**3. Results**

16 Hydrodynamics

In order to understand the interactions in this kind of systems, we have to note that the standard theory of colloidal interactions, the DLVO theory, fails to explain several experimental observations. For example, an attractive interaction is observed between the particles when the electrostatic potential is obtained. This is a effect that has been previously observed in experiments on suspensions of confined equally-charged microspheres (Behrens & Grier (2001a;b); Grier & Han (2004); Han & Grier (2003); Larsen & Grier (1997)). Grier and colleagues listed several experimental observations using suspensions of charged polystyrene particles with diameters around 0.65 microns at low ionic strength and strong spatial confinement. They note that such effects appear when a wall of glass or quartz is near the particles. Studying the *g*(*r*) function and its relation to the interaction potential, given by expression 36, they showed the appearance of a minimum on the potential located at *z* = 2.5 microns of the wall and a distance between centres to be *r*min = 3.5 microns. This attraction cannot be a Van der Waals interaction, because for this type of particle and with separations greater than 0.1 micrometres, this force is less than 0.01 *kBT* (Pailthorpe & Russel (1982)), while

The same group (Behrens & Grier (2001b)) extended this study using silica particle suspensions (silicon dioxide, SiO2) of 1.58 microns in diameter, with a high density of 2.2 g/cm3, using a cell of thickness *h* = 200 *μ*m. In this situation, even though the particles are deposited at a distance from the bottom edge of the particle to the bottom wall equal to *s* = 0.11 *μ*m, no minimum in the interaction energy between pairs appears, being the interaction purely repulsive, in the classical form of DLVO given by Eq.34. In that work, a methodology is also provided to estimate the Debye length of the system and the equivalent load *Z*∗ through a study of the presence of negative charge quartz wall due to the dissociation of silanol groups in presence of water (Behrens & Grier (2001a)). However, Han & Grier (2003) observed the existence of a minimum in the potential when they use polystyrene particles of 0.65 micron and density close to water, 1.05 g/cm3, with a separation between the walls of *h* = 1.3 microns. What is more, using silica particles from previous works, they observe a

The physical explanation of this effect is not clear (Grier & Han (2004)), being the main question how to explain the influence on the separation of the two walls in the confinement cell. However, some criticism has appeared about this results. For example, about the employment of a theoretical potential with a DLVO shape. An alternative is using a Sogami-Ise (SI) potential (Tata & Ise (1998)). Moreover, Tata & Ise (2000) contend that both the DLVO theory and the SI theory are not designed for situations in confinement, so interpreting the experimental data using either of these two theories may be wrong. Controversy on the use of a DLVO-type or SI potentials appears to be resolved considering that the two configurations represent physical exclusive situations (Schmitz et al. (2003)). In fact, simulations have been performed to explore the possibility of a potential hydrodynamic coupling with the bottom wall generated by the attraction between two particles (Dufresne et al. (2000); Squires & Brenner (2000)). However, the calculated hydrodynamic effects do not seem to explain the experimental minimum on the potential (Grier & Han (2004); Han & Grier (2003)). Other authors argue that this kind of studies should be more rigorous in the analysis of errors when extracting data from the images (Savin & Doyle (2005; 2007); Savin et al. (2007)) and other authors claim that the effect on the electrostatic potential may be an artefact (Baumgart et al. (2006)) that occurs because of

a incorrect extraction of the position of the particles (Gyger et al. (2008)).

**2.4.3 Anomalous effects**

this attractive interaction is about 0.7 *kBT*.

minimum separation between walls of *h* = 9 *μ*m.

Our experimental system is formed by a MRF composed of colloidal dispersions of superparamagnetic micron-sized particles in water. These particles have a radius of 485 nm and a density of 1.85 g/cm3, so they sediment to an equilibrium layer on the containing cell. They are composed by a polymer (PS) with nano-grains of magnetite dispersed into it, which provide their magnetic properties. The particles are also functionalized with carboxylic groups, so they have an electrical component, therefore, they repel each other, avoiding aggregation. This effect is improved by adding sodium dodecyl sulfate (SDS) in a concentration of 1 gr/l.

The containing cell consists on two quartz windows, one of them with a cavity of 100 *μ*m. The cell with the suspension in it is located in an experimental setup that isolate thermically the suspension and allows to generate a uniform external magnetic field in the centre of the cell. The particles and aggregates are observed using video-microscopy (see details for this experimental setup on (Domínguez-García et al. (2007))). Images of the fluid are saved on the computer and then analysed for extracting the relevant data by using our own developed software (Domínguez-García & Rubio (2009)) based on *ImageJ* ( *U. S. National Institutes of Health, Bethesda, Maryland, USA, http://rsb.info.nih.gov/ij/* (n.d.)). In Fig.3, we show an example of these microparticles and aggregates observed in our system.

The zeta potential of these particles is about −110 to −60 mV for a pH about 6 - 7. Therefore, the electrical content of the particles is relatively high and it is only neglected in comparison with the energy provided by the external magnetic field. However, the colloidal stability of these suspensions is not being controlled and it may have an effect on the dynamics of the clusters, specially when no magnetic field is applied. In any case, as we will see, even when a magnetic field is applied, it is observed a disagreement between theoretical aggregation times and experimental ones.

Fig. 3. Superparamagnetic microparticles observed when no external magnetic field is

where the exponent *ξ* is an exponent that depends on the dimensionality of the system, so if *d* ≥ 2, *ξ* = 1/2. Using Monte Carlo simulations they obtain that *ξ* � 0.51, and therefore that *z*

<sup>337</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

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

In the case of our experiments, we have experimentally obtained that the *z* exponent in aggregation is contained in the range of 0.43 − 0.67 (Domínguez-García et al. (2007)) with an average value of *z* ∼ 0.57 ± 0.03. These experimental values do not depend on the amplitude of the magnetic field nor on the concentration of particles, but they seem to depend on the ratio *R*1/*R*0, which is a sign of the more important regime of aggregation. The dependency on this ratio also appears when the morphology of the chains is studied (Domínguez-García, Melle & Rubio (2009); Domínguez-García & Rubio (2010)). Besides, the scaling behaviour given by Eq.7 is experimentally observed and checked. We have compared our experimental results with Brownian dynamics simulations based on a simple model which only included dipolar interaction between the particles, hard-sphere repulsion and Brownian diffusion, neglecting inertial terms and effects related with sedimentation or electrostatics. The results of these simulations agree with the theoretical prediction, whereas the experimental aggregation time, *t*ag, appears to be much longer than expected (Cernak et al. (2004); Domínguez-García et al. (2007)), about three orders of magnitude of difference. The formation of dimers (two-particles aggregates) in the experiments lapses *<sup>t</sup>* <sup>∼</sup> 102 seconds, but Brownian simulations show that this lapse of time is about *t* ∼ 0.1 s. This last value can be easily obtained by assuming that the equation for the movement between two particles with

*Mr*¨ + *γ*0*r*˙ + 3*μμ*0*m*2*r*−4*π*−<sup>1</sup> = 0 where *M* is the mass of the particles. Because of Reynolds number (Eq.19) is very low, we neglect the inertial term on this equation. If the particles are separated a initial distance *d* = *R*<sup>0</sup>

If we express this equation in function of the *λ* parameter 43 and of the diffusion coefficient

1 *<sup>λ</sup><sup>D</sup> <sup>φ</sup>*−5/2

<sup>15</sup>*μsμ*0*m*<sup>2</sup> *<sup>φ</sup>*−5/2

2*D*

<sup>2</sup>*<sup>D</sup>* (47)

*<sup>t</sup>*ag <sup>∼</sup><sup>=</sup> <sup>32</sup>*πγ*0*a*<sup>5</sup>

*<sup>t</sup>*ag <sup>∼</sup><sup>=</sup> <sup>2</sup>*a*<sup>2</sup> 15

applied (Left) and when it is applied (Right).

dipolar magnetic interaction is:

given by the Stokes-Einstein equation 13:

we can obtain that:

is *z* � 0.61.

#### **3.1 Control parameters**

We have already defined some important parameters as the Péclet number, Eq.29, and the Reynolds number Eq.19. However, in our system we need to define some external parameters related with the concentration of particles and the intensity of the magnetic field. The concentration of volume of particles in the suspension, *φ*, is defined as the fraction of volume occupied by the spheres relative to the total volume of the suspension. In a quasi-2D video-microscopy system is useful to take into account the surface concentration *φ*2*D*.

For measuring the influence of the magnetic interaction we used the *λ* parameter, defined as:

$$
\lambda \equiv \frac{\mathcal{W}\_m}{k\_B T} = \frac{\mu\_s \mu\_0 m^2}{16 \pi a^3 k\_B T} \tag{43}
$$

as the ratio of *Wm* = *U<sup>d</sup> ij*(*r* = 2*a*, *α* = 0), i.e., the magnetic energy, and the thermal fluctuations *kBT*. Here, *μ<sup>s</sup>* is the relative magnetic permeability of the solvent, *μ*<sup>0</sup> the magnetic permeability of vacuum and *m* the magnetic moment. The parameters *λ* y *φ*2*<sup>D</sup>* allow to define a couple of characteristic lengths. First, we define a distance *R*<sup>1</sup> for which the energy of dipolar interaction is equal to thermal fluctuations:

$$R\_1 \equiv 2a\lambda^{1/3} \tag{44}$$

Finally, we define a mean distance between particles:

$$R\_0 \equiv \sqrt{\pi} a \,\phi\_{2D}^{-1/2} \tag{45}$$

The comparative between these two quantities allows to distinguish between different aggregation regimes. When, *R*<sup>1</sup> < *R*0, the thermal fluctuations prevail over the magnetic interactions so diffusion is the main aggregation process. If *R*<sup>1</sup> > *R*0, the aggregation of the particles occurs mainly because of the applied magnetic field.

#### **3.2 Aggregation and disaggregation**

Studies about the dynamics of the irreversible aggregation of clusters under unidirectional constant magnetic fields have used a collection of experimental systems. For example, electro-rheological fluids (Fraden et al. (1989)), magnetic holes (non-magnetic particles in a ferrofluid) (Cernak et al. (2004); Helgesen et al. (1990; 1988); Skjeltorp (1983)), and magneto-rheological fluids and magnetic particles (Bacri et al. (1993); Bossis et al. (1990); Cernak (1994); Cernak et al. (1991); Fermigier & Gast (1992); Melle et al. (2001); Promislow et al. (1994)).

These studies focus their efforts in calculating the kinetic exponent *z* obtaining different values ranging *z* ∼ 0.4 − 0.7. The different methodologies employed can be the origin of these dispersed values. However, more recent studies (Domínguez-García et al. (2007); Martínez-Pedrero et al. (2007)) suggest that this value is approximately *z* ∼ 0.6 − 0.7 in accordance with experimental values reported for aggregation of dielectric colloids *z* ∼ 0.6 (Fraden et al. (1989)) and with recent simulations of aggregation of superparamagnetic particles (Andreu et al. (2011)). Regarding hydrodynamics interactions Miguel & Pastor-Satorras (1999) proposed and effective expression for explaining the dispersed value of the kinetic exponent based on logarithmic corrections in the diffusion coefficient (Eqs. 26 and 27):

$$\mathcal{S}(t) \sim (t \ln \left[ \mathcal{S}(t) \right])^{\tilde{\mathbb{S}}} , \tag{46}$$

18 Hydrodynamics

We have already defined some important parameters as the Péclet number, Eq.29, and the Reynolds number Eq.19. However, in our system we need to define some external parameters related with the concentration of particles and the intensity of the magnetic field. The concentration of volume of particles in the suspension, *φ*, is defined as the fraction of volume occupied by the spheres relative to the total volume of the suspension. In a quasi-2D

*kBT* <sup>=</sup> *<sup>μ</sup>sμ*0*m*<sup>2</sup>

fluctuations *kBT*. Here, *μ<sup>s</sup>* is the relative magnetic permeability of the solvent, *μ*<sup>0</sup> the magnetic permeability of vacuum and *m* the magnetic moment. The parameters *λ* y *φ*2*<sup>D</sup>* allow to define a couple of characteristic lengths. First, we define a distance *R*<sup>1</sup> for which the energy of dipolar

*<sup>R</sup>*<sup>0</sup> <sup>≡</sup> <sup>√</sup>*π<sup>a</sup> <sup>φ</sup>*−1/2

The comparative between these two quantities allows to distinguish between different aggregation regimes. When, *R*<sup>1</sup> < *R*0, the thermal fluctuations prevail over the magnetic interactions so diffusion is the main aggregation process. If *R*<sup>1</sup> > *R*0, the aggregation of the

Studies about the dynamics of the irreversible aggregation of clusters under unidirectional constant magnetic fields have used a collection of experimental systems. For example, electro-rheological fluids (Fraden et al. (1989)), magnetic holes (non-magnetic particles in a ferrofluid) (Cernak et al. (2004); Helgesen et al. (1990; 1988); Skjeltorp (1983)), and magneto-rheological fluids and magnetic particles (Bacri et al. (1993); Bossis et al. (1990); Cernak (1994); Cernak et al. (1991); Fermigier & Gast (1992); Melle et al. (2001);

These studies focus their efforts in calculating the kinetic exponent *z* obtaining different values ranging *z* ∼ 0.4 − 0.7. The different methodologies employed can be the origin of these dispersed values. However, more recent studies (Domínguez-García et al. (2007); Martínez-Pedrero et al. (2007)) suggest that this value is approximately *z* ∼ 0.6 − 0.7 in accordance with experimental values reported for aggregation of dielectric colloids *z* ∼ 0.6 (Fraden et al. (1989)) and with recent simulations of aggregation of superparamagnetic particles (Andreu et al. (2011)). Regarding hydrodynamics interactions Miguel & Pastor-Satorras (1999) proposed and effective expression for explaining the dispersed value of the kinetic exponent based on logarithmic corrections in the diffusion

<sup>16</sup>*πa*<sup>3</sup>*kBT* (43)

*<sup>R</sup>*<sup>1</sup> <sup>≡</sup> <sup>2</sup>*aλ*1/3 (44)

*<sup>S</sup>*(*t*) <sup>∼</sup> (*<sup>t</sup>* ln [*S*(*t*)])*<sup>ξ</sup>* , (46)

<sup>2</sup>*<sup>D</sup>* (45)

*ij*(*r* = 2*a*, *α* = 0), i.e., the magnetic energy, and the thermal

video-microscopy system is useful to take into account the surface concentration *φ*2*D*. For measuring the influence of the magnetic interaction we used the *λ* parameter, defined as:

*<sup>λ</sup>* <sup>≡</sup> *Wm*

**3.1 Control parameters**

as the ratio of *Wm* = *U<sup>d</sup>*

interaction is equal to thermal fluctuations:

**3.2 Aggregation and disaggregation**

Promislow et al. (1994)).

coefficient (Eqs. 26 and 27):

Finally, we define a mean distance between particles:

particles occurs mainly because of the applied magnetic field.

Fig. 3. Superparamagnetic microparticles observed when no external magnetic field is applied (Left) and when it is applied (Right).

where the exponent *ξ* is an exponent that depends on the dimensionality of the system, so if *d* ≥ 2, *ξ* = 1/2. Using Monte Carlo simulations they obtain that *ξ* � 0.51, and therefore that *z* is *z* � 0.61.

In the case of our experiments, we have experimentally obtained that the *z* exponent in aggregation is contained in the range of 0.43 − 0.67 (Domínguez-García et al. (2007)) with an average value of *z* ∼ 0.57 ± 0.03. These experimental values do not depend on the amplitude of the magnetic field nor on the concentration of particles, but they seem to depend on the ratio *R*1/*R*0, which is a sign of the more important regime of aggregation. The dependency on this ratio also appears when the morphology of the chains is studied (Domínguez-García, Melle & Rubio (2009); Domínguez-García & Rubio (2010)). Besides, the scaling behaviour given by Eq.7 is experimentally observed and checked. We have compared our experimental results with Brownian dynamics simulations based on a simple model which only included dipolar interaction between the particles, hard-sphere repulsion and Brownian diffusion, neglecting inertial terms and effects related with sedimentation or electrostatics. The results of these simulations agree with the theoretical prediction, whereas the experimental aggregation time, *t*ag, appears to be much longer than expected (Cernak et al. (2004); Domínguez-García et al. (2007)), about three orders of magnitude of difference. The formation of dimers (two-particles aggregates) in the experiments lapses *<sup>t</sup>* <sup>∼</sup> 102 seconds, but Brownian simulations show that this lapse of time is about *t* ∼ 0.1 s. This last value can be easily obtained by assuming that the equation for the movement between two particles with dipolar magnetic interaction is:

$$M\ddot{r} + \gamma\_0 \dot{r} + 3\mu\mu\_0 m^2 r^{-4} \pi^{-1} = 0$$

where *M* is the mass of the particles. Because of Reynolds number (Eq.19) is very low, we neglect the inertial term on this equation. If the particles are separated a initial distance *d* = *R*<sup>0</sup> we can obtain that:

$$t\_{\rm ag} \cong \frac{32\pi\gamma\_0 a^5}{15\mu\_s\mu\_0 m^2} \,\phi\_{2D}^{-5/2}$$

If we express this equation in function of the *λ* parameter 43 and of the diffusion coefficient given by the Stokes-Einstein equation 13:

$$t\_{\rm ag} \cong \frac{2a^2}{15} \frac{1}{\lambda D} \phi\_{2D}^{-5/2} \tag{47}$$

For completing this study, we also have shown results of disaggregation, that is, the process that occurs when the external magnetic field is switched off and the clusters vanish. For this process we study the kinetics in the same way that in aggregation, by searching power laws behaviours and calculating the kinetic exponents *z* and *z*� (Domínguez-García et al. (2011)). We have also developed Brownian dynamics simulations to be compared with the experiments. The Fig.4 summarizes some of our results in aggregation and disaggregation. The experimental kinetic exponents during disaggregation range from *z* = 0.44 to 1.12 and *z*� = 0.27 to 0.67, while simulations give very regular values, with *z* and *z*� ∼ 1. Then, the kinetic exponents do not agree, being also the process of disaggregation much faster in simulations. From these results, we conclude that remarkable differences exist between a simple theoretical model and the interactions in our experimental setup, differences that are

<sup>339</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

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

specially important when the influence of the applied magnetic field is removed.

suspension.

In all these experiments some data has been collected before any external field is applied. That allows us to study the microstructure of the suspensions by calculating the electrostatics potential using the methods previously explained. The inversion of the O-Z equation reveals an attractive well in the potential with a value in its minimum in the order of −0.2 *kBT*, similar to other observations of attractive interactions of sedimented particles in confinement situations. Moreover, these values of the minimum in the potential seems to depend of the concentration of particles (Domínguez-García, Pastor, Melle & Rubio (2009)), something which is expected, if it is related in some way with the electrical charge contained in the

Fig. 5. Left: *g*(*r*) function, Right: Electrostatic potential calculated by inverting the O-Z equation (all the approximations give the same result) Inset: a detail for *U*(*r*) in the region of

As a confirmation of these results, we show here a calculation of the electrostatic potential using a long set of images of charged superparamagnetic microparticles spreading in the experimental system described above. We have obtained images of the suspension during more than an hour, with a temporal lapse between images of 0.3 seconds. This data allow us to produce a very defined graph for the pair correlation function, showed in the Fig.5. In the right side of the Fig.5, we plot the electrostatic potential and in its inset we can see that the minimum has a value of about −0.1 *kBT*, confirming the previous results obtained in this

However, this result may be an effect of an imagining artefact. About that question, some of the studies which use particle tracking only apply some filters to the images for detecting brightness points and then extracting the position of the particles. Our image analysis

the minimum. Number density *n* = 0.0009

experimental setup.

For example, the aggregation processes for *S*(*t*) in the work of Promislow et al. (1994), show an aggregation time of 200 seconds. The paramagnetic particles used in that work have a diameter of 0.6 *μ*m and a 27% of magnetite content. Using the Stokes-Einstein expression, *D* = 0.86 *μ*m/s<sup>2</sup> is obtained, supposing that these particles do not sediment. Using *φ* = 0.0012 and *λ* = 8.6, we can obtain that *t*ag ∼ 122 seconds, in the order of their experimental result. In the case of our experiments, we obtain the same values using Eq.47 that using Brownian simulations.

These discrepancies may be related with hydrodynamic interactions which should affect the diffusion of the particles. From Eq.47, we see that some variation on the diffusion coefficient of the particles can modify the expected aggregation time for two particles. For testing that, we made some microrheology measurements using different types of isolated particles according to the theory of sedimentation and with the corrections on the values of the diffusion coefficient. The experimental values agree very well with the theoretical ones calculated from the expression 2.3.4 (Domínguez-García, Pastor, Melle & Rubio (2009)) but they imply a reduction on the diffusion coefficient a factor of three as a maximum, no being sufficient for explaining the discrepancy in the aggregation times.

Fig. 4. Experiments of aggregation and disaggregation. The experimental process of aggregation (a) begins with *λ* = 1718, *φ*2*<sup>D</sup>* = 0.088 while disaggregation is shown in (b). Brownian dynamics simulations results with *λ* = 100, *φ*2*<sup>D</sup>* = 0.03 are shown for aggregation (c) and disaggregation (d). Data from Refs.(Domínguez-García et al. (2007; 2011))

20 Hydrodynamics

For example, the aggregation processes for *S*(*t*) in the work of Promislow et al. (1994), show an aggregation time of 200 seconds. The paramagnetic particles used in that work have a diameter of 0.6 *μ*m and a 27% of magnetite content. Using the Stokes-Einstein expression, *D* = 0.86 *μ*m/s<sup>2</sup> is obtained, supposing that these particles do not sediment. Using *φ* = 0.0012 and *λ* = 8.6, we can obtain that *t*ag ∼ 122 seconds, in the order of their experimental result. In the case of our experiments, we obtain the same values using Eq.47 that using Brownian

These discrepancies may be related with hydrodynamic interactions which should affect the diffusion of the particles. From Eq.47, we see that some variation on the diffusion coefficient of the particles can modify the expected aggregation time for two particles. For testing that, we made some microrheology measurements using different types of isolated particles according to the theory of sedimentation and with the corrections on the values of the diffusion coefficient. The experimental values agree very well with the theoretical ones calculated from the expression 2.3.4 (Domínguez-García, Pastor, Melle & Rubio (2009)) but they imply a reduction on the diffusion coefficient a factor of three as a maximum, no

being sufficient for explaining the discrepancy in the aggregation times.

Fig. 4. Experiments of aggregation and disaggregation. The experimental process of aggregation (a) begins with *λ* = 1718, *φ*2*<sup>D</sup>* = 0.088 while disaggregation is shown in (b). Brownian dynamics simulations results with *λ* = 100, *φ*2*<sup>D</sup>* = 0.03 are shown for aggregation

(c) and disaggregation (d). Data from Refs.(Domínguez-García et al. (2007; 2011))

simulations.

For completing this study, we also have shown results of disaggregation, that is, the process that occurs when the external magnetic field is switched off and the clusters vanish. For this process we study the kinetics in the same way that in aggregation, by searching power laws behaviours and calculating the kinetic exponents *z* and *z*� (Domínguez-García et al. (2011)). We have also developed Brownian dynamics simulations to be compared with the experiments. The Fig.4 summarizes some of our results in aggregation and disaggregation. The experimental kinetic exponents during disaggregation range from *z* = 0.44 to 1.12 and *z*� = 0.27 to 0.67, while simulations give very regular values, with *z* and *z*� ∼ 1. Then, the kinetic exponents do not agree, being also the process of disaggregation much faster in simulations. From these results, we conclude that remarkable differences exist between a simple theoretical model and the interactions in our experimental setup, differences that are specially important when the influence of the applied magnetic field is removed.

In all these experiments some data has been collected before any external field is applied. That allows us to study the microstructure of the suspensions by calculating the electrostatics potential using the methods previously explained. The inversion of the O-Z equation reveals an attractive well in the potential with a value in its minimum in the order of −0.2 *kBT*, similar to other observations of attractive interactions of sedimented particles in confinement situations. Moreover, these values of the minimum in the potential seems to depend of the concentration of particles (Domínguez-García, Pastor, Melle & Rubio (2009)), something which is expected, if it is related in some way with the electrical charge contained in the suspension.

Fig. 5. Left: *g*(*r*) function, Right: Electrostatic potential calculated by inverting the O-Z equation (all the approximations give the same result) Inset: a detail for *U*(*r*) in the region of the minimum. Number density *n* = 0.0009

As a confirmation of these results, we show here a calculation of the electrostatic potential using a long set of images of charged superparamagnetic microparticles spreading in the experimental system described above. We have obtained images of the suspension during more than an hour, with a temporal lapse between images of 0.3 seconds. This data allow us to produce a very defined graph for the pair correlation function, showed in the Fig.5. In the right side of the Fig.5, we plot the electrostatic potential and in its inset we can see that the minimum has a value of about −0.1 *kBT*, confirming the previous results obtained in this experimental setup.

However, this result may be an effect of an imagining artefact. About that question, some of the studies which use particle tracking only apply some filters to the images for detecting brightness points and then extracting the position of the particles. Our image analysis

seems to be. Indeed, this "detaining" effect of the particles inside the clusters is not observed

<sup>341</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

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

What is more, we have also observed that the kinetic exponents during aggregation are different and slower if we add salt to the suspension (Domínguez-García et al. (2011)). This last effect may be related with an unexpected interaction of the particles with the charged quartz bottom wall by means of a a spontaneous macroscopic electric field. When the particles and clusters have no electrical component, they should be highly sedimented at the bottom of the quartz cell and the resistance to their the movement should be increased (Kutthe (2003)),

In this chapter, we have reviewed the main interactions, with focus on hydrodynamics and from a experimental point of view, that can be important in a confined colloidal system at low concentration of microparticles. We have used charged superparamagnetic microparticles dispersed in water in low-confinement conditions by means of a glass cell for the study of irreversible field-induced aggregation and disaggregation, as well as the microstructure of the suspension. Regarding aggregation characteristic times and basic behaviour on the disaggregation of the particles, we have observed significant discrepancies between the experimental results and the theory. Morover, anomalous effects in the electrostatic behaviour have been observed, showing that, in this kind of systems, the electro-hydrodynamics interactions are not well understood at present and deserve more theoretical and experimental

We wish to acknowledge Sonia Melle and J.M. Pastor for all the work done, J.M M. González, J.M. Palomares and F. Pigazo (ICMM) for the VSM magnetometry measurements and J.C. Gómez-Sáez for her proofreading of the English texts. This research has been partially supported by M.E.C. under Project No. FIS2006-12281-C02-02, M.C.I under FIS2009-14008-C02-02, C.A.M under S/0505/MAT/0227 and UNED by 2010V/PUNED/0010.

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in experiments with added salt.

**4. Conclusions**

investigations.

**6. References**

**5. Acknowledgements**

generating that the kinetic exponents reduce their value.

software (Domínguez-García & Rubio (2009)) employs open-sourced algorithms for detecting the centres of mass of the particles by detecting the borders of each object and then obtaining its geometrical properties. As an example, we have tried to evaluate how this border detection can have an influence on the result of the electrostatic potential. A measured apparent displacement Δ(*r*) = *r*� − *r* should affect to the radial distribution function in the following form: *g*(*r*) = *g*� (*r* + Δ(*r*))(1 + *d*Δ(*r*)/*dr*) (Polin et al. (2007)). From that expression, the variation in the electrostatic potential is:

$$
\beta l l'(r) - \beta l l(r) \cong -\beta \frac{d l l(r)}{dr} \Delta(r) + \frac{d \Delta(r)}{dr} \tag{48}
$$

For obtaining Δ(*r*) we have extracted a typical particle image and we have composed some set of images which consist on separating the two particles a known distance (*r*) in pixels. Next, we apply our methods of image analysis for obtaining the position of those particles and calculate the distances (*r*� ). Then, the apparent displacement, Δ(*r*) = *r*� − *r*, is observed to grow when the particles are very near. In Fig.6, we display the results of our calculations on the possible artefact in the analysis of the position of the particles by image binarization and binary watershed, a method for automatically separating particles that are in contact. The figure reveals that the correction on the electrostatic potential for this cause is basically negligible, because the correction in the potential is zero for distances *r* > 1.2 *μ*m. In the inset of the figure we can see some of the images we have employed for this calculation, showing the detected border of the particles among the images themselves.

Fig. 6. Estimation of a possible artefact in the analysis of the position of the particles. In the inset we have included some examples of the images used for this calculation.

In any case, the possibility of an artifact can be the cause of these observations in the electrostatic potential cannot be descarted. However, the direct or indirect presence and influence of these attractive wells has been detected in many other situations in these experiments. For example, the attractive interaction disappears when we added a salt, in our case KCl, to the suspensions, confirming the electrostatic nature of the phenomena (Domínguez-García, Pastor, Melle & Rubio (2009)). In disaggregation it is observed how the particles move inside the chains without leaving them (Domínguez-García et al. (2011)). The lapse of time that the particles are in this situation depends on the initial morphology of the aggregates, something which has been observed to depend on the ratio *R*1/*R*<sup>0</sup> (Domínguez-García, Melle & Rubio (2009); Domínguez-García & Rubio (2010)). Then, this effective lapse of time depends of how many particles are located near the other in a short distance. In that situation, the attractive interaction should play a role in disaggregation, as it

seems to be. Indeed, this "detaining" effect of the particles inside the clusters is not observed in experiments with added salt.

What is more, we have also observed that the kinetic exponents during aggregation are different and slower if we add salt to the suspension (Domínguez-García et al. (2011)). This last effect may be related with an unexpected interaction of the particles with the charged quartz bottom wall by means of a a spontaneous macroscopic electric field. When the particles and clusters have no electrical component, they should be highly sedimented at the bottom of the quartz cell and the resistance to their the movement should be increased (Kutthe (2003)), generating that the kinetic exponents reduce their value.
