The agarose required for electrophoresis (gel strength: 1800–2,300 g/cm³) was purchased from Kanto Chemical (Tokyo, Japan) and used without further purification. All aqueous solutions were prepared using deionized water that had been obtained from purifying tap water using a cartridge water purifier (G-10, Organo, Tokyo, Japan). In addition, Ti (≥99.5%) and Fe (≥99.5%) rods obtained from Nilaco (Tokyo, Japan) were used as the electrodes.

Sample tubes filled with agarose gel (called “gel holders” hereinafter) were prepared as further described. To check reproducibility, at least four gel holders were prepared under the same conditions for each experiment. Appropriate amounts of agarose powder were added to 30 ml deionized water at 25°C to form a 2.0 mass% agarose mixture. The resulting mixture was then stirred vigorously at 98°C for 30 s to produce a uniform agarose sol. Using a Pasteur pipette, the sol was transferred to 4 mm diameter and 45–75 mm long plastic straws, the bottoms of which were plugged with a Ti metal rod (used as the cathode) with a diameter of 4 mm and a length of ∼20 mm. Plastic straws were employed because they are simple to prepare and use, cheap, and support the easy introduction of viscous samples, such as organic sols (or organic colloidal solutions) [20]. The hot sol in the straw was left to cool to 25°C, and a solidified gel was formed, with the solidification time being approximately 10 min. The height of the resulting gel column in the gel holders (L) was 30–60 mm.

A 3 mm diameter and ∼30 mm long Fe metal rod was placed atop the prepared gel column as the anode. Because the Fe anode did not fit tightly inside the straw, it maintained contact with the gel surface even when the electric field had caused the gel column to contract. The cathode and anode were connected to a programmable power supply (PPS303, AS ONE, Osaka, Japan). During the application of constant or cyclic alternating voltages at 25°C (typical current ≈2 μA for 3 V and L = 50 mm), the precipitation patterns formed in the gel were monitored using a digital camera (Tough TG-6, Olympus, Tokyo, Japan).

After the voltage application experiments, some gel holders were re-plugged at both ends using styrene-resin stoppers covered with Parafilm for the Fe Kα intensity distribution measurements. These measurements were conducted using a homemade X-ray fluorescence (XRF) setup [21]. Briefly, the excitation source was Cu Kα₁ X-rays from an 18 kW generator (RU-300, Rigaku, Tokyo, Japan) operating at 40 kV and 60 mA, and the beam was focused to within 0.5 mm in the horizontal direction (resultantly, to have a linear shape) by a SiO₂ 10 1 ¯ 1 Johansson-type crystal monochromator. The gel holder to be measured was placed perpendicular to the linear X-ray beam on a computer-controlled X-Z stage. XRF signals from the gel holder were detected using a silicon PIN detector (XR-100CR, Amptek, Bedford, MA, United States), and the data were collected for 90 s using a multichannel analyzer (MCA8000A, Amptek). The distribution of the Fe Kα intensity from the gel holder in its axial direction was measured at 25°C by moving the holder in the same direction in 0.5 mm steps. The required collection time to obtain the Fe Kα intensity distribution was approximately 2 h. After subtracting the constant background from the Fe Kα peak, the integrated intensities over 6,214–6,647 eV were used to determine the Fe Kα distribution.

Subsequent to the Fe Kα distribution measurements, some gel columns were pulled out of the gel holder and cut into ∼1 mm-thick sections. These sections were allowed to dry for a few weeks in the ambient laboratory environment by being stuck on double-sided adhesive carbon tape and mounted on the aluminum stub of a scanning electron microscope (SU8220, Hitachi, Tokyo, Japan). Scanning electron microscopy (SEM) observation was conducted at 10.0 kV at a working distance of 15.2 mm. The XRF intensity map was also obtained using an XRF detector installed in SU8220 (EMAX X-Max^N, Oxford Instruments, Tokyo, Japan) and an analysis software (AZtec Live, Oxford Instruments).

Figure 2A shows the precipitation patterns formed in the gel columns with a length of 50 mm at 25°C under the applied constant voltages of 1, 2, 3, 4, and 5 V for 12 h. In these conditions, a continuous, broad, and diffusive precipitation band (not periodic or discrete) with a rust-brown color characteristic of Fe(OH)₃ was observed. As expected, applying higher voltages significantly lengthened the continuous band and deepened its rust-brown color, suggesting that such higher voltages promoted the generation of reactant ions and, consequently, the precipitation of Fe(OH)₃. Interestingly, the band positions also depended on the applied voltages; higher applied voltages tended to shift the precipitation band to the cathode side.

Figure 2B shows the spatiotemporal evolution of the continuous band formed under an applied constant voltage of 5 V for 12 h. After 5 h of the voltage application, a faint, broad rust-brown band formed at the position relatively near the cathode. This band expanded to the cathode side monotonously with time, deepening its rust-brown color without generating periodic bands. After 11 h, the edge of the band reached the cathode surface and its expansion stopped. Such a gradual extension toward the cathode with time was commonly observed under applied constant voltages.

During applications of voltages, the agarose gel gradually distorted with time, mainly at the anode side. Generally, at higher voltages (such as 5 V, Figure 2B) and shorter gel columns (such as 30 mm), such distortion occurred more quickly and largely. For instance, as shown in Figure 2B, the gel column began to shrink near the anode only 1 h after applying 5 V, and approximately half of the gel column was distorted after 12 h. The gel distortion prevented long-time observations even under applied lower voltages. Thus, the observation time in the current experiments, including experiments with cyclic alternating voltages, was limited up to 60 h. It should be noted that gel distortion or shrinkage under electric fields has been observed in various systems [22–24], and related theoretical approaches are being developed [25]; however, the details of the observed distortion are still unclear.

Figure 9 shows the Fe Kα intensity distribution of the gel column having periodic bands formed at 25°C under the applied cyclic alternating voltages: E _H = 3 V, E _L = 1 V, T _H = 1 h, T _L = 4 h, and N _C = 8; the horizontal axis indicates the distance from the cathode surface (X), where X > 0 is on the anode side, and the vertical axis is the relative Fe Kα intensity. As already shown in previous studies [26, 27], the Fe Kα distributions of hydrogels having Fe-related precipitates provide a good estimate of the Fe elemental distributions.

In Figure 9, six sharp Fe Kα distribution peaks can be observed, with their peak positions agreeing well with the positions of the periodic bands. This result strongly supported the assumption that rust-brown periodic bands comprised considerable amounts of Fe(OH)₃. Interestingly, the Fe Kα distribution was not observed near the cathode (X < 4 mm). This finding suggested that the Fe ions at the cathode side of the discrete band that emerged in each cycle (in Figure 9, the leftmost band at X = 5 mm, which emerged at N _C = 8), if any, were fully precipitated as Fe(OH)₃, promoting the growth of the band.

Figure 10 shows a typical SEM image of the rust-brown band formed in the gel columns with L = 50 mm at 25°C under the applied cyclic alternating voltages: E _H = 3 V, E _L = 1 V, T _H = 1 h, T _L = 4 h, and N _C = 8. The Fe Lα intensity map shown as pink dots in Figure 10, was overlapped for comparison. At the bottom of the SEM image, an XRF spectrum is provided. Here, all the measured XRF intensities (including Fe Lα intensities) have been added over the area of the SEM image. The estimated weight % values of the detected elements are tabulated at the right side of the spectrum.

Figure 10 indicates that several structures found in the SEM image mainly consist of C, O, and Fe atoms. Furthermore, it establishes that Fe atoms are uniformly distributed over the structures. The presence of the C and O atoms can be attributed to the agarose gel, while the Fe atoms come from the Fe compound(s), which is expected to be Fe(OH)₃ according to Eq. 4, generated by the RDR processes. The uniform distribution of Fe atoms, commonly observed for all the rust-brown bands obtained, strongly suggested that the precipitates formed in the current system essentially comprised not crystalline but gelatinous Fe(OH)₃. This finding is consistent with the well-known fact that Fe(OH)₃ is generally present as colloidal gel in aqueous conditions [28].

Sections 3.2.2 ‒ 3.2.4 demonstrated that the patterns of the periodic bands change largely by varying each parameter E _H, E _L, T _H, T _L, and L. The banding patterns also changed by simultaneously varying these parameters. Nevertheless, the basic features a) and b) described in Section 3.2.1 (that is, the immobile periodic bands at the cathode side, of which N _B values are basically given by N _C ‒ N _C1 + 1) generally held. The N _B values in the periodic bands obtained for arbitrary sets of E _H, E _L, T _H, T _L, L, and N _C can be estimated in the manner described below. Here, note that the N _B values can be decreased by the overlapping of the discrete bands owing to the inherent stochasticity of the band positions, as shown in Figure 4C; Figure 6A; Figure 7C; Figure 8. The decrease can also be achieved through the generation of the band close to the cathode surface, as suggested by Figure 8A. However, it should be noted that these effects were not considered in the following N _B estimation.

Figure 11 illustrates the linear correlation between A) N _C1 and E _H B) N _C1 and E _L C) N _C1 and T _H D) N _C1 and T _L, and E) N _C1 and L. Here, the results shown in Figures 3‒8 are considered. In this analysis, we considered the diffusive band that occasionally emerges in the N _C1 cycle and has more than 5-times broader width than the other periodic bands (see Figure 4A; Figure 4C; Figure 6A; Figure 6C; Figure 7A; and Figure 7C) as the “0.5 band”. Hence, some N _B and N _C1 values are half-integer values. Such half-integer N _B values, e.g., d.5, denote that there were d discrete bands plus one diffusive band (0.5) in that gel column. The half-integer N _C1 values, e.g., d’.5, denote that a diffusive band emerged in the d’ cycle, and subsequently, the first discrete band emerged in the d’+1 cycle.

Figure 11 indicates that the N _C1 values have good linear correlations with E _H, E _L, T _H, T _L, and L. Based on the results presented in Figure 11, the following linear relationships can be deduced: N C 1 = 4.5   –   0.5   E H , (5) N C 1 = 4   –   E L , (6) N C 1 = 4   –   T H , (7) N C 1 = 4   –   0.25   T L , (8) N C 1 = 0.05   L . (9)

Here, N _C1 values need to be positive, with these equations suggesting the upper limits of E _H, E _L, T _H, and T _L: E _H < 9 V, E _L < 4 V, T _H < 4 h, and T _L < 16 h. Furthermore, the slope values of these equations quantitatively suggest the extent of the influences of the E _H, E _L, T _H, and T _L values on the N _C1 values, and consequently, the N _B values: E _L = T _H (1) > E _H (0.5) > T _L (0.25). The relatively strong effects of E _L and T _H on N _B, as implied in Figures 5, 6, were quantitatively confirmed here.

By summarizing Eqs. 5‒9, Eq. 10 can be obtained as follows: N C 1 = 7.5   –   0.5   E H   –   E L   –   T H   –   0.25   T L × 0.02   L . (10)

Eq. 10 restricts the upper limits of E _H, E _L, T _H, and T _L more severely than Eqs. 5‒9 do. For instance, for E _H = 4 V, E _L = 2 V, and T _L = 6 h, T _H needs to be less than 2 h. By combining the relation N _B = N _C ‒ N _C1 + 1 with Eq. 10, Eq. 11 can be derived: N B = N C   –   7.5   –   0.5   E H   –   E L   –   T H   –   0.25   T L × 0.02   L + 1 . (11)

By Eq. 11, the N _B values for an arbitrary set of E _H, E _L, T _H, T _L, L, and N _C can be estimated within the limit imposed by Eq. 10. Here, the minus N _B values in Eq. 11 denote that no precipitation band emerges in the corresponding N _C cycle. For instance, regarding the condition of E _H = 3 V, E _L = 1 V, T _H = 1 h, T _L = 4 h, L = 50 mm, and N _C = 8, the corresponding N _C1 and N _B values are estimated to be three and six by Eqs 10, 11, respectively. These N _C1 and N _B values agree with the results shown in Figures 3A, D.

In the current RDR system, the band that emerges in the N _C1 cycle, called “the first-observed band” hereinafter, showed particularly diverse features, depending on the values of E _H, E _L, T _H, T _L, and L. More specifically, the first-observed band was qualitatively classified as either periodic, diffusive, or indistinguishable (as periodic or diffusive), or was nearly non-existent due to large thinness. This diversity of the first-observed band was included into the current N _B estimation as follows. In examining the linear correlations in Figure 11, the number of the first-observed band (N _B1) was set for 0, 0.5, and 1, respectively, when the band does not exist, is diffusive, or is periodic. By extension, the N _B1 values for the extremely faint first-observed band are considered as 0 (none) < N _B1 < 0.5 (diffusive). Similarly, the N _B1 values for the first-observed band difficult to be distinguished as either diffusive or periodic are considered as 0.5 (diffusive) < N _B1 < 1 (periodic). Thus, the diverse feature of the first-observed band is reflected as decimal values of N _B in Eq. 11. Based on this consideration, the decimal (non-half-integer) N _B value (d ₁. d ₂) can be interpreted as follows. The decimal N _B value with d ₂ < 5 denotes that, there are d ₁ periodic bands and possibly one faint band; the decimal N _B value with d ₂ > 5 denotes that, there are d ₁ periodic bands and one band difficult to be distinguished as either diffusive or periodic. Thus, the N _B values given by Eq. 11 can suggest not only the number of periodic bands but also the qualitative features of the first-observed band. Note that the suggestion by N _B for the features of the first-observed bands is not quantitative because a) the above discussion concerning N _B1 is qualitative and b) the criterion employed for the diffused band (more than 5-times broader bandwidths than those of other periodic bands) is temporary and lacks physical basis.

For instance, regarding the condition of E _H = 3 V, E _L = 1 V, T _H = 1 h, T _L = 4 h, and N _C = 5, the N _B values for L = 30, 40, 50, and 60 mm are estimated to be 4.2, 3.6, 3, and 2.4, respectively. Several values are decimal, suggesting that the features of some first-observed bands are undesirable in determining N _B and N _C1 values unambiguously. Although not presented here, our observation overall agreed with this suggestion. In contrast, regarding the condition of E _H = 4 V, E _L = 1 V, T _H = 1 h, T _L = 4 h, and N _C = 5, it was relatively easy to count the periodic bands, because the features of the first-observed bands were relatively manageable (Figure 8), as suggested by the integer/half-integer N _B values estimated by Eq. 11: 4.5, 4, 3.5, and 3 for L = 30, 40, 50, and 60 mm, respectively. Thus, the condition of E _H = 4 V was selected for examining the L-dependence of the periodic bands in Section 3.2.4.

As suggested by Eq. 11, the decimal values of E _H, E _L, T _H, T _L, and L generally resulted in the first-observed bands with features expressed by the decimal N _B values. For instance, regarding the condition of E _H = 3.2 V, E _L = 1.6 V, T _H = 1 h, T _L = 4 h, and N _C = 5, the first-observed band was difficult to be distinguished as either diffusive or periodic, as suggested by the decimal N _B value (3.7) estimated by Eq. 11. If the uncertainty of the N _B values within ±1 (owing to the features of the first-observed band) is less emphasized, there is no particular reason for ignoring the decimal values of E _H, E _L, T _H, T _L, and L in observation of the banding.

The simple RDR system proposed in this study can produce Liesegang-band-like, periodic precipitation bands under applied cyclic alternating voltages. Temporal oscillations, such as cyclic alternating voltages, occasionally mediate the formation of spatially extended structures such as periodic bands [10]. Therefore, the periodic banding shown in Figures 3‒8 might not seem a novel finding at first sight. However, this type of periodic banding has not been reported so far, and its formation mechanism is completely unknown. Many theoretical and experimental challenges regarding this type of banding remain across several scientific fields.

Currently, the reason why using cyclic alternating voltages can generate periodic bands that obey the relation N _B = N _C ‒ N _C1 + 1 is unknown. Investigating this question requires a deep understanding of the influences of thermal diffusion and electric field on the movement of the generated ions in agarose gel under applied cyclic alternating voltages. No mathematical model regarding this is currently available (such lack of theoretical model(s) has been occasionally part of a first experimental investigation of pattern formation by precipitation reactions [2, 6]). However, as a starting point toward a theoretical framework, a study on the electric-field effect on Liesegang banding [29] may be useful. Based on such theoretical efforts, it should be clarified what factors determine the optimal experimental condition to observe fine periodic bands less stochastically: E _H = 3 V, E _L = 1 V, T _H = 1 h, T _L = 4 h, and L = 50 mm.

Figure 9 shows the Fe Kα intensity distribution of the gel column having periodic bands after forming under an application of the cyclic alternating voltages. If in situ time-resolved Fe Kα measurements such as those previously conducted for a Liesegang banding system [27] could be conducted, the temporal variation of the Fe amounts in the precipitates under each voltage application condition will be deduced. Such information will be effective for examining the quantitative precipitation kinetics under both constant and cyclic alternating voltages, which may promote the theoretical efforts discussed above. Unfortunately, the timescale of the current T _H + T _L (typically 5 h) is comparable with that of the XRF measurements in our laboratory (∼2 h required). In this situation, the measurement time of XRF will strongly affect the periodic banding. This difficulty can be addressed by using synchrotron radiation (SR) because the timescale of the XRF measurements using SR is approximately 10 min, thereby resulting in plans toward conducting future SR experiments.

Time-resolved measurements of UV-visible (UV-Vis) absorption spectra are also interesting because the spectra can distinguish between Fe(II) and Fe(III) compounds. By combining the Fe Kα distributions, the emergence timing and temporal growth of the newly formed precipitation band will be more quantitatively studied, making a connection with the band characters, such as diffused, discrete, and intermediate. To obtain UV-Vis distributions in the gel column, a line-focusing UV-Vis source and a customized setup for moving the gel holder are required. Such experimental setup is currently in the design process.

The gel distortion at the anode side as shown in Figures 2, 6C should be further investigated to clarify its influence on periodic banding. Detailed electrochemical measurements at the anode side may help solve this issue. To conduct such electrochemical experiments, we are currently embedding a picoammeter and a reference electrode into the RDR setups. Further, the importance of pH for hydrogel distortion has been suggested in some studies [30, 31], and hence, in situ pH measurements over the gel column during the application of cyclic alternating voltages are intriguing. Hence, some pH probes installable into the current RDR system are being considered.

The generated precipitates did not consist of crystallites (such as Cu‒Fe PBA systems [16]) but gel, as suggested by Figure 10. Determining whether the observed periodic banding depends on the gelatinous nature of the reaction products is interesting. For this purpose, the generated rust-brown precipitates (Fe(OH)₃ gel) need to be more comprehensively characterized. We are currently planning to investigate the local structure around the Fe atoms in the gelatinous precipitates by measuring X-ray absorption fine structure (XAFS) at SR facilities [26]. Furthermore, the general importance of dissipation for self-assembled structures has recently been demonstrated [9]. Hence, concurrently with XAFS measurements, the energy and matter dissipation inherent to gel formation should be theoretically examined for the current RDR system.

A previous study [17] indicated that the substitution of electrodes in RDR system causes drastic changes in precipitation patterns. By substituting the Co, Ni, and Zn anodes for an Fe anode, the RDR patterning of Co(OH)₂, Ni(OH)₂, and Zn(OH)₂ can be examined using the current setup. The solubility-product constants of Co(OH)₂, Ni(OH)₂, and Zn(OH)₂ with values 5.92 × 10 ^–15, 5.48 × 10 ^–16, and 3 × 10 ^–17, respectively, are comparable with that of Fe(OH)₂ (4.87 × 10 ^–17) and considerably higher than that of Fe(OH)₃ (2.79 × 10 ^–39) [19]. Hence, such experiments may be helpful in elucidating the effects of low solubility of Fe(OH)₃ precipitates on the periodic banding under the RDR processes; preliminarily experiments are in progress.