**2.4.1 DLVO theory**

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.

The presence of this layer modifies the equation of the double-layer potential (Reiner & Radke

<sup>333</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

For calculating the electrostatic potential in a colloidal suspension, we can use the following methodology. This approach involves using the radial distribution function of the particles, *g*(*r*), knowing that it is related with the interaction energy of two particles in the limit of

*g*(*r*) = *e*

where *n* is the particle density and *β* ≡ 1/*kBT*. However, for finite concentrations, *g*(*r*) is influenced by the proximity between particles, so we can calculate the mean force potential,

But we do not know the relation between *w*(*r*) and *U*(*r*). Here, is usually defined a total correlation function *h*(*r*) ≡ *g*(*r*) − 1 and is used the Ornstein-Zernike (O-Z) equation for two

> � *c*(*r*� )*h*( � �**r**� − **r** �

−*βU*(*r*) MSA

The *c*(*r*) function is the direct correlation function between two particles. Now, it is necessary to close the integral equation by linking *h*(*r*), *c*(*r*) and *U*(*r*). For that, one of the following

named Mean Spherical Approximation (MSA), Hypernetted Chain (HNC) and Percus-Yevick

In the case of video-microscopy experiments, a more practical methodology is explained by Behrens & Grier (2001b) for obtaining the electrostatic potential. More explicitly, with the PY

*<sup>β</sup> <sup>I</sup>*(*r*) = <sup>−</sup> <sup>1</sup>

*<sup>β</sup>* ln [<sup>1</sup> <sup>+</sup> *n I*(*r*)] <sup>=</sup> <sup>−</sup> <sup>1</sup>

) � �*g*( � �**r**� − **r** � �) − 1 � *d*2*r*�

) − 1 − *n I*(*r*�

*β* �

ln � *<sup>g</sup>*(*r*) 1 + *n I*(*r*)

−*βU*(*r*) + *h*(*r*) − ln *g*(*r*) HNC (<sup>1</sup> <sup>−</sup> *<sup>e</sup>βU*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*(*r*)) PY

*<sup>w</sup>*(*r*) = <sup>−</sup> <sup>1</sup>

<sup>2</sup> <sup>+</sup> *<sup>s</sup>*�/*<sup>a</sup>* exp(−*κ<sup>s</sup>*

�

) (35)

<sup>−</sup>*<sup>β</sup> <sup>U</sup>*(*r*) (36)

*<sup>β</sup>* ln *<sup>g</sup>*(*r*) (37)

�)*dr*� (38)

*<sup>β</sup>* [ln *<sup>g</sup>*(*r*) <sup>−</sup> *n I*(*r*)] , (40)

�� , (41)

, (42)

(39)

*Udl*(*s*) = <sup>2</sup>*πε*(*ψ*)<sup>2</sup> <sup>2</sup>

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

lim *n*→0

*h*(*r*) = *c*(*r*) + *n*

⎧ ⎨ ⎩

*<sup>U</sup>*(*r*) = *<sup>w</sup>*(*r*) + *<sup>n</sup>*

*g*(*r*�

*<sup>U</sup>*(*r*) = *<sup>w</sup>*(*r*) + <sup>1</sup>

which can be calculated numerically.

In both cases, *I*(*r*) is the convolution integral defined as: *<sup>I</sup>*(*r*) = � �

*c*(*r*) =

(1993); Shen et al. (2001)):

**2.4.2 Ornstein-Zernike equation**

particles in a two-dimensional fluid:

assumptions is employed:

approximation we have:

and with the HC:

infinite dilution by means of the Boltzmann distribution:

where *s*� = *s* − 2*δ*.

*w*(*r*):

(PY).

Thus, the method to control the aggregation is to vary the ionic strength of medium, i.e., the pH. In most applications in colloids, it is enormously important to control aggregation of particles, for example, for purification treatments of water.

The situation around a negatively charged colloidal particle is approximately described by the double layer model. This model is used to display the ionic atmosphere in the vicinity of the charged colloid and explain how the repulsive electrical forces act. Around the particle, the negative charge forms a rigid layer of positive ions from the fluid, usually called Stern layer. This layer is surrounded by the diffuse layer that is formed by positive ions seeking to approach the colloidal particle and that are rejected by the Stern layer. In the diffuse layer there is a deficit of negative ions and its concentration increases as we left the colloidal particle. Therefore, the diffuse layer can be viewed as a positively charged atmosphere surrounding the colloid.

The two layers, the Stern layer and diffuse layer, form the so-called double layer. Therefore, the negative particle and its atmosphere produce a positive electrical potential associated with the solution. The potential has its maximum value on the surface of the particle and gradually decreases along the diffuse layer. The value of the potential that brings together the Stern layer and the diffuse layer is known as the Zeta potential, whose interest mainly lies in the fact that it can be measured. This Zeta potential measurement, is commonly referred as *ζ* and measured in mV. The Zeta potential is usually measured using the Laser Doppler Velocimeter technique. This device applies an electric field of known intensity of the suspension, while this is illuminated with a laser beam. The device measures the rate at which particles move so that the Zeta potential, *ζ*, can be calculated by several equations that relate the Zeta potential electrophoretic mobility, *μe*.

In a general way, it is possible to use the following expression, known as the Hückel equation:

$$
\mu\_{\ell} = \frac{2}{3} \frac{\varepsilon \mathbb{J}}{\eta} f(\kappa a) \tag{33}
$$

where *ε* is the dielectric constant of the medium, *η* its viscosity, *a* the radius of the particle and where 1/*κ* is the width of the double layer, known as the Debye screening length and where *f*(*κa*) is the named Henry function. In the case of 1 < *κa* < 100, the Zeta potentials can be calculated by means of some analytic expression of the Henry function (Otterstedt & Brandreth. (1998)). Summarising, the higher is the Zeta potential, the more intense will be the Coulombian repulsion between the particles and the lower will be the influence of the Van der Waals force in the colloid.

The Van der Walls potential, which can provide a strong attractive interaction, is usually neglected because its influence is limited to very short surface-to-surface distances in the order of 1 nm. Therefore, the DLVO electrostatic potential between two particles located a radial distance *r* one from the other is usually given by the classical expression:

$$\mathcal{U}(r) = \frac{(Z^\*e)^2}{\varepsilon} \frac{\exp\left(2a\kappa\right)}{(1+a\kappa)^2} \frac{\exp\left(-\kappa r\right)}{r} \tag{34}$$

where *Z*<sup>∗</sup> is the effective charge of the particles and *σ*eff = *Z*∗*e*/4*πa*<sup>2</sup> is their density of effective charge. Therefore, in this theory, two spherical like-charged colloidal particles suffered a purely electrostatic repulsion between them. The colloidal particle can have carboxylic groups (*COOH*) attached to their surfaces, creating a layer of negative charge of length *δ* in the order of nanometers surrounding the colloidal particles (Shen et al. (2001)). The presence of this layer modifies the equation of the double-layer potential (Reiner & Radke (1993); Shen et al. (2001)):

$$\mathcal{U}\_{dl}(\mathbf{s}) = 2\pi\epsilon(\psi)^2 \frac{2}{2+s'/a} \exp(-\kappa s') \tag{35}$$

where *s*� = *s* − 2*δ*.

14 Hydrodynamics

Thus, the method to control the aggregation is to vary the ionic strength of medium, i.e., the pH. In most applications in colloids, it is enormously important to control aggregation of

The situation around a negatively charged colloidal particle is approximately described by the double layer model. This model is used to display the ionic atmosphere in the vicinity of the charged colloid and explain how the repulsive electrical forces act. Around the particle, the negative charge forms a rigid layer of positive ions from the fluid, usually called Stern layer. This layer is surrounded by the diffuse layer that is formed by positive ions seeking to approach the colloidal particle and that are rejected by the Stern layer. In the diffuse layer there is a deficit of negative ions and its concentration increases as we left the colloidal particle. Therefore, the diffuse layer can be viewed as a positively charged atmosphere surrounding the

The two layers, the Stern layer and diffuse layer, form the so-called double layer. Therefore, the negative particle and its atmosphere produce a positive electrical potential associated with the solution. The potential has its maximum value on the surface of the particle and gradually decreases along the diffuse layer. The value of the potential that brings together the Stern layer and the diffuse layer is known as the Zeta potential, whose interest mainly lies in the fact that it can be measured. This Zeta potential measurement, is commonly referred as *ζ* and measured in mV. The Zeta potential is usually measured using the Laser Doppler Velocimeter technique. This device applies an electric field of known intensity of the suspension, while this is illuminated with a laser beam. The device measures the rate at which particles move so that the Zeta potential, *ζ*, can be calculated by several equations that relate the Zeta potential

In a general way, it is possible to use the following expression, known as the Hückel equation:

where *ε* is the dielectric constant of the medium, *η* its viscosity, *a* the radius of the particle and where 1/*κ* is the width of the double layer, known as the Debye screening length and where *f*(*κa*) is the named Henry function. In the case of 1 < *κa* < 100, the Zeta potentials can be calculated by means of some analytic expression of the Henry function (Otterstedt & Brandreth. (1998)). Summarising, the higher is the Zeta potential, the more intense will be the Coulombian repulsion between the particles and the lower will be the

The Van der Walls potential, which can provide a strong attractive interaction, is usually neglected because its influence is limited to very short surface-to-surface distances in the order of 1 nm. Therefore, the DLVO electrostatic potential between two particles located a

> exp (2*aκ*) (1 + *aκ*)<sup>2</sup>

where *Z*<sup>∗</sup> is the effective charge of the particles and *σ*eff = *Z*∗*e*/4*πa*<sup>2</sup> is their density of effective charge. Therefore, in this theory, two spherical like-charged colloidal particles suffered a purely electrostatic repulsion between them. The colloidal particle can have carboxylic groups (*COOH*) attached to their surfaces, creating a layer of negative charge of length *δ* in the order of nanometers surrounding the colloidal particles (Shen et al. (2001)).

exp (−*κr*)

radial distance *r* one from the other is usually given by the classical expression:

*ε*

*<sup>U</sup>*(*r*) = (*Z*∗*e*)<sup>2</sup>

*<sup>η</sup> <sup>f</sup>*(*κa*) (33)

*<sup>r</sup>* (34)

*<sup>μ</sup><sup>e</sup>* <sup>=</sup> <sup>2</sup> 3 *ε ζ*

particles, for example, for purification treatments of water.

colloid.

electrophoretic mobility, *μe*.

influence of the Van der Waals force in the colloid.

#### **2.4.2 Ornstein-Zernike equation**

For calculating the electrostatic potential in a colloidal suspension, we can use the following methodology. This approach involves using the radial distribution function of the particles, *g*(*r*), knowing that it is related with the interaction energy of two particles in the limit of infinite dilution by means of the Boltzmann distribution:

$$\lim\_{n \to 0} \lg(r) = e^{-\beta \, \mathcal{U}(r)} \tag{36}$$

where *n* is the particle density and *β* ≡ 1/*kBT*. However, for finite concentrations, *g*(*r*) is influenced by the proximity between particles, so we can calculate the mean force potential, *w*(*r*):

$$w(r) = -\frac{1}{\beta} \ln g(r) \tag{37}$$

But we do not know the relation between *w*(*r*) and *U*(*r*). Here, is usually defined a total correlation function *h*(*r*) ≡ *g*(*r*) − 1 and is used the Ornstein-Zernike (O-Z) equation for two particles in a two-dimensional fluid:

$$h(r) = c(r) + n \int c(r')h(|\mathbf{r'} - \mathbf{r}|)d\mathbf{r'}\tag{38}$$

The *c*(*r*) function is the direct correlation function between two particles. Now, it is necessary to close the integral equation by linking *h*(*r*), *c*(*r*) and *U*(*r*). For that, one of the following assumptions is employed:

$$\mathcal{L}(r) = \begin{cases} -\beta \mathcal{U}(r) & \text{MSA} \\ -\beta \mathcal{U}(r) + h(r) - \ln \mathcal{g}(r) & \text{HNC} \\ (1 - e^{\beta \mathcal{U}})(1 + h(r)) & \text{PY} \end{cases} \tag{39}$$

named Mean Spherical Approximation (MSA), Hypernetted Chain (HNC) and Percus-Yevick (PY).

In the case of video-microscopy experiments, a more practical methodology is explained by Behrens & Grier (2001b) for obtaining the electrostatic potential. More explicitly, with the PY approximation we have:

$$\mathcal{U}I(r) = w(r) + \frac{n}{\beta}I(r) = -\frac{1}{\beta} \left[ \ln g(r) - nI(r) \right],\tag{40}$$

and with the HC:

$$M(r) = w(r) + \frac{1}{\beta} \ln\left[1 + nI(r)\right] = -\frac{1}{\beta} \left[\ln\left(\frac{g(r)}{1 + nI(r)}\right)\right],\tag{41}$$

In both cases, *I*(*r*) is the convolution integral defined as:

$$I(r) = \int \left[ \mathcal{g}(r') - 1 - nI(r') \right] \left[ \mathcal{g}(|\mathbf{r'} - \mathbf{r}|) - 1 \right] d^2 r' \,\tag{42}$$

which can be calculated numerically.

Polin et al. (2007) realized that some minimums in the electrostatic potential can be eliminated by measuring the error on the displacement of the particles. However, this is not a double implication and other experimental minimums in the potential remain there. In that work, the authors take into account all the proposed artefacts to date for their measurements, demonstrating that charged glass surfaces really induce attractions between charged colloidal spheres. Moreover, Tata et al. (2008) claim that their observations using confocal laser scanning of millions of charged colloidal particles establish the existence of an attractive

<sup>335</sup> Hydrodynamics on Charged Superparamagnetic Microparticles

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

Moreover, other possible electrostatic variations in these systems may appear for several reasons. For instance, the emergence of a spontaneous macroscopic electric field in charged colloids (Rasa & Philipse (2004)). Moreover, according to several studies, changes in the fluid due to, for example, environmental pollution with atmospheric CO2, can be relatively easy and are not negligible at low concentrations, being able to radically change the electrical properties on the fluid (Carrique & Ruiz-Reina (2009)). Thus, interactions related to colloidal stability can produce anomalous effects and significant changes in, for example, sedimentation kinetics (Buzzaccaro et al. (2008)) or sedimentation-diffusion profiles (Philipse & Koenderink (2003)). Then, these electrostatic effects can affect the dynamics of

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

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

aggregation and influence the mobility of the particles and clusters.

behaviour in the electrostatic potential.

**3. Results**

a concentration of 1 gr/l.

and experimental ones.

#### **2.4.3 Anomalous effects**

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 this attractive interaction is about 0.7 *kBT*.

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

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)).

Polin et al. (2007) realized that some minimums in the electrostatic potential can be eliminated by measuring the error on the displacement of the particles. However, this is not a double implication and other experimental minimums in the potential remain there. In that work, the authors take into account all the proposed artefacts to date for their measurements, demonstrating that charged glass surfaces really induce attractions between charged colloidal spheres. Moreover, Tata et al. (2008) claim that their observations using confocal laser scanning of millions of charged colloidal particles establish the existence of an attractive

behaviour in the electrostatic potential. Moreover, other possible electrostatic variations in these systems may appear for several reasons. For instance, the emergence of a spontaneous macroscopic electric field in charged colloids (Rasa & Philipse (2004)). Moreover, according to several studies, changes in the fluid due to, for example, environmental pollution with atmospheric CO2, can be relatively easy and are not negligible at low concentrations, being able to radically change the electrical properties on the fluid (Carrique & Ruiz-Reina (2009)). Thus, interactions related to colloidal stability can produce anomalous effects and significant changes in, for example, sedimentation kinetics (Buzzaccaro et al. (2008)) or sedimentation-diffusion profiles (Philipse & Koenderink (2003)). Then, these electrostatic effects can affect the dynamics of aggregation and influence the mobility of the particles and clusters.
