Traveling-Wave Electrophoresis For Microfluidic Separations

Models and microfluidic experiments are presented of an electrophoretic separation technique in which charged particles whose mobilities exceed a tunable threshold are trapped between the crests of a longitudinal electric wave traveling through a stationary viscous fluid. The wave is created by applying periodic potentials to electrode arrays above and below a microchannel. Predicted average velocities agree with experiments and feature chaotic attractors for intermediate mobilities.

Figure 1

(a) Exploded view of sandwich architecture for traveling-wave electrophoresis. A microchannel is formed between two glass plates (transparent) and two polydimethylsiloxane (PDMS) spacers of height $h=15\mu \mathrm{m}$ (gray). Photolithographically patterned interdigitated arrays of gold electrodes on the bottom surface of the top plate are held at oscillating electrical potentials ${\Phi }_0(t)$ and ${\Phi }_2(t)$. Similar arrays atop the bottom plate are held at potentials ${\Phi }_1(t)$ and ${\Phi }_3(t)$. (b) Side view of the channel showing two replicates of the four-electrode pattern, of wavelength $\lambda =80\mu \mathrm{m}$. The experimental device has 50 replicates of this pattern and electrode cross sections of half width $a=1\mu \mathrm{m}$ and thickness $b=150\mathrm{nm}$. The fabrication process leaves a PDMS layer (gray) of thickness comparable to $b$ on the bottom plate. The potential of each electrode leads its neighbor to the right in phase by 90°, producing a wave propagating to the right that can trap ions whose mobilities exceed a tunable threshold. (c) Detailed view of the cylindrical electrodes of radius $a=1\mu \mathrm{m}$ used for the 2D model, whose four-electrode pattern is replicated indefinitely. Positions on electrode surfaces (points A and B) are specified by contact angles $\theta$ satisfying $0\le \theta \le \pi$.

Figure 2

Simulated separation of two ions with different mobilities and the same initial positions based on the 2D model, obtained by applying the potentials of Eq. (5) to the electrode pattern of Fig. 1c. Shown are snapshots taken over one period $\tau$ of the wave motion, at times $t/\tau =0/8$, $1/8$, etc. The color bar represents values of the electric potential $\varphi (x,y,t)$ relative to its amplitude ${\varphi }_0$. Dashed lines identify crests of the potential that travel along the channel at average speed $c=f\lambda$, where $f=1/\tau$ is the electrode oscillation frequency. A “trapped” high-mobility anion with responsiveness $R=-3$ [A, Eq. (3)] keeps pace with this crest, on average, traveling one wavelength during each period of the wave motion. A “localized” low-mobility anion with $R=-1$ (B) is unable to keep pace with the wave, oscillating instead about a single electrode. Cations with $R=3$ and $R=1$ follow the same trajectories as these anions if released at the same initial position after a phase shift of 180°. The ion velocity vector $\mathbf{v}$ has a time average $\overline{\mathbf{v}}=c\widehat{\mathrm{x}}$ for trapped cations and anions and $\overline{\mathbf{v}}=0$ for localized ions.

Figure 3

Ratio of the steady-state average axial ion velocity ${\overline{v}}_x$ to the wave speed $c$ vs the responsiveness $R$, from Eq. (4) for the 1D model (dashed trace), for the 2D model (solid trace), and for microfluidic experiments (data points accompanied by numerical values of the frequency $f$). In the experiments, ions are trapped for $f=1$, 1.5, and 2 Hz, partially trapped for $f=3$, 4, and 8 Hz, and localized for $f=16$ and 32 Hz. The 2D model captures the localization threshold and underestimates the trapping threshold.

Figure 4

(a) Details of ${\overline{v}}_x/c$ vs $R$ from the 2D model (heavy trace, A) and steady-state values of the electrode contact angle $\theta$ for simulations at fixed $R$ (B). Chaotic attractors feature scattered values of $\theta$ whose specific positions vary from simulation to simulation. Periodic attractors feature one or more isolated values. (b) Detail near $|R|=2$ showing a period-doubling cascade with ${\overline{v}}_x/c=1/5$ (I), narrow band chaos with periodic windows with ${\overline{v}}_x/c=1/5$ (II), and broadband chaos with incommensurate velocities (III). Inset: Steady-state trajectories wrapped to the interval $0\le x\le \lambda$, including a period-1 attractor for $|R|=1.94$, a narrow band chaotic attractor for $|R|=1.97$, and a broadband chaotic attractor with ${\overline{v}}_x/c=0.283$ for $|R|=2.00$.