Self-similar nested sequences on a chaotic attractor for traveling-wave electrophoresis

Oscillating electric potentials are applied to interdigitated arrays of cylindrical electrodes above and below a stationary conducting viscous fluid. The phases of these potentials are chosen to produce a longitudinal traveling wave that traps high-mobility ions and partially traps intermediate-mobility ions in periodic and narrowband chaotic attractors with average velocities that are commensurate with the wave speed. Stable periodic attractors have periods up to 101 times the wave period. Incommensurate broadband chaotic attractors are described by one-dimensional iterated contact-angle return maps, which feature self-similar nested sequences that converge geometrically at unstable trapped orbits. Sequences of singular angles and sequences of step transitions are characterized by distinct convergence factors. A criterion for allowed interelectrode orbits is developed. Experiments are suggested to evaluate the applicability of the theory to microfluidic separations.

Figure 1

(Color online) Channel geometry and periodic four-electrode pattern of wavelength $\lambda$ considered herein, a pattern replicated indefinitely in the $\pm x$ directions. Shown are two replicates of this pattern including electrode numbers $i=0,1,2,\dots ,8$ located at $x_i=i\lambda /4$. Electrodes are treated as impenetrable conducting half-cylinders of radius $a⪡\lambda$ ($a\approx \lambda$ in the figure for clarity) with axes perpendicular to the plane of the figure. Oscillating electric potentials ${\Phi }_0\left(t\right)$, ${\Phi }_1\left(t\right)$, ${\Phi }_2\left(t\right)$, and ${\Phi }_3\left(t\right)$ given by Eq. (1) are applied to the electrodes and are synchronized so that each electrode leads its neighbor to the right in phase by $\pi /2$, producing a wave propagating to the right that traps high-mobility ions. Positions on electrode surfaces (points A and B) are denoted by contact angles $\theta$ measured in radians and satisfying $0\le \theta \le \pi$, with $\theta =0$, $\pi /2$, and $\pi$ corresponding to the leftmost, middle, and rightmost contact points.

Figure 2

Steady-state phases from Eq. (11) (circles) for ions trapped by a 1D electric potential wave from Eq. (9) (solid trace) traveling through a stationary viscous fluid. Trapped cations with dimensionless mobilities $R\ge 1$ (numerical values accompanying cation phases) travel on the trailing sides of potential wells, with lower-mobility cations requiring stronger electric fields to overcome Stokes drag. Trapped anions with $R\le -1$ travel on the trailing sides of peaks. Ions with $\left|R\right|<1$ are not trapped by the wave, but execute periodic oscillations at a frequency given by Eq. (12).

Figure 3

(Color online) Periodic electrode pattern used in computing $\varphi \left(x,y,t\right)$ from Eq. (19), with electrodes held at potentials ${\Phi }_0$, ${\Phi }_1$, ${\Phi }_2$, and ${\Phi }_3$ located respectively at positions ${\mathbf{r}}_0^{\left(l,m\right)}$, ${\mathbf{r}}_1^{\left(l,m\right)}$, ${\mathbf{r}}_2^{\left(l,m\right)}$, and ${\mathbf{r}}_3^{\left(l,m\right)}$ from Eq. (18). Dashed lines separate replicates of this pattern identified by their values of $l$ and $m$. Replicates for $m=0$ correspond to the physical electrode pattern of the channel (Fig. 1). Replicates for $m\ne 0$ correspond to image charges needed to ensure $\partial \varphi /\partial y=0$ at the channel boundaries $y=0$ and $y=h$. The diagram applies to calculations of $\varphi \left(x,y,t\right)$ within the central computational area (shaded) defined by $0\le x<\lambda /2$ and $0\le y<h$, and for the truncations $L=M=2$. Contributions of the peripheral electrodes (also shaded) with $l=\pm L$ and $m=\pm M$ receive reduced weights according to Eq. (20) in order to satisfy the boundary conditions and to ensure symmetry and continuity, while the contributions of the remaining interior electrodes are included fully in the calculations.

Figure 4

(a) Ratio of the steady-state average axial ion velocity ${\overline{v}}_x$ to the wave speed $c$ vs the responsiveness $R$ of Eq. (3). Shown are results for the 1D model from Eq. (13) (trace A) and numerical results for the 2D model for electrode pattern wavelength $\lambda =80\mu \text{m}$, electrode radius $a=1\mu \text{m}$, and channel heights $h=10$, 15, 20, 30, and $40\mu \text{m}$ giving aspect ratios $\Gamma =\lambda /h=8$, 5.3, 4, 2.7, and 2 (traces B, C, D, E, and F). Traces for the 2D model feature commensurate velocity plateaus satisfying Eq. (22) (horizontal dashed traces and integer ratios). Frames (b) and (c) show small-velocity detail.

Figure 5

(Color online) Periodic attractors associated with velocity plateaus of Fig. 4, trace D, for aspect ratio $\Gamma =\lambda /h=4$. Each frame shows one wavelength of the electrode pattern of Fig. 1, with numbered electrodes of radius $a=\lambda /80$ shown to scale as small semicircles. Shown are steady-state trajectories of ions moving through a stationary viscous solution in response to the 2D traveling-wave potential of Eq. (19) after the decay of transients. Trajectories that disappear through the right side of the frames reappear on the left. (a) Two attractors of velocity 1 for $R=3.6$ involving successive identical 2:2 interelectrode orbits, one attractor involving the upper electrodes 0,2,4,… and the other involving the lower electrodes 1,3,5,…. (b) Attractor of velocity 1/5 for $R=2.5$ involving 1:5 orbits, with one lag cycle per orbit [$m=1$ in Eq. (22)] during which the ion lags one cycle behind the traveling wave. (c) Attractor of velocity 1/9 for $R=1.92$ involving 1:9 orbits with two lag cycles each $\left(m=2\right)$.

Figure 6

(Color online) Periodic attractors associated with velocity plateaus of Fig. 4, trace F, for aspect ratio $\Gamma =2$ after the manner of Fig. 5, with velocities 1/13, 1/53, and 1/101 for $R=3.94$ (a), 2.28 (b), and 1.80 (c), respectively. Ion trajectories beginning at arbitrary initial positions quickly relax to these highly stable attractors.

Figure 7

Details of ${\overline{v}}_x/c$ vs. $R$ for aspect ratio $\Gamma =4$ (heavy trace A, corresponding to trace D in Fig. 4), with localization and trapping thresholds $R_l=1.260$ and $R_t=3.554$. Also shown (b) are steady-state values of the electrode contact angle $\theta$ encountered for simulations at fixed $R$. Chaotic attractors (at $R=3.2$, for example) feature scattered random values of $\theta$ whose specific positions vary from simulation to simulation. Periodic attractors (at $R=2.4$, for example) feature one or more isolated values. Dashed lines identify commensurate velocity plateaus with ${\overline{v}}_x/c=1/5$, 1/9, and 1/13.

Figure 8

(a) Velocity and contact-angle detail for the velocity 1/5 plateau of Fig. 7. (b) Contact angle detail near $R=2.3$ showing four period-doubling cascades to chaos, the first for increasing $R$ near $R=2.28$, the second for decreasing $R$ near $R=2.31$, the third for increasing $R$ near $R=2.31$, and the fourth for decreasing $R$ near $R=2.34$. Whereas only narrowband chaos is found between the first two cascades, a window of broadband chaos occurs above $R=2.32$ between the third and fourth. The transition from periodicity to broadband chaos below $R=2.27$ is abrupt.

Figure 9

(Color online) (a) Velocity-1/5 periodic attractors for $R=2.35$ (solid) and 2.34 (dashed). These attractors are associated with the fourth period-doubling cascade in Fig. 8b and involve only 2:10 interelectrode orbits. (b) Velocity-1/5 narrowband chaotic attractor for $R=2.33$ involving only 2:10 orbits. (c) Velocity-0.6836 broadband chaotic attractor for $R=3.2$ involving a variety of orbits.

Figure 10

(Color online) (a) Contact angle return maps associated with the attractors of Figs. 9a, 9b including a period-1 attractor (circle, $R=2.35$), a period-2 attractor (squares, $R=2.34$), a period-4 attractor (triangles, $R=2.335$), and a narrowband chaotic attractor (solid trace, $R=2.33$). (b) Return map for the broadband chaotic attractor (solid trace, $R=3.2$) of Fig. 9c.

Figure 11

(a) Velocity (a) and contact-angle (b) detail for the velocity-1/9 plateau from Fig. 7, showing periodic attractors (at $R=1.86$, for example) and narrowband chaotic attractors (at $R=1.88$, for example) associated with commensurate velocities ${\overline{v}}_x/c=1/9$ and broadband attractors (at $R=1.94$, for example) with incommensurate velocities. (b) The left side of the plateau features a period 2 attractor that bifurcates, near $R=1.86$, to a period 4 attractor, and back again. (c) The right side of the plateau features a classic period-doubling cascade.

Figure 12

(Color online) Extended contact-angle return map for a broadband chaotic attractor with $R=3.2$ [Figs. 9c, 10b] showing the approach to a singular accumulation angle ${\theta }^{\infty }$ via an ascending sequence of singular angles ${\theta }^k$ (labeled), with $k=1,2,\dots$, $\infty$, and the approach to the same angle via a descending sequence of step-transition angles ${\theta }_k^{\infty }$. Frames (a)–(c) show increasing detail near ${\theta }^{\infty }$. Gray rectangles identify regions that are shown in greater detail elsewhere.

Figure 13

(Color online) Interelectrode orbits that contribute to the broadband chaotic attractor for $R=3.2$ [see Figs. 9c, 12]. The closed half-disk represents electrode 0, the electrode of origin. (a) Trapped orbits associated with a descending sequence of step transitions, showing three 2:2 orbits that depart from the electrode of origin at the contact angles $\theta /\pi =0.8$, 0.6, and 0.4, one 4:4 orbit for $\theta /\pi =0.3$, and one 6:6 orbit for $\theta /\pi =0.28$. (b) Untrapped $-1:3$, 0:4, 1:5, 2:6, and 3:7 orbits that participate in an ascending sequence of singular angles, with $\theta /\pi =0.05$, 0.1, 0.15, 0.2, and 0.22. (c) Two 1:5 orbits, for $\theta /\pi =0.0105$ and 0.071, flanking the $-1:3$ orbit at $\theta /\pi =0.05$ in Fig. 14a. These 1:5 orbits are distinct from each other and from the 1:5 orbit shown in (b).

Figure 14

(Color online) Small-angle details of Fig. 12a showing how the singular angle ${\theta }^1$ resolves into a doublet of closely spaced accumulation angles ${\theta }^{1-}$ and ${\theta }^{1+}$ that enclose ascending and descending sequences of singular angles. Shown also are sequences of step-transition angles ${\theta }_k^{1\pm }$ that approach these angles, for $k=1,2,3,\dots$, $\infty$.

Figure 15

(Color online) (a) Details of the ascending sequence of singular angles ${\theta }^{1+,k}$ that approach the accumulation angle ${\theta }^{1+}$ of Fig. 14c, with $k=1,2,3,\dots$, $\infty$. This sequence is similar to the sequence ${\theta }^k$ that approaches ${\theta }^{\infty }$, as shown in Fig. 12. (b) Fine detail near ${\theta }^{1+,1}$, which resolves into a doublet of closely spaced accumulation angles ${\theta }^{1+,1-}$ and ${\theta }^{1+,1+}$ that enclose ascending and descending sequences of singular angles similar to Fig. 14c.

Figure 16

(Color online). Log-linear plot of singular-angle margins $\Delta {\theta }^k={\theta }^{\infty }-{\theta }^{k+}$ [circles A, see Fig. 12a], doublet separations $\Delta {\theta }^{k^{\prime }}={\theta }^{k+}-{\theta }^{k-}$ [circles B, see Fig. 14b], and level-2 singular-angle margins $\Delta {\theta }^{k^{″}}={\theta }^{\infty }-{\theta }^{1+,k+}$ [circles C, see Fig. 15a] vs $k$, all of which vanish geometrically as $k\to \infty$ according to Eqs. (27, 28) (traces A, B, and C). Also shown are step-transition margins $\Delta {\theta }_k={\theta }_k^{\infty }-{\theta }^{\infty }$ [squares D, see Fig. 12b] and the level-2 step-transition margins $\Delta {\theta }_k^{\prime }={\theta }_k^{1+}-{\theta }^{1+}$ [squares E, see Figs. 14b, 14c] vs $k$, both of which vanish geometrically as $k\to \infty$ according to Eqs. (29, 30) (traces D and E).