**2.1.1 Magnetic dipolar interaction.**

2 Hydrodynamics

consequences, as an one million-fold increase in the viscosity of the fluid, leading to practical and industrial applications, such as mechanical devices of different types (*Lord Corporation, http://www.lord.com/* (n.d.); *N*akano & Koyama (1998); Tao (2000)). This magnetic particle technology has been revealed as useful in other fields such as microfluidics (Egatz-Gómez et al. (2006)) or biomedical techniques (Komeili (2007); Smirnov et al. (2004);

In our case, we investigate the dynamics of the aggregation of magnetic particles under a constant and uniaxial magnetic field. This is useful not only for the knowledge of aggregation properties in colloidal systems, but also for testing different models in Statistical Mechanics. Using video-microscopy (Crocker & Grier (1996)), we have measured the different exponents which characterize this process during aggregation (Domínguez-García et al. (2007)) and also in disaggregation (Domínguez-García et al. (2011)), i.e., when the chains vanishes as the external field is switched off. These exponents are based on the temporal variation of the aggregates' representative quantities, such as the size *s* or length *l*. For instance, the main dynamical exponent *<sup>z</sup>* is obtained through the temporal evolution of the chains length *<sup>s</sup>* <sup>∼</sup> *<sup>t</sup>z*. Our experiments analyse the microestructure of the suspensions, the aggregation of the particles under external magnetic fields as well as disaggregation when the field is switched off. The observations provide results that diverge from what a simple theoretical model says. These differences may be related with some kind of electro-hydrodynamical interaction,

In this chapter, we would like first to summarize the basic theory related with our system of magnetic particles, including magnetic interactions and Brownian movement. Then, hydrodynamic corrections and the Boltzmann sedimentation profile theory in a confined suspension of microparticles will be explained and some fundamentals of electrostatics in colloids are explained. In the next section, we will summarize some of the most recent remarkable studies related with the electrostatic and hydrodynamic effects in colloidal suspensions. Finally, we would like to link our findings and investigations on MRF with the theory and studies explained herein to show how the modelization and theoretical comprehension of these kind of systems is not perfectly understood at the present time.

In this section, we are going to briefly describe the theory related with the main interactions and effects which can be suffered by colloidal magnetic particles: magnetic interactions, Brownian movement, hydrodynamic interactions and finally electrostatic interactions.

By the name of "colloid" we understand a suspension formed by two phases: one is a fluid and another composed of mesoscopic particles. The mesoscopic scale is situated between the tens of nanometers and the tens of micrometers. This is a very interesting scale from a physical point of view, because it is a transition zone between the atomic and molecular scale and the

When the particles have some kind of magnetic property, we are talking about magnetic colloids. From this point of view, two types of magnetic colloids are usually considered: ferromagnetic and magneto-rheologic. The ferromagnetic fluids or ferrofluids (FF) are colloidal suspensions composed by nanometric mono-domain particles in an aqueous or organic solvent, while magneto-rheological fluids (MRF) are suspensions of paramagnetic micro or nanoparticles. The main difference between them is the permanent magnetic moment

Vuppu et al. (2004); Wilhelm, Browaeys, Ponton & Bacri (2003); Wilhelm et al. (2005)).

which has not been taken into account in the theoretical models.

**2. Theory**

**2.1 Magnetic particles**

purely macroscopic one.

Fig. 1. Left: Two magnetic particles under a magnetic field *H* . The angle between the field direction and the line that join the centres of the particles is named as *α*. Right: The attraction cone of a magnetic particle. Top and bottom zones are magnetically attractive, while regions on the left and on the right have repulsive behaviour.

As it has been said before, the main interest of MRF are their properties in response to external magnetic fields. These properties can be optical (birefringence (Bacri et al. (1993)), dichroism (Melle (2002))) or magnetical or rheological. Under the action of an external magnetic field, the particles acquire a magnetic moment and the interaction between the magnetic moments generates the particles aggregation in the form of chain-like structures. More in detail, when a magnetic field *H* is applied, the particles in suspension acquire a dipolar moment:

$$
\vec{m} = \frac{4\pi a^3}{3}\vec{M} \tag{1}
$$

where *M* = *χH* and *a* are respectively the particle's imanation and radius, whereas *χ* is the magnetic susceptibility of the particle.

The most simple way for analysing the magnetic interaction between magnetic particles is through the dipolar approximation. Therefore, the interaction energy between two magnetic dipoles *m <sup>i</sup>* and *m <sup>j</sup>* is:

$$\mathcal{U}\_{ij}^{d} = \frac{\mu\_0 \mu\_s}{4\pi r^3} \left[ (\vec{m}\_i \cdot \vec{m}\_j) - \Im(\vec{m}\_i \cdot \boldsymbol{\hbar})(\vec{m}\_j \cdot \boldsymbol{\hbar}) \right] \tag{2}$$

where*ri* is the position vector of the particle *i*,*r* =*rj* −*ri* joins the centre of both particles and *r*ˆ =*r*/*r* is its unitary vector.

Basically, the aggregation process has two stages: first, the chains are formed on the basis of the aggregation of free particles, after that, more complex structures are formed when chains aggregate by lateral interaction. When the applied field is high and the concentration of particles in the fluid is low, the interactions between the chains are of short range. Under this situation, there are two regions of interaction between the chains depending on the lateral distance between them: when the distance between two strings is greater than two diameters of the particle, the force is repulsive; if the distance is lower, the resultant force is attractive, provided that one of the chains is moved from the other a distance equal to one particle's radius in the direction of external field (Furst & Gast (2000)). In this type of interactions, the temperature fluctuations and the defects in the chains morphology are particularly important. Indeed, variations on these two aspects generate different types of theoretical models for the interaction between chains. The model that takes into account the thermal fluctuations in the structure of the chain for electro-rheological fluids is called *HT* (Halsey & Toor (1990)), and was subsequently extended to a modified HT model (MHT) (Martin et al. (1992)) to include dependence on field strength. The latter model shows that only lateral interaction occurs between the chains when the characteristic time associated with their thermal relaxation is greater than the characteristic time of lateral assembling between them. Possible defects in the chains can vary the lateral interaction, mainly through perturbations in the local field.

<sup>323</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

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

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

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

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

**2.1.3 Irreversible aggregation**

DLA or RLCA.

science (Lin et al. (1989); Tirado-Miranda (2001)).

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

$$\vec{F}\_{ij}^{d} = \frac{3\mu\_{0}\mu\_{s}}{4\pi r^{4}} \left\{ \left[ (\vec{m}\_{l} \cdot \vec{m}\_{\dot{j}}) - \mathbf{5} (\vec{m}\_{\dot{l}} \cdot \hat{r}) (\vec{m}\_{\dot{j}} \cdot \hat{r}) \right] \hat{r} + (\vec{m}\_{\dot{j}} \cdot \hat{r}) \vec{m}\_{\dot{l}} + (\vec{m}\_{\dot{l}} \cdot \hat{r}) \vec{m}\_{\dot{j}} \right\} \tag{3}$$

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:

$$\mathcal{U}\_{ij}^d = \frac{\mu\_0 \mu\_s m^2}{4\pi r^3} (1 - 3\cos^2 a) \tag{4}$$

$$\vec{F}\_{ij}^{d} = \frac{3\mu\_0\mu\_s m^2}{4\pi r^4} \left[ (1 - 3\cos^2 a)\mathbb{1} - \sin(2a)\mathbb{1} \right] \tag{5}$$

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

From the above equations, it follows that the radial component of the magnetic force is attractive when *α* < *α<sup>c</sup>* and repulsive when *α* > *αc*, where *α<sup>c</sup>* = arccos √ 1 <sup>3</sup> � <sup>55</sup>◦, so 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 oriented in the direction of *H*ˆ .

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

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

#### **2.1.2 Magnetic interaction between chains**

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

Basically, the aggregation process has two stages: first, the chains are formed on the basis of the aggregation of free particles, after that, more complex structures are formed when chains aggregate by lateral interaction. When the applied field is high and the concentration of particles in the fluid is low, the interactions between the chains are of short range. Under this situation, there are two regions of interaction between the chains depending on the lateral distance between them: when the distance between two strings is greater than two diameters of the particle, the force is repulsive; if the distance is lower, the resultant force is attractive, provided that one of the chains is moved from the other a distance equal to one particle's radius in the direction of external field (Furst & Gast (2000)). In this type of interactions, the temperature fluctuations and the defects in the chains morphology are particularly important. Indeed, variations on these two aspects generate different types of theoretical models for the interaction between chains. The model that takes into account the thermal fluctuations in the structure of the chain for electro-rheological fluids is called *HT* (Halsey & Toor (1990)), and was subsequently extended to a modified HT model (MHT) (Martin et al. (1992)) to include dependence on field strength. The latter model shows that only lateral interaction occurs between the chains when the characteristic time associated with their thermal relaxation is greater than the characteristic time of lateral assembling between them. Possible defects in the chains can vary the lateral interaction, mainly through perturbations in the local field.
