**2.4 Electrostatics**

12 Hydrodynamics

with Δ*ρ* = *ρ<sup>p</sup>* − *ρ*. We can define the Péclet number as the ratio between the sedimentation

Then, the vertical distance travelled by gravity for a cluster in a time equal to that a particle

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

ln *<sup>ρ</sup>*(*z*) <sup>∝</sup> <sup>−</sup> *<sup>z</sup>*

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

*LG*

*e*−*z*/*<sup>L</sup> <sup>e</sup>*−*a*/*<sup>L</sup>* <sup>−</sup> *<sup>e</sup>*(*a*−*h*)/*<sup>L</sup>*

*zPB*(*z*)*dz* = (32)

<sup>=</sup> <sup>4</sup>*πa*4*g*Δ*<sup>ρ</sup>*

<sup>3</sup>*kBT* (29)

(30)

(31)

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

time *ts* and diffusion *td* using a fixed distance, for instance, 2*a*:

*Pe* <sup>≡</sup> *td ts*

Boltzmann profile as Faucheux & Libchaber (1994) showed:

*L* ≡ *kBT* (*g*Δ*M*)

*PB*(*z*) = <sup>1</sup>

<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>*).

 *h*−*a a*

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

*zm* =

*L*

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

> <sup>=</sup> *<sup>e</sup>*−*a*/*L*[*aL* <sup>+</sup> *<sup>L</sup>*2] <sup>−</sup> *<sup>e</sup>*(*a*−*h*)/*L*[(*<sup>h</sup>* <sup>−</sup> *<sup>a</sup>*)*<sup>L</sup>* <sup>+</sup> *<sup>L</sup>*2] *<sup>L</sup>*[*e*−*a*/*<sup>L</sup>* <sup>−</sup> *<sup>e</sup>*(*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

the balance on the thermal forces and gravity:

<sup>=</sup> *Mga kBT*

spread a distance equal to the diameter of the particle *d* is *dc* = *vc td* = *PeNd*.

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 particles.
