>1</sup>

(20)

(21)

<sup>∼</sup><sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>9</sup> 8 *a z* + *O a z* 3

two expressions are practically equal when *z* ≥ 1.5.

is obtained, where *D*<sup>0</sup> is the diffusion coefficient given by Eq.13.

*n*(*n* + 1) (2*n* − 1)(2*n* + 3)

> *<sup>D</sup>*⊥(*z*) *D*<sup>0</sup>

∞ ∑ *n*=1

**2.3.1 Particle-wall interaction.**

(1961)):

*<sup>D</sup>*⊥(*z*) *D*<sup>0</sup>

= 4 <sup>3</sup> sinh *<sup>α</sup>*

of the particle and the wall.

direction it is found:

of micro-nano particles (regardless their poralization). Thus, it is possible to obtain the viscosity of the medium simply by studying the displacement of the particle in the fluid. The microrheology has been widely used since the late nineties of last century (Waigh (2005)). Due to microrheology needs and for the sake of the analysis of the thermal fluctuation spectrum of probe spheres in suspension, the generalized Stokes-Einstein equation (Mason & Weitz (1995)) was developed. This expression is similar to Eq.13, but introducing Laplace transformed quantities:

$$
\tilde{D}(s) = \frac{k\_B T}{6\pi a s \,\tilde{\eta}\_s} \tag{18}
$$

where *s* is the Laplace frequency, and *η*˜*<sup>s</sup>* and *D*˜ (*s*) are the Laplace transformed viscosity and diffusion coefficient. The dynamics of the Brownian particles can be very different depending on the mechanical properties of the fluid. This equation is the base for the rheological study, by obtaining its viscoelastic moduli (Mason (2000)), of the complex fluid in which the particles are immersed.

If we only track the random motion of colloidal spheres moving freely in the fluid, we are talking of "passive" microrheology, but there are variations on this technique named "active" microrheology, for example, using optical tweezers (Grier (2003)). This technique allows to study the response of colloidal particles in viscoelastic fluids and the structure of fluids in the micro-nanometer scales (Furst (2005)), measure viscoelastic properties of biopolymers (like DNA) and the cell membrane (Verdier (2003)). Other useful methodologies are the two-particles microrheology (Crocker et al. (2000)) which allows to accurately measure rheological properties of complex materials, the use of rotating chains following an external rotating magnetic field (Wilhelm, Browaeys, Ponton & Bacri (2003); Wilhelm, Gazeau & Bacri (2003)) or magnetic bead microrheometry (Keller et al. (2001)).

#### **2.3 Hydrodynamics**

When we are talking about hydrodynamics in a colloidal suspension of particles we need to introduce the Reynolds number, Re, defined as:

$$\text{Re} \equiv \frac{\rho\_r \upsilon \, a}{\eta} \tag{19}$$

where *ρr* is the relative density, *a* is the particle radius, *v* is the velocity of the particle in the fluid which has a viscosity *η*. This number reflects the relation between the inertial forces and the viscous friction. If we are in a situation of low Reynolds number dynamics, as it usually happens in the physical situation here studied, the inertial terms in the Newton equations can be neglected, and *mx*¨ ∼= 0.

However, even in the case of low Reynolds number, the diffusion coefficient of particles in a colloidal system may have certain deviations from the expressions explained above. The diffusion coefficient can vary due to hydrodynamic interactions between particles, the morphology of the clusters, or because of the enclosure containing the suspension. When a particle moves near a "wall", the change in the Brownian dynamics of the particle is remarkable. The effective diffusion coefficient then varies with the distance of the particle from the wall (Russel et al. (1989)), the closer is the particle to the wall, the lower the diffusion coefficient. The interest of the modification on Brownian dynamics in confinement situations is quite large, for example to understand how particles migrate in porous media, how the macromolecules spread in membranes, or how cells interact with surfaces.

Fig. 2. Comparative analysis between the relative diffusion coefficient for the Brenner equation (Eq.20) and the first order approximation (Eq.21), as a function of the distance to the wall *z* for a particle of diameter 1 (*z*-unit are in divided by the diameter of the particle). These two expressions are practically equal when *z* ≥ 1.5.

#### **2.3.1 Particle-wall interaction.**

8 Hydrodynamics

of micro-nano particles (regardless their poralization). Thus, it is possible to obtain the viscosity of the medium simply by studying the displacement of the particle in the fluid. The microrheology has been widely used since the late nineties of last century (Waigh (2005)). Due to microrheology needs and for the sake of the analysis of the thermal fluctuation spectrum of probe spheres in suspension, the generalized Stokes-Einstein equation (Mason & Weitz (1995)) was developed. This expression is similar to Eq.13, but introducing Laplace transformed

*<sup>D</sup>*˜ (*s*) = *kB <sup>T</sup>*

where *s* is the Laplace frequency, and *η*˜*<sup>s</sup>* and *D*˜ (*s*) are the Laplace transformed viscosity and diffusion coefficient. The dynamics of the Brownian particles can be very different depending on the mechanical properties of the fluid. This equation is the base for the rheological study, by obtaining its viscoelastic moduli (Mason (2000)), of the complex fluid in which the particles

If we only track the random motion of colloidal spheres moving freely in the fluid, we are talking of "passive" microrheology, but there are variations on this technique named "active" microrheology, for example, using optical tweezers (Grier (2003)). This technique allows to study the response of colloidal particles in viscoelastic fluids and the structure of fluids in the micro-nanometer scales (Furst (2005)), measure viscoelastic properties of biopolymers (like DNA) and the cell membrane (Verdier (2003)). Other useful methodologies are the two-particles microrheology (Crocker et al. (2000)) which allows to accurately measure rheological properties of complex materials, the use of rotating chains following an external rotating magnetic field (Wilhelm, Browaeys, Ponton & Bacri (2003); Wilhelm, Gazeau & Bacri

When we are talking about hydrodynamics in a colloidal suspension of particles we need to

Re <sup>≡</sup> *<sup>ρ</sup><sup>r</sup> v a*

where *ρr* is the relative density, *a* is the particle radius, *v* is the velocity of the particle in the fluid which has a viscosity *η*. This number reflects the relation between the inertial forces and the viscous friction. If we are in a situation of low Reynolds number dynamics, as it usually happens in the physical situation here studied, the inertial terms in the Newton equations can

However, even in the case of low Reynolds number, the diffusion coefficient of particles in a colloidal system may have certain deviations from the expressions explained above. The diffusion coefficient can vary due to hydrodynamic interactions between particles, the morphology of the clusters, or because of the enclosure containing the suspension. When a particle moves near a "wall", the change in the Brownian dynamics of the particle is remarkable. The effective diffusion coefficient then varies with the distance of the particle from the wall (Russel et al. (1989)), the closer is the particle to the wall, the lower the diffusion coefficient. The interest of the modification on Brownian dynamics in confinement situations is quite large, for example to understand how particles migrate in porous media, how the

macromolecules spread in membranes, or how cells interact with surfaces.

*<sup>η</sup>* (19)

(2003)) or magnetic bead microrheometry (Keller et al. (2001)).

introduce the Reynolds number, Re, defined as:

6*πas η*˜*s*

(18)

quantities:

are immersed.

**2.3 Hydrodynamics**

be neglected, and *mx*¨ ∼= 0.

When a particle diffuses near a wall, thanks to the linearity of Stokes equations, the diffusion coefficient can be separated into two components, one parallel to the wall *D* and the other perpendicular *<sup>D</sup>*⊥. In the literature, several studies in this regard can be found (Crocker (1997); Lin et al. (2000); Russel et al. (1989)). One particularly important is the study of Faucheux & Libchaber (1994) where measurements of particles confined between two walls are reported. This work provides a table with the diffusion coefficients obtained (theoretical and experimental) for different samples (different radius and particles) and different distances from the wall, from 1 to 12 *μ*m. For example, for a particle diameter 2.5 *μ*m, a distance of 1.3 *μ* m from the wall and with a density 2.1 times that of water, a diffusion coefficient *D*/*D*<sup>0</sup> = 0.32 is obtained, where *D*<sup>0</sup> is the diffusion coefficient given by Eq.13.

There are no closed analytical solutions for this type of problem, with the exception of that obtained for a sphere moving near a flat wall in the direction perpendicular to it (Brenner (1961)):

$$\frac{D\_{\perp}(z)}{D\_{0}} = \left\{ \frac{4}{3} \sinh a \sum\_{n=1}^{\infty} \frac{n(n+1)}{(2n-1)(2n+3)} \left[ \frac{2 \sinh[(2n+1)a] + (2n+1) \sinh[2a]}{4 \sinh^{2}[(n+1/2)a] - (2n+1)^{2} \sinh^{2}[a]} - 1 \right] \right\}^{-1} \tag{20}$$

where *α* ≡ arccosh (*z*/*a*) and *a* is the radius particle and *z* is the distance between the centre of the particle and the wall.

Theoretical calculations in this regard are generally based on the methods of reflections, which involves splitting the hydrodynamic interaction between the wall and the particle in a linear superposition of interactions of increasing order. Using this method, it is possible to obtain a iterative solution for this problem in power series of (*a*/*z*). In the case of the perpendicular direction it is found:

$$\frac{D\_\perp(z)}{D\_0} \cong 1 - \frac{9}{8} \left(\frac{a}{z}\right) + O\left(\frac{a}{z}\right)^3 \tag{21}$$

The effect in the perpendicular direction is much lower and negligible (*O*(*ρ*−6)).

in Water Suspension: Effects of Low-Confinement Conditions and Electrostatics Interactions

and perpendicular to the axis of the chain, as follows (Doi & Edwards (1986)):

perpendicular, parallel and rotational to the axis of the chains (*<sup>D</sup>*, *<sup>D</sup>*⊥, *Dr*).

the others mentioned above by means of the relationship:

*vc* <sup>=</sup> *MgN γ*0

 <sup>1</sup> <sup>−</sup> *<sup>ρ</sup> ρp* 

*<sup>D</sup>* <sup>=</sup> *kBT* 2*πηa*

This result is based assuming point particles, but similar expressions are obtained by modelling the aggregates in the form of cylinders of length *L* and diameter *d* = 2*a*. Tirado & García (1979; 1980) provide diffusion coefficients for this objects in the directions

By using mesaurements of Dynamic Light Scattering (DLS), an effective diffusion coefficient, *D*eff, of the aggregates can be extracted (Koppel (1972)). This effective coefficient is related to

which is correct if *qL* >> 1 where *q* is the scattering wave vector defined as: *q* = 4*π*/*λ<sup>l</sup>* sin(*θ*/2), *λ<sup>l</sup>* is the wave length of the laser over the suspension and *θ* is the scattering

A particularly important effect is the sedimentation of the clusters or aggregates. It is essential, when a colloidal system is studied, determine the position of the aggregates from the wall, as well as knowing what the deposition rate by gravity is and when the equilibrium in a given layer of fluid is reached. The velocity *vc* experienced by a cluster composed of *N* identical spherical particles of radius *a* and mass *M* falling by gravity in a fluid without the presence of

where *g* is the value of the gravity acceleration, *ρ* is the fluid density, *ρp* is the density of the particles, *γ*<sup>0</sup> is t the drag coefficient and *D* the diffusion coefficient. If we have only one spheric particle, the last equation yields the classic result for the sedimentation velocity:

> *vp* <sup>=</sup> <sup>2</sup>*a*2*g*Δ*<sup>ρ</sup>* 9*η*

*L*2

<sup>=</sup> *MgDN kBT*

 <sup>1</sup> <sup>−</sup> *<sup>ρ</sup> ρp* 

*<sup>D</sup>*eff = *<sup>D</sup>*<sup>⊥</sup> +

ln *s*

*<sup>s</sup>* (26)

<sup>12</sup> *Dr* (28)

*<sup>D</sup>*<sup>⊥</sup> = *<sup>D</sup>*/2 (27)

When the aggregates are formed in the suspensions, their way of spreading in the fluid is expected to change. By analogy with the Stokes-Einstein equation, in which the diffusion coefficient depends on the inverse of particle diameter (*<sup>D</sup>* <sup>∼</sup> *<sup>a</sup>*−1), Miyazima et al. (1987) suggested that the diffusion coefficient depends on the inverse cluster size *s* in the form *<sup>D</sup>*(*s*) <sup>∼</sup> *<sup>s</sup>γ*, where *<sup>γ</sup>* is the coefficient that marks the degree of homogeneity of the kernel on the Smoluchowski equation (Eq. 6). The result for the diffusion coefficient *γ* = −1 is considered to be strictly valid for spherical particles that not interact hydrodynamically among them. However, in the case of an anisotropic system, as is the case of chain aggregates, the diffusion coefficient varies due to the hydrodynamic interaction in the direction parallel

<sup>329</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

**2.3.3 Anisotropic friction**

angle.

**2.3.4 Cluster sedimentation**

walls is (González et al. (2004)):

In the Fig.2 a comparison between the exact equation 20 and this first order expression 21 is plotted. These two expressions provide similar results when *z* ≥ 1.5.

In the case of the parallel direction to the wall we have the following approximation :

$$\frac{D\_{\mathbb{H}}(z)}{D\_{0}} \cong 1 - \frac{9}{16}\frac{a}{z} + \frac{1}{8}\frac{a^{3}}{z^{3}} - \frac{45}{256}\frac{a^{4}}{z^{4}} - \frac{1}{16}\frac{a^{5}}{z^{5}} + \dots \tag{22}$$

which is commonly used in their first order:

$$\frac{D\_{\parallel}(z)}{D\_{0}} \cong 1 - \frac{9}{16} \left(\frac{a}{z}\right) + O\left(\frac{a}{z}\right)^{3} \tag{23}$$

If we are thinking about one particle between two close walls, Dufresne et al. (2001) showed how it is possible to deduce, using the Stokeslet method (Liron & Mochon (1976)), a very complicated closed expression for the diffusion coefficients when *a* � *h*, being *h* the distance between the two walls. However, the method of reflections gives approximated theoretical expressions. Basically, there are three approximations that provide good results and which are different because of small modifications in the drag force. The first of these methods is the Linear Superposition Approximation (LSA) where the drag force over the sphere is chosen as the sum of the force that makes all the free fluid over the sphere. A second method is the Coherent Superposition Approximation (CSA) whose modification proposed by Bensech & Yiacoumi (2003) was named as Modified Coherent Superposition Approximation (MCSA) and gives the following expression:

$$\frac{D(z)}{D\_0} = \left\{ 1 + \left[ \mathbb{C}(z) - 1 \right] + \left[ \mathbb{C}(h - z) - 1 \right] + \sum\_{n=1}^{\infty} (-1)^n \frac{nh - z - a}{nh - z} \left[ \mathbb{C}(nh + z) - 1 \right] \right.$$

$$+ \sum\_{n=1}^{\infty} (-1)^n \frac{(n - 1)h + z - a}{(n - 1)h + z} \left[ \mathbb{C}((n + 1)h - z) - 1 \right] \right\}^{-1} \quad \text{(24)}$$

where the function *C*(*z*) is the inverse of the normalized diffusion coefficient (*D*0/*D*(*z*)) in the only one wall situation.

Another interesting physical configuration is the hydrodynamic coupling of two Brownian spheres near to a wall. Dufresne et al. (2000) showed that the collective diffusion coefficients in the directions parallel and perpendicular to the surface are related by a hydrodynamical coupling because of the fact that the surrounded fluid moved by one of the particles affects the other. This wall-induced effect may have an influence in the origin of some anomalous effects in experiments of confined microparticles in suspension.

#### **2.3.2 Particle-particle interaction**

Another effect of considerable importance, or at least, that we must take into account, is the hydrodynamic interaction between two particles. This effect is quantified by the parameter *ρ* = *r*/*a* where *r* is the radial distance between the centres of the particles and *a* is their radius. Crocker (1997) showed how the modification of the diffusion coefficient due to the mutual hydrodynamic interaction between the two particles varies in the directions parallel or perpendicular to the line joining the centres of mass. Finally, they obtained that the predominant effect is the one that occurs in the radial direction and which is given by:

$$\frac{D}{D\_0} \cong -\frac{15}{4\rho^4} \tag{25}$$

The effect in the perpendicular direction is much lower and negligible (*O*(*ρ*−6)).

## **2.3.3 Anisotropic friction**

10 Hydrodynamics

In the Fig.2 a comparison between the exact equation 20 and this first order expression 21 is

*<sup>z</sup>*<sup>5</sup> <sup>+</sup> ... (22)

(23)

In the case of the parallel direction to the wall we have the following approximation :

<sup>∼</sup><sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>9</sup> 16 *a z* + *O a z* 3

If we are thinking about one particle between two close walls, Dufresne et al. (2001) showed how it is possible to deduce, using the Stokeslet method (Liron & Mochon (1976)), a very complicated closed expression for the diffusion coefficients when *a* � *h*, being *h* the distance between the two walls. However, the method of reflections gives approximated theoretical expressions. Basically, there are three approximations that provide good results and which are different because of small modifications in the drag force. The first of these methods is the Linear Superposition Approximation (LSA) where the drag force over the sphere is chosen as the sum of the force that makes all the free fluid over the sphere. A second method is the Coherent Superposition Approximation (CSA) whose modification proposed by Bensech & Yiacoumi (2003) was named as Modified Coherent Superposition Approximation

> ∞ ∑ *n*=1

(−1)*<sup>n</sup>* (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*<sup>h</sup>* <sup>+</sup> *<sup>z</sup>* <sup>−</sup> *<sup>a</sup>*

where the function *C*(*z*) is the inverse of the normalized diffusion coefficient (*D*0/*D*(*z*)) in the

Another interesting physical configuration is the hydrodynamic coupling of two Brownian spheres near to a wall. Dufresne et al. (2000) showed that the collective diffusion coefficients in the directions parallel and perpendicular to the surface are related by a hydrodynamical coupling because of the fact that the surrounded fluid moved by one of the particles affects the other. This wall-induced effect may have an influence in the origin of some anomalous

Another effect of considerable importance, or at least, that we must take into account, is the hydrodynamic interaction between two particles. This effect is quantified by the parameter *ρ* = *r*/*a* where *r* is the radial distance between the centres of the particles and *a* is their radius. Crocker (1997) showed how the modification of the diffusion coefficient due to the mutual hydrodynamic interaction between the two particles varies in the directions parallel or perpendicular to the line joining the centres of mass. Finally, they obtained that the predominant effect is the one that occurs in the radial direction and which is given by: *D D*<sup>0</sup>

<sup>∼</sup><sup>=</sup> <sup>−</sup> <sup>15</sup>

(−1)*<sup>n</sup> nh* <sup>−</sup> *<sup>z</sup>* <sup>−</sup> *<sup>a</sup>*

*nh* <sup>−</sup> *<sup>z</sup>* [*C*(*nh* <sup>+</sup> *<sup>z</sup>*) <sup>−</sup> <sup>1</sup>]

<sup>4</sup>*ρ*<sup>4</sup> (25)

<sup>−</sup><sup>1</sup>

(24)

(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*<sup>h</sup>* <sup>+</sup> *<sup>z</sup>* [*C*((*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*<sup>h</sup>* <sup>−</sup> *<sup>z</sup>*) <sup>−</sup> <sup>1</sup>]

plotted. These two expressions provide similar results when *z* ≥ 1.5.

<sup>∼</sup><sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>9</sup> 16 *a z* + 1 8 *a*3 *<sup>z</sup>*<sup>3</sup> <sup>−</sup> <sup>45</sup> 256 *a*4 *<sup>z</sup>*<sup>4</sup> <sup>−</sup> <sup>1</sup> 16 *a*5

*D*(*z*) *D*<sup>0</sup>

*D*(*z*) *D*<sup>0</sup>

which is commonly used in their first order:

(MCSA) and gives the following expression:

1 + [*C*(*z*) − 1] + [*C*(*h* − *z*) − 1] +

+ ∞ ∑ *n*=1

effects in experiments of confined microparticles in suspension.

*D*(*z*) *D*<sup>0</sup> = 

only one wall situation.

**2.3.2 Particle-particle interaction**

When the aggregates are formed in the suspensions, their way of spreading in the fluid is expected to change. By analogy with the Stokes-Einstein equation, in which the diffusion coefficient depends on the inverse of particle diameter (*<sup>D</sup>* <sup>∼</sup> *<sup>a</sup>*−1), Miyazima et al. (1987) suggested that the diffusion coefficient depends on the inverse cluster size *s* in the form *<sup>D</sup>*(*s*) <sup>∼</sup> *<sup>s</sup>γ*, where *<sup>γ</sup>* is the coefficient that marks the degree of homogeneity of the kernel on the Smoluchowski equation (Eq. 6). The result for the diffusion coefficient *γ* = −1 is considered to be strictly valid for spherical particles that not interact hydrodynamically among them. However, in the case of an anisotropic system, as is the case of chain aggregates, the diffusion coefficient varies due to the hydrodynamic interaction in the direction parallel and perpendicular to the axis of the chain, as follows (Doi & Edwards (1986)):

$$D\_{\parallel} = \frac{k\_B T}{2\pi \eta a} \frac{\ln s}{s} \tag{26}$$

$$D\_{\perp} = D\_{\parallel}/2\tag{27}$$

This result is based assuming point particles, but similar expressions are obtained by modelling the aggregates in the form of cylinders of length *L* and diameter *d* = 2*a*. Tirado & García (1979; 1980) provide diffusion coefficients for this objects in the directions perpendicular, parallel and rotational to the axis of the chains (*<sup>D</sup>*, *<sup>D</sup>*⊥, *Dr*).

By using mesaurements of Dynamic Light Scattering (DLS), an effective diffusion coefficient, *D*eff, of the aggregates can be extracted (Koppel (1972)). This effective coefficient is related to the others mentioned above by means of the relationship:

$$D\_{\rm eff} = D\_{\perp} + \frac{L^2}{12} D\_r \tag{28}$$

which is correct if *qL* >> 1 where *q* is the scattering wave vector defined as: *q* = 4*π*/*λ<sup>l</sup>* sin(*θ*/2), *λ<sup>l</sup>* is the wave length of the laser over the suspension and *θ* is the scattering angle.

#### **2.3.4 Cluster sedimentation**

A particularly important effect is the sedimentation of the clusters or aggregates. It is essential, when a colloidal system is studied, determine the position of the aggregates from the wall, as well as knowing what the deposition rate by gravity is and when the equilibrium in a given layer of fluid is reached. The velocity *vc* experienced by a cluster composed of *N* identical spherical particles of radius *a* and mass *M* falling by gravity in a fluid without the presence of walls is (González et al. (2004)):

$$v\_c = \frac{M\text{g}N}{\gamma\_0} \left(1 - \frac{\rho}{\rho\_p}\right) = \frac{M\text{g}DN}{k\_B T} \left(1 - \frac{\rho}{\rho\_p}\right).$$

where *g* is the value of the gravity acceleration, *ρ* is the fluid density, *ρp* is the density of the particles, *γ*<sup>0</sup> is t the drag coefficient and *D* the diffusion coefficient. If we have only one spheric particle, the last equation yields the classic result for the sedimentation velocity:

$$v\_p = \frac{2a^2 g \Delta \rho}{9\eta}.$$

are not negligible. Specifically, they calculated the friction coefficient *γ<sup>N</sup>* depending on *N* (number of particles) for linear chains located at a distance *z* of the wall and applying a transverse velocity *Vx* = 0.08 diameters per second. The friction coefficient *γN*, to reach a

<sup>331</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

in Water Suspension: Effects of Low-Confinement Conditions and Electrostatics Interactions

*<sup>γ</sup><sup>N</sup>* <sup>=</sup> *Fx*

because of the hydrodynamic interactions between the rod and the wall.

*PB*(*z*)

(Domínguez-García, Pastor, Melle & Rubio (2009); Faucheux & Libchaber (1994)):

 *z*+*δ*(*z*,*η*) *z*−*δ*(*z*,*η*)

where *Fx* is the force over the chain and *d* the diameter of the particle. Then, they obtain that, far away from the wall, *γ*<sup>30</sup> ∼ 6 for a chain formed by 30 particles. But, near enough from the wall, the friction coefficient grows to a value *γ*<sup>30</sup> ∼ 200. Recently, Paddinga & Briels (2010) showed simulation results for translational and rotational friction components of a colloidal rod near to a planar hard wall. They obtained a enhancement friction tensor components

In any case, when we are thinking on one spherical Brownian particle, we can estimate the diffusion coefficient using the Boltzmann profile by calculating the mean position of the particle using Eq.32. Then, if we can calculate the experimental diffusion coefficient when sedimentation affects to the particles, we can employ the following expression

*D*(*z*�

where *PB*(*z*) is the Boltzmann probability distribution, *NB*(*z*) is the normalization of that

to the wall. This expression introduces a correction because of the vertical movement: during each time window of span *τ*, the particle typically explores a region of size 2*δ* with

wall. The height of the particle from the bottom, *z*, is calculated by assuming the Boltzmann

In a colloidal system, there are usually present not only external forces or hydrodynamic interaction of particles with the fluid, but also electrostatic interactions of various kinds. Moreover, as we shall see, many of the commercial micro-particles have carboxylic groups (−*COOH*) to facilitate their possible use, for example, in biological applications. These groups provide for electrolytic dissociation, a negative charge on the particle surface, so that we can see their migration under a constant and uniaxial electric field using the technique of electrophoresis. Therefore, these groups generate an electrostatic interaction between the

DLVO theory (Derjaguin & Landau (1941); Verwey & Overbeek (1948)) is the commonly used classical theory to explain the phenomena of aggregation and coagulation in colloidal particle systems without external fields applied. Roughly speaking, the theory considers that the colloidal particles are subject to two types of electrical forces: repulsive electrostatic forces due to same-sign charged particles and, on the other hand, Van der Waals forces which are of attractive nature and appear due to the interaction between the molecules that form the colloid. According to the intensity relative to each other, the particles will aggregate or repel.

, *<sup>η</sup>*) *PB*(*z*�

, *η*) is the corrected diffusion coefficient of the particle for the motion parallel

<sup>2</sup>*τD*⊥(*z*, *<sup>η</sup>*), where *<sup>D</sup>*<sup>⊥</sup> is the diffusion coefficient for the motion normal to the

*NB*(*z*� , *η*) *dz*� *dz*

)

3*πηdVx*

velocity *Vx* in the transverse direction was obtained as:

*Dδ* = *L* 0

function, *D*(*z*�

2

**2.4 Electrostatics**

probability distribution.

*δ*(*z*, *η*) = <sup>1</sup>

particles.

**2.4.1 DLVO theory**

with Δ*ρ* = *ρ<sup>p</sup>* − *ρ*. We can define the Péclet number as the ratio between the sedimentation time *ts* and diffusion *td* using a fixed distance, for instance, 2*a*:

$$P\_{\ell} \equiv \frac{t\_d}{t\_s} = \frac{Mga}{k\_B T} \left(1 - \frac{\rho}{\rho\_p}\right) = \frac{4\pi a^4 g \Delta \rho}{3k\_B T} \tag{29}$$

Then, the vertical distance travelled by gravity for a cluster in a time equal to that a particle spread a distance equal to the diameter of the particle *d* is *dc* = *vc td* = *PeNd*.

The above expressions are satisfied when sedimentation occurs in an unconfined fluid. If there is a bottom wall, then it provides a spatial distribution of particles *ρ* which depends on the relative height with respect to the bottom wall. If the system is in an equilibrium state and with low concentration of particles, we can use the Boltzmann density profile, which measures the balance on the thermal forces and gravity:

$$
\ln \rho(z) \propto -\frac{z}{L\_G} \tag{30}
$$

where *LG* ∼ *kBT*/*Mg*. As mentioned, this density profile is valid when the interactions between the colloidal particles are neglected. However, experimental situations can be much more complicated, resulting in deviations from this profile, so theoretical research is still in development about this question (Chen & Ma (2006); Schmidt et al. (2004)). In fact, it has been discovered experimentally that the influence of the electric charge of silica nanoparticles in a suspension of ethanol may drastically change the shape of the density profile (Rasa & Philipse (2004)). We will here assume the expression 30 to be correct, so that the average height *zm* of a particle of radius *a*, between two walls separated by a distance *h*, can be determined by the Boltzmann profile as Faucheux & Libchaber (1994) showed:

$$P\_B(z) = \frac{1}{L} \left( \frac{e^{-z/L}}{e^{-a/L} - e^{(a-h)/L}} \right) \tag{31}$$

where *z* is the position of the particle between the two walls, where the bottom wall is at *z* = 0 and the top is located at *z* = *h*, *L* is the characteristic Boltzmann length defined as *L* ≡ *kBT* (*g*Δ*M*) <sup>−</sup><sup>1</sup> where <sup>Δ</sup>*<sup>M</sup>* <sup>≡</sup> (4/3)*πa*3(*ρ<sup>p</sup>* <sup>−</sup> *<sup>ρ</sup>*).

Therefore, the mean distance *zm* can be calculated:

$$z\_{\mathfrak{M}} = \int\_{a}^{h-a} z P\_{\mathfrak{B}}(z) dz = \tag{32}$$

$$= \frac{e^{-a/L} [aL + L^2] - e^{(a-h)/L} [(h-a)L + L^2]}{L[e^{-a/L} - e^{(a-h)/L})}$$

With that expression and the equations for the diffusion coefficient near a wall (Eqs. 20 to 25) we can estimate the effective diffusion coefficient of a sedimented particle. However, when we have a set of particles, clusters or aggregates near the walls of the enclosure, the evaluation of hydrodynamic effects on the diffusion coefficient and their dynamics is not an easy problem to evaluate theoretically or experimentally. In fact, this problem is very topical, for example, focused on polymer science (Hernández-Ortiz et al. (2006)) or more specifically, in the case of biopolymers, such as DNA strands, moving by low flows in confined enclosures (Jendrejack et al. (2003)). Kutthe (2003) performed Stokestian dynamics simulations (SD) of chains, clusters and aggregates in various situations in which hydrodynamic interactions

are not negligible. Specifically, they calculated the friction coefficient *γ<sup>N</sup>* depending on *N* (number of particles) for linear chains located at a distance *z* of the wall and applying a transverse velocity *Vx* = 0.08 diameters per second. The friction coefficient *γN*, to reach a velocity *Vx* in the transverse direction was obtained as:

$$\gamma\_N = \frac{F\_\chi}{3\pi\eta dV\_\chi}$$

where *Fx* is the force over the chain and *d* the diameter of the particle. Then, they obtain that, far away from the wall, *γ*<sup>30</sup> ∼ 6 for a chain formed by 30 particles. But, near enough from the wall, the friction coefficient grows to a value *γ*<sup>30</sup> ∼ 200. Recently, Paddinga & Briels (2010) showed simulation results for translational and rotational friction components of a colloidal rod near to a planar hard wall. They obtained a enhancement friction tensor components because of the hydrodynamic interactions between the rod and the wall.

In any case, when we are thinking on one spherical Brownian particle, we can estimate the diffusion coefficient using the Boltzmann profile by calculating the mean position of the particle using Eq.32. Then, if we can calculate the experimental diffusion coefficient when sedimentation affects to the particles, we can employ the following expression (Domínguez-García, Pastor, Melle & Rubio (2009); Faucheux & Libchaber (1994)):

$$D\_{\parallel}^{\delta} = \int\_{0}^{L} P\_{B}(z) \left[ \int\_{z-\delta(z,\eta)}^{z+\delta(z,\eta)} D\_{\parallel}(z',\eta) \frac{P\_{B}(z')}{N\_{B}(z',\eta)} dz' \right] dz'$$

where *PB*(*z*) is the Boltzmann probability distribution, *NB*(*z*) is the normalization of that function, *D*(*z*� , *η*) is the corrected diffusion coefficient of the particle for the motion parallel to the wall. This expression introduces a correction because of the vertical movement: during each time window of span *τ*, the particle typically explores a region of size 2*δ* with *δ*(*z*, *η*) = <sup>1</sup> 2 <sup>2</sup>*τD*⊥(*z*, *<sup>η</sup>*), where *<sup>D</sup>*<sup>⊥</sup> is the diffusion coefficient for the motion normal to the wall. The height of the particle from the bottom, *z*, is calculated by assuming the Boltzmann probability distribution.
