# Like-Charge Attraction and Hydrodynamic Interaction

**Introduction**

When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics.

In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hydrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well.

**Theory**

This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere.

When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move closer to one another as they move upward. This is depicted below in Figure 2.

Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be

<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math>

Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1.

If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is

<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math>

If we assumed that displacement was due to <math>F_{eff}</math>, a closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math>

<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math>

Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math>

The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>):

<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \, \frac{\exp(-\kappa r)}{r}, </math>

where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.

The wall force (<math>F_w</math>) is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by

<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \, \exp(-\kappa h), </math>

where <math>\sigma_g</math> is the surface charge density at the wall.

The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent.

From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below.

As shown in Figure 3, at distances close to the wall, the spheres appear to be attracted to one another, although this effect is kinematic in nature. The authors also compare their theory to Brownian dynamics simulations as shown in Figure 4 below. Their model fits the experimental data well.

**Conclusions**

The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.

**Reference**

T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)