# Electrokinetics and microfluidics

## Dynamics and directed assembly in colloidal suspensions

**Collaborators: Joshua Kuntz and Andrew Pascall (Lawrence Livermore National Laboratory)
Research Assistant: Jae Sung Park (spherical particles and electrophoretic deposition)**

Electric fields are commonly used for the manipulation of colloidal particles in suspension, where they can induce particle motions in different ways: for instance, charged particles in a uniform field undergo electrophoretic migration, while polarizable particles in a non-uniform field undergo dielectrophoresis. Recent theoretical work also showed that polarizable particles (such as metallic particles) in an electric field are subject to non-linear induced electroosmotic flows driven on the particle surfaces. This phenomenon, coined induced-charge electrophoresis, can lead to hydrodynamic interactions between suspended particles, the effects of which on particle dynamics have yet to be fully understood.

Over the last several years, we have developed both theoretical models and computational tools to analyze the dynamics in suspensions of particles undergoing these various phenomena. Motivated by applications involving sub-micrometer metallic rods carrying a barcode in the form of metallic stripes (which can be used for biochemical detection as well as DNA and protein bioassays), we first studied the dynamics in suspensions of infinitely polarizable slender rods. We derived a simple slender-body formulation that accounts for induced-charge electrophoresis based on a thin double layer approximation, and showed that the effects of the electric field on a single rod can be modeled by a linear slip velocity along the rod axis, which causes particle alignment and drives a stresslet flow in the surrounding fluid. Based on this slender-body model, the hydrodynamic interactions between a pair of aligned rods were studied, and we identified domains of attraction and repulsion, suggesting that particle pairings may occur as a result of hydrodynamic interactions. We then performed large-scale simulations of suspensions of hydrodynamically interacting rods, including the effects of induced-charge electrophoresis. Simulations were performed for both non-Brownian and Brownian suspensions. In the non-Brownian case, the simulations confirmed the occurrence of particle pairings in the suspension, as demonstrated by sharp peaks in the pair distribution function. The superposition of all the electrophoretic flows driven on the rod surfaces is also observed to result in a diffusive motion at long times, and hydrodynamic dispersion coefficients were calculated. The case of Brownian suspensions was also investigated, and Brownian fluctuations are found to hinder particle pairing and alignment. Comparisons with experiments were also performed, and showed excellent agreement.

More recently, that study was extended to the case of spherical particles, which is somewhat more subtle as both dielectrophoresis and induced-charge electrophoresis have to be taken into account. First, we performed a detailed calculation of pair interactions using both asymptotic methods and boundary-integral simulations. The calculation of pair interactions was then used to perform direct simulations of semi-dilute suspensions. Our simulations showed similar results as in the case of rods, namely the existence of transient pairings that result in an effective diffusion. In the case where polarizability is weak, these pairings no longer take place and instead the particles aggregate into chains that coalesce into complex cellular patterns, in agreement with recent experiments. We also recently investigated the effect of surface contamination, and discovered a transition from diffusive dynamics to chaining and aggregation as contamination becomes more significant and reduces effective polarizability.

**Figure 1:** Chaining and cellular pattern formation in confined suspensions of dielectric particles in an AC electric field.

In more recent work, we have investigated the use of electrophoretic deposition as a technique for the manufacturing of novel materials with specified microstructural or functional properties (e.g. functionally graded materials). An ideal manufacturing technique should provide detailed control on the material structure and composition down to the nanoscale. To this end, we have been developing quantitative, detailed models and algorithms for the electrophoretic deposition of colloidal suspensions, with the aim of helping the design and optimization of experiments. Our simulations include linear electrophoresis, electrostatic repulsion based on DLVO theory, dipolar interactions, van-der-Waals interactions, Brownian motion, and excluded volume interactions. Using this method, we have simulated the deposition of colloidal suspensions under various conditions, and have shown the existence of a transition from regular fcc crystals to amorphous deposits as field strength is increased, in agreement with experiments performed at Lawrence Livermore National Laboratory (see figure 2). We are currently investigating the use of patterned electrodes for the design and manufacturing of colloidal crystals with three-dimensional features.

**Figure 2:** Electrophoretic deposition of collodal suspensions. Top: experiments by Joshua Kuntz and Andrew Pascall (Lawrence Livermore National Laboratory). Bottom: direct numerical simulations. Field strength is increased from left to right, resulting in a transition from crystalline to amorphous deposits.

###### References

**Direct numerical simulations of electrophoretic deposition of charged colloidal suspensions**

J. S. Park, D. Saintillan, *Proceedings of the 4th International Conference on Electrophoretic Deposition*, in *Key Engineering Materials*, **507** 47 (2012).

**From diffusive motion to local aggregation: Effect of surface contamination in dipolophoresis**

J. S. Park, D. Saintillan, *Soft Matter*, **7** 10720 (2011). [reprint]

**Electric-field-induced ordering and pattern formation in colloidal suspensions**

J. S. Park, D. Saintillan, *Phys. Rev. E*, **83** 041409 (2011). [reprint]

**Dipolophoresis in large-scale suspensions of ideally polarizable spheres**

J. S. Park, D. Saintillan, *J. Fluid Mech.*, **662** 66 (2010). [reprint]

**Hydrodynamic interactions in metal rod-like particle suspensions due to induced charge electroosmosis**

K. A. Rose, B. Hoffman, D. Saintillan, E. S. G. Shaqfeh, J. G. Santiago, *Phys. Rev. E*, **79** 011402 (2009). [reprint]

**Nonlinear interactions in electrophoresis of ideally polarizable particles**

D. Saintillan, *Phys. Fluids*, **20** 067104 (2008). [reprint]

**Stabilization of a suspension of sedimenting rods by induced-charge electrophoresis**

D. Saintillan, E. S. G. Shaqfeh, E. Darve, *Phys. Fluids*, **18** 121701 (2006). [reprint]

**Hydrodynamic interactions in the induced-charge electrophoresis of colloidal rod dispersions**

D. Saintillan, E. Darve, E. S. G. Shaqfeh, *J. Fluid Mech.*, **563** 223 (2006). [reprint]

## Interactions in Quincke electrorotation

**Research Assistant: Debasish Das**

Quincke rotation, a well-observed phenomenon in particle suspensions, denotes the spontaneous rotation of dielectric particles immersed in a slightly dielectric liquid when subjected to a strong enough DC electric field. Quincke rotation occurs when the charge relaxation time of the particles is greater than that of the fluid medium, causing the particles to become polarized in a direction opposite to that of the electric field and therefore giving rise to an unstable equilibrium orientation. When slightly perturbed, the particles start to rotate, and if the applied electric field exceeds a critical value this perturbation does not decay and the particle rotation reaches a steady state with a constant angular velocity obtained by balancing the viscous torque with the electric torque due to the induced dipole. The dynamics of a particle undergoing Quincke rotation have been previously shown to obey the classic Lorenz oscillator equations with two bifurcations. When the applied electric field exceeds the critical electric field for spontaneous rotation, the angular velocity undergoes a supercritical pitchfork bifurcation, by which the zero angular velocity state becomes unstable and a stable state is reached where the angular velocity depends on the ratio of the applied electric field to the critical field. Upon increasing the electric field further, the angular velocity undergoes a subcritical Hopf bifurcation and becomes erratic indicating Lorenz chaos.

A potentially useful application of Quincke rotation lies in its ability to modify the effective rheological properties of suspensions under flow. When a suspension undergoing Quincke rotation is subjected to a steady shear flow, its apparent viscosity has been shown to decrease as a result of the enhanced rotation rate of the particles with respect to the flow vorticity, which has the effect of increasing the flow rate and therefore of decreasing the suspension viscosity. An apparent increase in the effective conductivity of the suspension has also been reported. Most previous studies of these two effects have focused on the dynamics of a single isolated spherical particle and therefore are unable to capture interactions between particles in semi-dilute or concentrated suspensions. In particular, experiments in suspensions typically exhibit weaker angular velocities than predicted by the theory for an isolated sphere. Also, the critical field above which rotation takes place has been found to be slightly higher than that predicted by the single sphere theory.

To analyze some of these effects, we have used a combination of numerical simulations and asymptotic theory to study the effect of electrohydrodynamic interactions between particles on Quincke rotation. We have studied the prototypical case of two equally sized spheres carrying no net charge and interacting with each other both electrically and hydrodynamically. We have used the classic method of reflections to capture far-field interactions, which yields a coupled system of time-dependent ordinary differential equations for the dipole moments, angular velocities, and positions of the two spheres. Our results show that spheres held in place tend to synchronize at a steady angular velocity (see figure 3), which differs from that of a single sphere owing to interactions. We also find that the critical field above which rotation takes place is higher than that predicted by the single sphere theory, in agreement with experimental observations.

**Figure 3:** Synchronization of two electrohydrodynamically interacting spheres undergoing Quincke rotation.

###### Reference

**Electrohydrodynamic interaction of spherical particles under Quincke rotation**

D. Das, D. Saintillan, in preparation (2012).

## Optimization of mixing by chaotic advection

**Research Assistant: Qizheng Yan**

Fluid mixing plays a central role in many technological applications, ranging from pharmaceutical processes to polymer science to biotechnology, where it may be useful to homogenize two distinct chemical species or reagents to facilitate a reaction. Fundamental applications that rely on mixing are as varied as biochemical analysis in lab-on-chip devices, genomic analysis using so-called polymerase chain reaction, among many others. With the recent emergence of efficient and powerful micro- and nanofabrication techniques, there has been much interest in scaling down these processes to small-scale fluidic devices, which only require very tiny sample sizes. Mixing at small scales is, however, a significant challenge owing to the predominance of viscous effects and to the slow diffusion that typically characterize these flows. In particular, traditional methods for mixing, which are based on turbulence, do not carry over to these devices where all the flows are smooth and laminar. These observations have driven much research over the last two decades aimed at designing efficient ways of enhancing mixing in small-scale flows. These techniques are varied and include: complex device geometries, flow instabilities, mixing by active liquids, and mixing by external fields. Many successful devices have been constructed, but in most cases their design has been driven by trial-and-error with only limited quantitative guidelines, in part owing to our poor understanding of the fundamental characteristics of 'good' mixing flows.

To address this problem, we have used finite-horizon optimal control theory to numerically determine a general class of optimal mixing flows in a two-dimensional periodic geometry as truncated sums of time-modulated Fourier modes satisfying the Stokes equations (this approach was originally proposed by Mathew *et al.*, *J. Fluid Mech.* 2009). These flows are obtained to minimize the final value of a multiscale mixing variance for a passive scalar field transported by the fluid. Excellent mixing by the optimal flow fields is observed (see figure 4), as demonstrated by an exponential decay of the mixing variance with time. The time-averaged kinetic energy spectra of these flows were calculated to investigate the effect of scale, and we find that optimal mixing flows with prescribed kinetic energy contain a wide range of spatial scales but are strongly dominated by high-wavenumber modes, whereas those with prescribed power input are always dominated by large scales. Finite-time Liapunov exponents were also calculated for the optimized flow fields, and show that efficient mixing is directly correlated with large values of the exponents.

**Figure 4:** Evolution of the scalar field as it is advected by optimal mixing flows with different numbers of spatial modes: *N* = 2, 10, and 24. The calculations were performed at constant kinetic energy.

###### Reference

**Optimal chaotic mixing by two-dimensional Stokes flows**

D. Saintillan, Q. Yan, submitted (2012).

## Dynamics of autonomous catalytic nanomotors

**Collaborator: Dr. Yiping Zhao (University of Georgia)
Research Assistant: Shrenik Kothari**

Catalytic nanomotors are nano-to-microsized actuators converting chemical energy into kinetic energy through a catalyzed reaction using an on-board catalyst. Nature provides researchers with models by which to design inorganic artificial nanomotors, and several avenues have been traversed to mimic naturally occurring biomotors. In this work, we have collaborated with the Zhao research group at University of Georgia to use a combination of experiments (performed at UGA) and simulations (performed at the University of Illinois) to study the relationship between nanomotor shape and resulting dynamics. This study has specifically focused on asymmetric nanomotors consisting of a spherical microbead with an arm extending to different lengths and angles. Such nanomotors are fabricated in the Zhao lab using a manufacturing technique known as glancing angle deposition (GLAD), and exhibit interesting dynamics in hydrogen peroxyde solutions, where they trace nearly spherical trajectories with speeds and curvatures that depend on particle shape and on the intensity of the catalytic reaction fueling the motion.

To understand these dynamics more precisely, we have developed a detailed algorithm to simulate the dynamics of these nanomotors. These simulations include an accurate representation of the nanomotor geometry, full hydrodynamic interactions between components of the nanomotor and the supporting wall, a model for frictional forces with the wall, as well as Brownian forces to account for thermal fluctuations. A sample nanomotor geometry is illustrated in figure 5, and consists of a rigid sphere connected to a section of an ellipsoid to model the fan-like shape of the nanomotor arm in the experiments. By removing sections of different lengths on the free end of the ellipsoid, various arm lengths have been modeled for direct comparison with experiments. Particle dynamics have been modeled using the method of regularized Stokeslets, allowing for the direct numerical calculation of the resistance matrix for the nanomotor, which relates the force and torque on a particle to its linear and angular velocities. Detailed comparisons with the experiments have been performed, and showed very good agreement for the dependence of the trajectory curvature and angular frequency of the nanomotors upon arm lengths and linear velocity (which is related to the propulsion force magnitude).

**Figure 5:** (a) Images of the nanomotors used in the experiments. (b) Model of a nanomotor, as a sperical bead to which a portion of a spheroid is attached. The figure also shows the mesh used in the regularized Stokeslet simulations. (c) Comparison between experiments and simulations for the curvature of the trajectories vs arm length.

###### Reference

**Geometrically designing the kinematic behavior of catalytic nanomotors**

J. G. GIbbs, S. Kothari, D. Saintillan, Y.-P. Zhao, *Nano Lett.*, **11** 2543 (2011). [reprint]