# Hydrodynamics within the Electric Double Layer on Slipping Surfaces

Seventh entry by Kelly Miller, AP225 Fall 2011

## Hydrodynamics within the Electric Double Layer on Slipping Surfaces

Authors: Laurent Joly, Christophe Ybert, Emmanuel Trizac, Lyde´ric Bocquet

Journal: Physical Review Letters, 93 (25), 2004 [1]

## Background

The electric double layer (EDL) is fundamental to understanding the properties of charged colloidal systems. The EDL idea was introduced by Gouy, Debye, and Huckel to describe how microions distributed themselves close to a charged surface. The width of the EDL determines the electric interaction range between macromolecules and therefore, controls the static phase behavior of these systems.

On the other hand, the dynamical level of EDL is fundamental to many electrokinetic effects:

-electrophoresis

-electro-osmosis

-streaming current or potential

These phenomena are governed by the surface via the EDL and therefore, provide good ways to drive flows in microfluidic devices where surface effects are dominant.

## Introduction

This paper uses molecular dynamics simulations to show that the dynamics of the electric double layer (EDL) is dependent on the wettability of the charged surface on which the EDL forms. Traditionally, EDL dynamics have been described using the Poisson-Boltzmann theory and on continuum hydrodynamics for the flow fields. These two theories are captured in the zeta potential ($\zeta$) which is traditionally defined as the electric potential (V(z)) computed at the surface of shear where the fluid velocity disappears. However - this definition of ($\zeta$) relies on the assumption of a no-slip boundary condition of the solvent at the solid surface.

Recently, there has been progress made in understanding rheology of fluids at small scales. This is partly a result of computer simulations (such as the molecular dynamics system used by the authors of this paper) but is mainly due to the development of new experimental techniques such as the atomic force microscopy (AFM). Using these techniques, continuum hydrodynamics are found to remain valid up to very small length scales, BUT the "no-slip" boundary condition for the fluid velocity at the solid surface seems to be violated in many situations. Interestingly, this violation is found to be controlled by the wetting properties of the fluid on the solid surface. The no-slip boundary condition is upheld on hydrophilic surfaces but not on hydrophobic surfaces.

The focus of this paper is to show that a slip effect for the solvent at a charged surface considerably enhances the measured electrokinetic effects and results in an enhanced ($\zeta$) potential due to the dynamics of the solvent at the surface.

## The Model

The fluid system (solvent and microions) is confined between two parallel solid substrates, composed of individual atoms fixed on an fcc lattice.

The interactions between solvent and solid substrate particles are governed by the following potential:

The tuning parameter: cij allows for the wetting properties of the fluid on the surface to be adjusted. The substrate exhibits hydrophilic behavior and for large fluid-solid cohesivity, Cfs, and a hydrophobic behavior for small Cfs.

Through this tuning parameter, the model includes the wettability of the surface (which is traditionally neglected in traditional descriptions of electrokinetics).

## Results

The measured velocity profiles for the situation corresponding to the wetting substrate is shown in the main plot of Figure 2 (below) and for the nonwetting case is shown in the inset of the figure.

For the wetting substrate the velocity profile exhibits the shape as predicted by continuum hydrodynamics (even at the small scale of the EDL). The no-slip BC is found to apply inside the liquid, at a distance of about one layer of solvent particle, in agreement with the theory. This immobile layer coincides with the Stern layer of stationary ions close to the charged surface.

For the nonwetting substrate the velocity profile shows a very different behavior. In this case, the first layer of ions now contributes a large amount to the global streaming current (in sharp contrast to the wetting case). The remobilization of the Stern layer adds on to the slippage effect and therefore increases the $\zeta$ potential. Also, the slip length has been measured to hardly depend on the screening factor, which means that the ions do not affect the fluid-solid friction in the nonwetting case.

Figure 3 depicts the overall conclusion that nonwettability strongly amplifies the electrokinetic effects, that is - the ratio between the $\zeta$ potential and the surface potential is much larger in the hydrophobic case than it is in the hydrophillic case.

In this figure, the simulation points for $\zeta$ are compared to the Poisson-Boltzmann estimate for the electric potential. Figure 2 shows that for the wetting case, the agreement is good but, for the nonwetting case, the agreement is not good due to the fact that the $\zeta$ potential is dominated by the slip effect and the immobile Stern layer is absent.

## Conclusion

This paper has used simulations to show that the $\zeta$ potential (the essence of EDL dynamics) depends on the wettability of the charged substrate. In the wetting situation, the $\zeta$ potential can be related to surface charge properties, confirming the traditional Stern Layer situation but, for nonwetting substrates, electric and slip effects are important.

This was a very interesting paper as it explored the interaction of two important soft-matter topics: wettability and charged interfaces.