**Hydrodynamics on Charged Superparamagnetic Microparticles in Water Suspension: Effects of Low-Confinement Conditions and Electrostatics Interactions**

P. Domínguez-García1 and M.A. Rubio<sup>2</sup>

<sup>1</sup>*Dep. Física de Materiales, UNED, Senda del Rey 9, 28040. Madrid* <sup>2</sup>*Dep. Física Fundamental, UNED, Senda del Rey 9, 28040. Madrid Spain*

#### **1. Introduction**

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The study of colloidal dispersions of micro-nano sized particles in a liquid is of great interest for industrial processes and technological applications. The understanding of the microstructure and fundamental properties of this kind of systems at microscopic level is also useful for biological and biomedical applications.

However, a colloidal suspension must be placed somewhere and the dynamics of the micro-particles can be modified as a consequence of the confinement, even if we have a low-confinement system. The hydrodynamics interactions between particles and with the enclosure's wall which contains the suspension are of extraordinary importance to understanding the aggregation, disaggregation, sedimentation or any interaction experienced by the microparticles. Aspects such as corrections of the diffusion coefficients because of a hydrodynamic coupling to the wall must be considered. Moreover, if the particles are electrically charged, new phenomena can appear related to electro-hydrodynamic coupling.

Electro-hydrodynamic effects (Behrens & Grier (2001a;b); Squires & Brenner (2000)) may have a role in the dynamics of confined charged submicron-sized particles. For example, an anomalous attractive interaction has been observed in suspensions of confined charged particles (Grier & Han (2004); Han & Grier (2003); Larsen & Grier (1997)). The possible explanation of this observation could be related with the distribution of surface's charges of the colloidal particles and the wall (Lian & Ma (2008); Odriozola et al. (2006)). This effect could be also related to an electrostatic repulsion with the charged quartz bottom wall or to a spontaneous macroscopic electric field observed on charged colloids (Rasa & Philipse (2004)). In this work, we are going to describe experiments performed by using magneto-rheological fluids (MRF), which consist (Rabinow (1948)) on suspensions formed by water or some organic solvent and micro or nano-particles that have a magnetic behaviour when a external magnetic field is applied upon them. Then, these particles interact between themselves forming aggregates with a shape of linear chains (Kerr (1990)) aligned in the direction of the magnetic field. When the concentration of particles inside the fluid is high enough, this microscopic behaviour turns to significant macroscopic

of the first type: while in a FF, magnetic aggregation is possible without an external magnetic field, this does not occur in a MRF. The magnetic particles of a MRF are usually composed by a polymeric matrix with small crystals of some magnetic material embedded on it, for example, magnetite. When the particles are superparamagnetic, the quality of the magnetic response is

<sup>321</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

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

Another point of view for classifying these suspensions is the rheological perspective. By rheology, we name the discipline which study deformations and flowing of materials when some stress is applied. In some ranges, it is possible to consider the magnetic colloids as Newtonian fluids because, when an external magnetic field is applied, the stress is proportional to the velocity of the deformation. On a more global perspective, these fluids can be immersed on the category of complex fluids (Larson (1999)) and are studied as complex

Now we are going to briefly provide some details about magnetic interactions: magnetic

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

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:

*<sup>m</sup>* <sup>=</sup> <sup>4</sup>*πa*<sup>3</sup>

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

The most simple way for analysing the magnetic interaction between magnetic particles is through the dipolar approximation. Therefore, the interaction energy between two magnetic

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

(*m <sup>i</sup>* · *m <sup>j</sup>*) − 3(*m <sup>i</sup>* · *r*ˆ)(*m <sup>j</sup>* · *r*ˆ)

<sup>3</sup> *<sup>M</sup>* (1)

(2)

improved because the imanation curve has neither hysteresis nor remanence.

dipolar interaction, interaction between chains and irreversible aggregation.

systems (Science. (1999)).

**2.1.1 Magnetic dipolar interaction.**

on the left and on the right have repulsive behaviour.

*Ud*

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

magnetic susceptibility of the particle.

dipoles *m <sup>i</sup>* and *m <sup>j</sup>* is:

*r*ˆ =*r*/*r* is its unitary vector.

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

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

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.

### **2. Theory**

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.

#### **2.1 Magnetic particles**

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 purely macroscopic one.

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 of the first type: while in a FF, magnetic aggregation is possible without an external magnetic field, this does not occur in a MRF. The magnetic particles of a MRF are usually composed by a polymeric matrix with small crystals of some magnetic material embedded on it, for example, magnetite. When the particles are superparamagnetic, the quality of the magnetic response is improved because the imanation curve has neither hysteresis nor remanence.

Another point of view for classifying these suspensions is the rheological perspective. By rheology, we name the discipline which study deformations and flowing of materials when some stress is applied. In some ranges, it is possible to consider the magnetic colloids as Newtonian fluids because, when an external magnetic field is applied, the stress is proportional to the velocity of the deformation. On a more global perspective, these fluids can be immersed on the category of complex fluids (Larson (1999)) and are studied as complex systems (Science. (1999)).

Now we are going to briefly provide some details about magnetic interactions: magnetic dipolar interaction, interaction between chains and irreversible aggregation.
