## 3 Results
### Formation of Multicolored Periodic Bands under Cyclic Alternating Voltage in the RDR Setup  
Figure 8 shows the spatiotemporal evolution of a typical multicolored pattern formed in the sample tube under an applied cyclic alternating voltage: i.e., 4 V for 1 h and then 1 V for 4 h per cycle for a total of 20 cycles. In this cyclic voltage sequence, fewer reactant ions will be produced at 1 V and diffusion should exert a stronger effect on the residual ions to induce the formation of wide, diffuse precipitation patterns. In contrast, more reactant ions will be produced at 4 V and the stronger electric field should have a greater influence on the ions, thus producing narrow, thick precipitation bands, as suggested by the constant voltage experiments. After 15 h of cyclic voltage application (3 cycles), the wide region at the anode side became slightly green. However, no marked reddish-brown band was observed, which is unlike the observations at constant voltages of 2 and 4 V (see Figures 4 and 7). After 40 h (8 cycles), several short, faint, and reddish-brown bands were formed near the cathode (-10 mm from its surface). After 65 h (13 cycles), the number of thin reddish-brown bands increased. It should be noted that the formation of these periodic bands was stochastic with a probability of \(\sim\)50% (e.g., Figure 9a,b); in the other \(\sim\)50% of cases, an almost continuous band was formed (e.g., Figure 9c,d). Either way, during the formation of the reddish-brown band(s), the green region became gradually deeper in color. Interestingly, even after stopping the voltage application, occasionally the formation of periodic bands continued (compare the patterns obtained after 90 (18 cycles) and 135 h (35 h after 20 cycles) in Figure 8). Figure 8: Spatiotemporal evolution of a typical multicolored pattern formed at 25 \({}^{\circ}\)C in the sample tube under 20 cycles of 4 V (1 h)–1 V (4 h). The initial conditions are as follows: [[Fe(CN)\({}_{6}\)]\({}^{3-}\)] = 0.050 M and 2.0 mass% agarose. The elapsed time after initial voltage application is indicated at the right of each image. Bottom: the pattern after removing unneacted ions. The arrow at the bottom indicates the region where periodic bands were formed. A scale bar is also provided at the bottom of the image. Top: illustration of the charges of the electrodes and the movement directions of reactant ions induced by the electric field. These stochastically formed periodic bands showed the following common features. (1) The periodic bands were formed within a relatively narrow region (<5 mm). (2) The number of residual bands after removing unreacted ions was within 12 \(\pm\) 5. (3) The spaces between the adjacent bands were within 0.30 \(\pm\) 0.25 mm after removing unreacted ions (details will be discussed later). (4) The periodic bands maintained their positions during the observation time, similar to the unusual Liesegang bands previously found in a Co-Fe PBA system [30]. (5) After removing unreacted ions, several periodic bands became blue instead of reddish-brown, strongly suggesting the presence of Cu(OH)\({}_{2}\) precipitates (as already suggested by Figures 4 and 7). Figure 10a is an enlarged image of the multicolored periodic bands after removing unreacted ions (bottom image in Figure 8) together with 0.5 mm graduations for ease of comparison. These periodic bands are numbered from \(n\) = 1 (the cathode side) to 15 (the anode side). Their positions (\(X_{n}\)) were measured using the graduations with an accuracy of \(\pm\)0.05 mm to determine the band spacing \(d_{n}=X_{n+1}-X_{n}\), which is plotted in Figure 10b. It is well known that Liesegang bands tend to follow an empirical scaling law, the so-called spacing law, irrespective of the electrolyte pair and the geometry of the system: \(X_{n+1}/X_{n}\) = 1 + \(p\), where \(p\) > 0 for most systems [1; 2; 3; 4; 5].