Agate-like stripe pattern formation via cyclic-voltage-driven reaction-transport -reaction process

The Liesegang phenomenon, conventionally used to explain geological banding, does not account for key features of natural agate, such as irregular band spacing and horizontal onyx structures, leaving its formation mechanism a long-standing enigma. This study demonstrates a two-dimensional (2D) periodic precipitation patterning system governed by a cyclic-voltage-driven reaction–transport–reaction (RTR) process to replicate and elucidate these complex architectures. By cyclically alternating between high and low potentials across diverse electrode configurations in agarose hydrogels, we tracked the spatiotemporal evolution of metal hydroxide precipitates. Systematically adjusting the configuration, number, and metal species (Fe and Cu) of the anodes enabled us to engineer a range of unique 2D striped morphologies. The generated patterns resemble natural agate and onyx, structures that were previously difficult to replicate. Our findings establish the 2D-RTR process as a powerful physicochemical framework complementary to the Liesegang phenomenon, providing key insights into geological morphogenesis driven by intermittent hydrothermal influxes.

The pattern formation of agate remains incompletely understood in modern earth sciences and mineralogy and has been a “continuing enigma” for over a century [1]. The conventionally prevailing hypothesis suggests that hydrothermal fluids infiltrate silica gel within rock cavities, generating characteristic banded structures via the Liesegang phenomenon [2–5]. Recent studies, however, have demonstrated that agate formation involves multiple distinct mechanisms beyond the Liesegang phenomenon [1, 6], such as textural self-organization [7] and intermittent hydrothermal influx [8]. While the Liesegang phenomenon is crucial for band formation [3], it does not adequately account for several features commonly observed in natural agates, including irregular band spacing [9], variation in crystal structures between layers [10], and the presence of horizontal bands induced by gravity (onyx structures) [11]. Given these limitations, investigating periodic precipitation phenomena in gels that deviate from the classical Liesegang model provides an effective approach for elucidating the underlying mechanisms of agate-pattern generation.

Recently, we discovered a new class of periodic precipitation banding systems characterized by the formation of metal hydroxide bands within a gel column, governed by the cyclic switching between a high potential (E H) and a low potential (E L) for intervals of T H and T L, respectively [12, 13]. Interfacial electrochemical reactions at the anode (M → M n + + n e −, where M is a metal) and cathode (n H 2 O + n e − → n OH − + (n /2)H 2) generate reactant ions that migrate toward their respective counter electrodes, subsequently reacting (M n + + n OH − → M(OH) n) to yield precipitates within the gel matrix. Because this system integrates ionic transport with two distinct reaction regimes—electrochemical and precipitation—we designated it a reaction–transport–reaction (RTR) process [14]. Scanning electron microscopy–energy dispersive X-ray spectroscopy (SEM–EDX) analyses have revealed that the products of the one-dimensional RTR systems consist of a gel-like matrix composed of M and O [12, 13]. Since metal hydroxides formed in aqueous solutions at ambient temperature typically exhibit gel-like morphology, these findings strongly indicate that the RTR products are indeed M(OH) n. The chemical identity of the precipitate is fundamentally independent of the spatial dimensionality of the framework; hence, this characterization can be directly extended to higher-dimensional domains.

In typical Liesegang experiments [2, 4, 5], the generated patterns vary significantly with the spatial dimensions of the experimental setup, forming periodic bands in one dimension [15], rings in two dimensions [16], or shells in three dimensions [17]. Such dimensional dependence of the Liesegang phenomenon has not only enhanced the understanding of its underlying mechanisms [4, 5] but also extended its potential applications [18]. Inspired by these precedents, the present study demonstrates how the RTR process can generate two-dimensional precipitation patterns that closely mimic agate architectures, documenting several characteristic morphologies observed in this process.

Materials

Round bars of Fe (99.5%) and Cu (99.9%) were purchased from Nilaco Corporation (Tokyo, Japan) for use as electrodes. Analytical reagent-grade NaNO 3, used as a supporting electrolyte, and agarose (gel strength: 1800–2300 g cm −2) were obtained from Fujifilm Wako Pure Chemical Corporation (Osaka, Japan) and Kanto Chemical Co., Inc. (Tokyo, Japan), respectively. All chemicals were used as received without further purification. Aqueous solutions were prepared using deionized water purified through a cartridge water purification system (G-10, Organo, Tokyo, Japan).

Sample preparation

Perforated plastic cases (29 mm width × 60 mm length × 15 mm height) were utilized to cast the agarose gel containing NaNO 3 as a supporting electrolyte. Initially, NaNO 3 powder was dissolved in deionized water at 298 K to prepare aqueous solutions (30 mL) with concentrations (C se) ranging from 0.005 to 0.010 mol/L. Subsequently, 2.0 wt% agarose was introduced, and the mixture was vigorously stirred at 373 K for 45 s to ensure the formation of a homogeneous NaNO 3 –agarose sol. The resulting hot sol was transferred using a Pasteur pipette into the sample cases, where the two bottom holes had been previously plugged with copper rods (4 mm diameter × 10 mm length) to serve as cathodes. The sol-filled cases were allowed to cool to 298 K, resulting in a solidified gel (29 mm width × 60 mm length × 10 mm height). Following solidification, another metal rod (3 mm diameter × 22 mm length) was inserted into the two top holes of the case to serve as the anode. To ensure reproducibility, duplicate samples prepared under identical conditions were used in each experimental run (Figure 1A), and each run was repeated at least twice.

Two-dimensional RTR patterning

To achieve two-dimensional RTR patterning, voltage was applied between the anode(s) and cathode(s) using programmable power supplies (PPS303, AS ONE, Osaka, Japan); a schematic illustration of this configuration is depicted in Figure 1B. The applied voltage was cyclically switched between a higher voltage E H for a duration of T H and a lower voltage E L for a duration of T L for N C cycles (Figure 1C). The experimental runs were performed under the following specific parameters: E H = 3–4 V, E L = 2 V, T H = 1 h, T L = 2 h, and N C = 14–16 (see Table 1 for details). All patterning processes were carried out at a constant temperature of 298 K. During the process, the in situ evolution of the precipitation patterns was monitored using a digital camera (TG-7, OM Digital Solutions, Tokyo, Japan).

Following the removal of the voltage, the final hydrogel samples were placed on a white, backlit background and photographed under LED illumination. A physical ruler with 1-mm graduations was included in each photograph to serve as a scale reference. The captured images were subsequently analyzed using technical graphics software (Canvas X; Canvas GFX, Inc., Boston, MA, USA) to determine the structural parameters of the patterns with a measurement uncertainty of ±0.01 mm. Specifically, the radii of curvature (r) of the deflected bands, the spatial distances between adjacent bands (d 1 and d 2), and the widths (w) of the individual bands were quantified.

Diverging stripes

The experimental setup illustrated in Figure 2A generated diverging stripe patterns characterized by overlapping precipitation bands with varying degrees of curvature (Figure 2B). Although such overlapping, diverging bands with distinct curvatures are not reported in conventional Liesegang phenomena, they frequently appear in natural agates and other minerals. Figure 2C exemplifies this structure in banded calcite (reproduced from a specimen photograph [19]).

A comprehensive study of one-dimensional RTR systems [13] established that intermittent ion supply from electrodes is a prerequisite for the formation of periodic precipitation bands during the RTR process. Furthermore, the setup in Figure 2A features a small electrode-to-gel contact area (electrode diameter of 3–4 mm relative to a gel width of 29 mm), and the electrode configuration is asymmetric with respect to the central axis of the gel (i.e., the interelectrode axis is off-center). These findings suggest that the intermittent supply of reactant ions from a localized, asymmetrically positioned ion source contributes to the formation of these diverging stripe patterns.

The divergence of the precipitation pattern in Figure 2B exhibits a distinctive geometric feature. While the precipitation bands near the interelectrode axis (left side of the figure) are aligned almost in parallel, similar to those observed in one-dimensional systems [12, 13], bands distant from the interelectrode axis—particularly near the anode—curve significantly toward the anode. Figure 2D demonstrates that this curvature closely correlates with the growth dynamics of the stripe pattern. The first precipitation band initially emerged perpendicular to the interelectrode axis during the T L interval at N C = 2, then propagated rightward with pronounced curvature until the end of the T L interval at N C = 3. Upon completion of the E H application at N C = 4, a second precipitation band formed on the cathodic side of the first band, and the overall color intensity of the bands increased during the subsequent E L application. At N C = 5, a third band appeared on the cathodic side of the second band. This sequential addition of precipitation bands with each cycle remained consistent from N C = 6 onward. The sequential, cycle-dependent expansion toward the cathodic side aligns with results obtained in one-dimensional RTR systems [12, 13]. Notably, the curvature of the band generated at N C = 6 was significantly reduced. As the cycles progressed, the band curvature decreased further; beyond N C = 9, almost-parallel bands formed sequentially on the cathodic side, resembling the one-dimensional systems [12, 13].

Distorted stripes

An experimental setup featuring an additional anode positioned in the direction of the deflected precipitation pattern in Figure 2A (specifically, the upper-right region of Figure 2B) is illustrated in Figure 3A. This configuration produced “distorted stripes,” consisting of overlapping precipitation bands that were convexly warped toward the anodes, with varying degrees of distortion (Figure 3B). Although these striped patterns—similar to the “diverging stripes” (Figure 2B)—are unprecedented in conventional Liesegang phenomena, they frequently occur in natural agates, as exemplified by the blue lace agate shown in Figure 3C (reproduced from a specimen photograph [20]).

Figure 3D illustrates the growth process of these distorted stripes. The first precipitation band, which exhibited a significantly convex distortion, fully emerged at the end of the T H interval at N C = 3. Interestingly, this distorted band formed nearly perpendicular to the two interelectrode axes connecting the cathode to the dual anodes. Consistent with the diverging pattern observed in Figure 2B, the completion of the E H application at N C = 4 induced the formation of a second precipitation band on the cathodic side of the first band, which subsequently darkened during the T L interval. At N C = 5, a third band appeared on the cathodic side of the second. For N C ≥ 6, the convex distortion progressively decreased, while the pattern maintained a steady trend where each increment in N C added one band toward the cathode. Consequently, the overall growth dynamics closely mirrored those shown in Figure 2B, except for the deflection direction. These findings indicate that the configuration and number of electrodes (ion sources) critically determine the morphology of precipitation patterns generated by the RTR process. By contrast, their influence on the N C dependence of the band count is relatively minor.

Arching stripes

Incorporating an additional electrode into the configuration shown in Figure 3A and reversing the electrode polarity yielded a novel setup (Figure 4A). This arrangement produced arch-shaped stripes (“arching stripes”) oriented nearly perpendicular to the three interelectrode axes connecting the single anode and three cathodes (Figure 4B). Due to their unique geometric beauty, natural agates featuring stripes of this morphology are often highly valued as gemstones; a representative example of such a black-banded agate is shown in Figure 4C (reproduced from a specimen photograph [21]).

The growth dynamics of the arching stripes are depicted in Figure 4D. Expanding the electrode count to four significantly modified both the pattern morphology and the N C dependence of the band count. The shorter interelectrode distances (and consequently higher local electric fields) in Figure 4A compared to the setups in Figures 2A, 3A led to the early formation of a narrow precipitation band between the lower-side electrodes by the T L interval at N C = 1. By the end of the N C = 2 cycle, a well-defined arch-shaped band developed nearly perpendicular to the three interelectrode axes. For N C ≥ 3, the bands broadened toward the three cathodes but did not form discrete periodic bands in that region. In the rightmost domain, however, periodic bands progressing toward the cathode (upper) side in accordance with N C were observed from N C = 6 onward. Furthermore, from N C = 7, a faint, wide precipitation zone emerged, propagating upward and to the right from the interior of the arch-shaped band. These results in Figure 4 indicate that the formation of highly warped structures, such as arching stripes, requires a relatively higher density of pulsating ion sources.

Meeting stripes at a boundary

In the setup shown in Figure 1B, varying the types of parallel anode metals enables multicolor precipitation patterns composed of multiple metal hydroxides. As an illustrative example, Figure 5A depicts a configuration in which iron and copper anodes are positioned on the left and right sides, respectively. This setup produced multicolored periodic precipitation bands of reddish-brown (iron hydroxide) and blue (copper hydroxide, which appears greenish in Figure 5B due to the backlighting illumination) that aligned approximately in parallel while curving downward beneath their respective anodes, ultimately meeting at a distorted interface (Figure 5B). Such multicolored stripe structures are also frequently observed in natural agates, as exemplified by the zebra agate shown in Figure 5C (reproduced from a specimen photograph [22]).

Figure 5D illustrates the growth process of the multicolored stripes meeting at the interface (referred to as the “meeting stripe”). Until the end of N C = 2, no notable precipitation bands were observed beneath the iron anode. In contrast, beneath the copper anode, a thin blue precipitation band formed immediately after initiating E H application at N C = 2 and extended toward the iron anode (the left side) throughout the N C = 2 cycle. Upon starting E H application at N C = 3, a reddish-brown precipitate characteristic of iron hydroxide emerged near the tip of the advancing blue precipitate. For N C > 4, the respective periodic precipitation bands developed toward the cathode while maintaining contact at their boundary. Concurrently, a nearly uniform, faint precipitation region—particularly on the iron anode side—began to expand toward the copper anode, sharpening the segregation between the copper- and iron-hydroxide formation zones. The N C dependence of the copper hydroxide band formation closely corresponded with Figure 2D; fundamentally, one band formed per N C toward the cathode with progressively decreasing curvature. Conversely, unlike in one-dimensional systems [12], no distinct N C dependence was observed for the iron hydroxide precipitation bands. These results in Figure 5 suggest that when multiple pulsating ion sources supply distinct reactant ions, (i) the resulting precipitates tend to form individual periodic precipitation bands in regions adjacent to their respective ion sources rather than intermixing; however, (ii) the N C dependence of the band formation may be lost under certain conditions.

Reproducibility of precipitation patterns and characteristic parameters

Figure 6 compares the precipitation patterns of duplicate samples for each experimental mode. These patterns reflect the complex and subtle spatiotemporal distribution of ions within the gel; thus, perfect morphological identity was not achieved even under identical experimental conditions. However, the qualitative features of the patterns were consistently preserved across all modes. In the “Diverging” bands (Figure 6A) and “Distorted” bands (Figure 6B), minor differences in the curvature of the deflected bands and the degree of overlap were observed, yet both replicates consistently exhibited bending in the upper band (anode side). They share identical deflection orientations—convex downward for “Diverging” and convex upward for “Distorted” bands—and display a consistent trend where the curvature diminishes toward the lower band (cathode side). For the “Arching” bands in Figure 6C, despite variations in curvature, band thickness, and the number of periodic precipitation bands (visible in the upper right), the qualitative morphology of the pattern—characterized by a branched, arched structure—remained consistent. Similarly, in the “Meeting” bands (Figure 6D), although the number, spacing, and thickness of the periodic bands, along with the curvature of the deflected bands, showed differences, distinct commonalities emerged: reddish-brown precipitation bands developed on the left and blue ones (appearing greenish due to backlighting) on the right, with the reddish-brown bands progressively shortening relative to the greenish bands toward the bottom. Despite such phenotypic variability, the qualitative reproducibility of these precipitation patterns enables the quantification of characteristic structural parameters for each mode (as indicated on the right side of Figure 6). Table 2 summarizes these structural parameters for each pattern.

As summarized in Table 2, no significant differences were found in the interband distance near the interelectrode axis (d 2) or the band width (w) between the “Diverging” and “Distorted” modes. These values were also consistent, within experimental error, with the results previously reported for a one-dimensional system using a Cu anode (d 2 = 1.20 ± 0.40 mm, w = 0.33 ± 0.17 mm) [13]. For the “Meeting” pattern, the d 2 value expanded to roughly twice that of the aforementioned patterns, while the w value remained unchanged. Conversely, the “Arching” pattern exhibited a roughly tenfold increase in the w value in specific areas. The temporal evolution of the precipitation patterns in Figure 4 suggests that the localized widening of the “Arching” bands is partially attributable to the coalescence or overlapping of periodic precipitation bands.

In the “Diverging” and “Distorted” patterns, the distance (d 1) between the curved bands formed away from the electrode axis was approximately 2.6 and 1.4 times greater than d 2, respectively. Furthermore, the radius of curvature decreased in the order of “Diverging,” “Distorted,” and “Arching” bands.

Although the structural parameters in Table 2 exhibit relatively large standard deviations, the observed trends should prove useful for quantitative comparisons across different experiments and for validating numerical simulation models.

Mechanism of precipitate formation via two-dimensional RTR patterning

By synthesizing the empirical findings of the present two-dimensional system with established insights from one-dimensional configurations [13], we establish a qualitative model to unravel two-dimensional precipitate patterning. Although numerical simulations for one-dimensional architectures have been achieved by extending the Nernst–Planck equation [13], two-dimensional configurations present a critical challenge due to the spatially non-uniform potential gradient. In contrast, the gradient in one-dimensional systems is treated as uniform throughout the bulk medium, except near the electrodes. Due to this multidimensional complexity, computational modeling of the current system is still under development. Consequently, we offer a conceptual interpretation of the underlying mechanism at this stage, as schematically illustrated in Figure 7.

To delineate the mechanism, we first summarize the governing principles of RTR systems using a simplified single anode–cathode pair, which are applicable to both dimensions. When the applied voltage exceeds a threshold (E thre), metal ions (M n +) are assumed to be generated at a constant surface concentration (C M) at the anode, with generation ceasing below this threshold. Likewise, OH − ions are generated at a surface concentration of nC M at the cathode when the voltage exceeds E thre. Accordingly, cyclic voltage switching between a high potential (E H > E thre) and a low potential (E L < E thre) drives the intermittent production of M n + and OH − ions, which are generated during the T H interval and halt during the T L interval.

Upon migrating to the counter electrodes, M n + ions are assumed to be consumed via the formation of M(OH) n and/or M at the cathode surface, while OH − ions are similarly removed by forming M(OH) n at the anode. Although the accumulation of M(OH) n on the electrode surfaces could potentially diminish the effective electrochemical area, this effect was neglected for simplicity.

The emergence of a precipitation band requires the local concentration product at the intersection of the counter-directional fluxes to exceed the solubility product (K sp) of M(OH) n. Below this threshold, the opposing ionic fluxes simply bypass each other, resulting in unreacted dispersion within the gel matrix. Importantly, in aqueous media and hydrogels, the diffusion coefficient (D) and the subsequent ionic mobility—defined as nFD /(RT), where F is the Faraday constant, R is the gas constant, and T is the absolute temperature (298 K)—of OH − are inherently greater than those of metal ions due to the Grotthuss mechanism. This transport asymmetry is visually represented in Figure 7, where the spatial profile of the blue (OH −) flux is markedly broader than that of the dark yellow (M n +) flux.

The Grotthuss mechanism proceeds via the cooperative rearrangement of bonds within the hydrogen-bonded network of H 2 O molecules, specifically through the cleavage of an O–H bond in one molecule and the concomitant rapid formation of a new bond with an adjacent molecule [23, 24]. Within the RTR framework, when the M n + and OH − fluxes intersect, this rapid structural translocation enables OH − ions to penetrate the advancing M n + front, temporarily restricting instantaneous M(OH) n precipitation. Due to this tunneling-like transport, the reactant ions undergo incomplete precipitation within the reaction zone. Consequently, the unreacted M n + and OH − populations bypass the nascent precipitation band and continue migrating toward their respective counter electrodes during the T L interval.

Accounting for this tunneling-like transport rationalizes the periodic banding propagated toward the cathode—a phenomenon widely documented in one-dimensional architectures [13] and replicated in the “Diverging,” “Distorted,” and “Meeting” modes of two-dimensional configurations (Figures 2, 3, 5). Under E L conditions, since electrochemical ion generation is suspended, the local concentrations beyond the primary band remain insufficient to trigger additional M(OH) n nucleation within that specific cycle. Nevertheless, driven by their enhanced mobility via the Grotthuss mechanism, OH − ions migrate toward the anode, where they can undergo a surface reaction to form M(OH) n deposits. Because this interfacial reaction is heavily suppressed at this stage (E L < E thre), OH − ions accumulate in the vicinity of the anode. Upon transitioning to the T H interval of the subsequent cycle, the elevated potential accelerates this side reaction, leading to the rapid consumption of the accumulated OH − ions. Concurrently, E H application drives the re-emission of M n + and OH − ions from their respective electrodes. Owing to their superior transport velocity, most OH − ions from the preceding cycle reach the anode and are depleted before the completion of the subsequent T H interval. By contrast, the less mobile M n + ions generated during the previous cycle remain in transit within the gel bulk. These residual M n + ions subsequently encounter the newly emitted OH − flux, triggering the formation of a distinct new M(OH) n precipitation band shifted toward the cathode side. The sequential repetition of this synchronized cycle thus dictates the sustained propagation of periodic precipitation bands toward the cathode.

The primary governing factors of the one-dimensional RTR growth process—namely, (i) the suspension of ion generation during the T L interval and (ii) the Grotthuss-driven transport asymmetry enabling tunneling-like behavior—are inherently dimension-independent and thus applicable to two-dimensional domains. Nevertheless, two-dimensional configurations possess a distinct characteristic absent in their one-dimensional counterparts: a spatially non-uniform potential gradient. Because this spatial variation cannot be readily quantified at present, a rigorous quantitative evaluation of the ion and precipitate distributions lies beyond the scope of this work. To bypass this limitation, the subsequent discussion establishes a conceptual framework to interpret the diverse band morphologies documented in Figures 2–5. Specifically, as a first approximation, the spatial distribution of ions emitted along the interelectrode axis is modeled as a fan-shaped profile that expands under diffusion-controlled transport and contracts when dominated by the localized potential gradient.

Regarding the “Diverging” mode, Figure 7A shows the junction where two counter-directional ion flows intersect, initiating Cu(OH) 2 precipitation during the T H interval at N C = 2. The newly formed precipitation bands are delineated by dark red curves. This panel schematically illustrates the process in which OH − ions, owing to their superior diffusivity and mobility, encounter Cu 2+ ions on the anode side of the interelectrode axis, thereby establishing a localized Cu(OH) 2 precipitation zone. This conceptual model aligns well with the empirical observations documented in Figure 2.

Figure 7B schematically captures the initial phase of the T L interval at N C = 3 of the “Diverging” mode. Because electrochemical ion emission is suspended, both Cu 2+ and OH − ions undergo transport under a highly attenuated potential gradient while diffusing laterally perpendicular to the interelectrode axis. This spatial spreading, coupled with counter-directional transport, is significantly accelerated for OH − ions due to their higher diffusivity. Conversely, the less mobile Cu 2+ ions coalesce with residual ion populations from preceding cycles, inducing localized precipitation in the vicinity of the pre-existing band, which serves as a preferential nucleation template. In this region, the resulting precipitation zone exhibits an upward deflection, mirroring the geometric asymmetry in the ion diffusion profiles. In other words, the upward curvature documented in Figure 2 is rationalized by the retarded lateral expansion of the Cu 2+ flux relative to the OH − front.

Figure 7C schematically illustrates the ion distribution during the late T L interval at N C = 3. Beyond the upward-deflected precipitation zone—within which virtually all previous-cycle-derived Cu 2+ ions have been consumed—OH - ions penetrate the precipitates via a tunneling-like transport mechanism and accumulate near the anode, while Cu 2+ ions spread laterally. This leads to the formation of Cu 2+ and OH − depletion regions on the anode and cathode sides, respectively.

Figure 7D schematically illustrates the ion distribution during the T H interval at N C = 4. A fraction of the previous-cycle-derived Cu 2+ ions that bypassed the precipitation zone react with OH − ions newly released from the cathode, thereby establishing a subsequent precipitation band on the cathode side of the pre-existing one. Although the Cu 2+ ions emitted from the anode at N C = 4 have not yet reached the reaction front in this schematic, the precipitation band is anticipated to propagate rightward with an upward deflection upon the intersection of this ion flux with the OH − front, mirroring the mechanisms detailed in Figures 7B,C. Within this cycle, the reaction of OH − ions with the residual Cu 2+ ions from the previous cycle to form precipitates is expected to attenuate the degree of deflection relative to the previous cycle (N C = 3).

Figure 7D indicates that the ion distribution at this stage is remarkably complex. Such intricate profiles are expected to be highly susceptible to subtle variations in experimental parameters. This sensitivity likely underlies the observed variations in the radius of curvature (r) of the precipitation bands, as well as their mutual overlap and spacing (d 1), as exemplified in Figure 6A, even among samples prepared under identical experimental conditions.

With a further increase in the cycle number, reactant ions are expected to undergo diffusive accumulation within the peripheral region to the right, where electrodes are absent. This process gradually homogenizes the lateral ion distribution. Once this distribution becomes uniform, the deflection of the precipitation bands is mitigated, leading to the development of planar banded patterns reminiscent of those documented in one-dimensional RTR systems. Strikingly, the formation of such highly parallel banded patterns over extended cycles becomes clearly discernible in the regime of N C ≥ 8 in Figure 2D.

Furthermore, regarding the “Distorted,” “Arching,” and “Meeting” modes, which involve greater number of electrodes and more complex ion profiles, it is still possible to provide a rational qualitative explanation of their underlying mechanisms, albeit in a conceptual manner.

Figure 7E schematically illustrates the ion distribution during the T H interval at N C = 3 of the “Distorted” mode. The experimental configuration of this mode is distinguished from the “Diverging” mode by the inclusion of an additional Cu anode in the upper-right region. During this cycle, because the Cu 2+ flux from this upper-right anode has not yet intersected the counter-directional OH − front, the precipitation band develops exclusively near the left-side interelectrode axis. Consequently, a pronounced structural divergence between the “Distorted” and “Diverging” modes is unlikely to manifest at this stage, a prediction that aligns well with the experimental results (compare Figures 2, 3). In subsequent cycles, however, the Cu 2+ flux from the upper-right anode encounters the OH − front at a lower vertical position relative to the left-side Cu 2+ flux, thereby establishing a novel precipitation band absent in the “Diverging” mode. The precipitation band deflecting in the direction opposite to that of the “Diverging” mode—documented in Figure 3—presumably emerges as this nascent band propagates and coalesces with the band generated in the preceding cycle. At extended cycles, analogous to the “Diverging” mode, lateral ionic diffusion facilitates the formation of planar banded patterns, a phenomenon distinctly observed in the regime of N C ≥ 8 in Figure 3D.

Figure 7F schematically illustrates the ion distribution during the T H interval at N C = 1 of the “Arching” mode. Because the interelectrode distance between the underlying anode and cathode is small, the resulting potential gradient along this axis is relatively high, driving rapid ionic accumulation. Aligned with this expectation, a precipitation band extending toward the anode—approximately perpendicular to the lower electrode axis—develops at N C = 1 (Figure 4D). While a rigorous elucidation of the complex migration–precipitation dynamics involving OH − ions from the three cathodes and the opposing Cu 2+ flux remains challenging for N C > 1, qualitative insights can still be gleaned. Owing to the high mobility of OH − ions, their lateral distribution in the upper region (toward the two cathodes) likely approaches near-uniformity at an early stage. Consequently, the precipitation bands formed in this domain do not exhibit the pronounced deflection characteristic of the “Diverging” mode. This prediction aligns qualitatively with the empirical results in Figures 4, 6C.

Figure 7G schematically illustrates the ion distribution during the T H interval at N C = 2 of the “Meeting” mode. In its simplest terms, this mode can be conceptualized as a superposition of two independent “Diverging” modes. Analogous to Figure 7A, this simplified model predicts the formation of relatively short precipitation zones on the anode side of both electrode axes at N C = 2. However, within the empirical “Meeting” mode, no precipitation zone was detected on the left side (the Fe anode side) at N C = 2 (Figure 5). This result indicates that under these specific experimental conditions, the formation of Fe(OH) 3 precipitates is less favorable than that of Cu(OH) 2. In other words, Fe(OH) 3 precipitation is suppressed because the OH − ions required for its nucleation are preferentially consumed by the competitive Cu(OH) 2 precipitation process. The disappearance of the “cycle-dependent increase in the number of precipitate bands”—a signature trait of the one-dimensional Fe(OH) 3 RTR system [12]—under the “Meeting” mode configuration can likewise be ascribed to the depletion of OH − ions by the coexisting Cu 2+ flux.

Two-dimensional RTR system as a physicochemical model for studying stripe patterns of agate

Within contemporary mineral science [1, 6], agate banding originates from a synergistic interplay of mechanisms, including the episodic influx and subsequent accumulation of hydrothermal fluids from external reservoirs [8, 25], spontaneous fluctuations of crystal growth or interfacial self-organization [7, 9], and the Liesegang phenomenon. A comprehensive understanding of agate banding requires numerical simulations [9, 26, 27] and experimental reproduction and validation of “reaction–transport coupling”—the intricate interaction among diffusion, chemical reactions, and directional transport that underlie mineral morphogenesis. Nevertheless, an experimental platform capable of fully capturing the multifaceted formation of agate banding patterns remains underdeveloped. However, targeted physicochemical model systems that isolate individual aspects of this geological process do exist; platforms utilizing hydrogel matrices at ambient temperatures (hereafter the “hydrogel system”) are predominantly exploited. While the Liesegang phenomenon serves as a canonical paradigm, the proposed two-dimensional RTR patterning architecture offers a novel alternative. Table 3 delineates the comparative characteristics between the current two-dimensional RTR approach and the typical two-dimensional Liesegang phenomenon.

The physicochemical conditions of ambient hydrogel systems differ significantly from those of natural agate morphogenesis—specifically, a silica-rich medium under elevated temperatures and pressures driven by episodic hydrothermal fluid influx and efflux. Consequently, hydrogel-based approaches cannot fully capture the geologically realistic thermodynamics and material transformations underlying agate genesis. These low-temperature systems fail to replicate the solution convection and rapid phase transitions characteristic of hydrothermal regimes, nor can they trace the dynamic structural trajectories—encompassing dehydration and condensation—that dictate crystallization within silica gels. Furthermore, the temporal scale of hydrogel systems is limited to days or weeks, compared to the millennia required for natural agate accretion. Although modeling with a hydrogel system effectively provides a kinetically accelerated window into agate formation, such extreme temporal compression risks altering or oversimplifying the complex mechanisms that dictate natural banding patterns.

Despite these apparent limitations, hydrogel-based platforms offer a critical advantage: they enable the validation of universal traits in the underlying nonlinear dynamics governing pattern formation across disparate physicochemical regimes. Whether the matrix is a natural geological cavity or a synthetic hydrogel, precipitation architectures emerge fundamentally from the synergistic coupling between transport and precipitation reactions. Therefore, mathematical frameworks formulated for ambient hydrogel systems can be mapped onto high-temperature, high-pressure silica gel regimes by judiciously scaling transport and kinetic parameters. Furthermore, striking geometric analogies exist between the morphologies documented in both the Liesegang phenomenon and the current two-dimensional RTR configurations (Figures 2–5) and the bands characteristic of natural agate. This morphological convergence strongly indicates that many geometric hallmarks of precipitation patterns are invariant features, independent of external variables such as temperature, pressure, or the driving force of an electric field. Consequently, elucidating patterning mechanics within hydrogel systems provides a viable pathway to decode the fundamental geometric rules dictating agate morphogenesis.

Furthermore, hydrogel-based approaches also offer exceptional experimental ease and superior controllability. In contrast to hydrothermal or high-pressure simulations that require specialized instrumentation and preclude real-time in situ monitoring, ambient hydrogel systems allow precise tuning of ionic influx rates and source concentrations. This capability enables granular tracking of the elementary processes dictating pattern formation, serving as a simplified first-order approximation of the complex geological phenomenon.

Based on this rationale, hydrogel-based approaches should be conceptualized not as platforms engineered to replicate the precise geological synthesis of agate, but as abstract model systems designed to decode the fundamental mathematical and physical frameworks underlying pattern morphogenesis. As a specialized hydrogel-based architecture, the two-dimensional RTR system outlined in Table 3 facilitates intermittent, localized introduction of ionic fluxes from multiple positions within the gel network. This system readily establishes diverse striped precipitation zones on a spatial scale reminiscent of the canonical Liesegang phenomenon, utilizing a fundamentally analogous precipitation mechanism. Crucially, the properties of this two-dimensional RTR system complement the Liesegang phenomenon when evaluated as physicochemical models. While the Liesegang phenomenon statically generates concentric architectures (agate-like profiles) within a reactant-abundant environment driven by a concentration gradient, the two-dimensional RTR patterning dynamically constructs linear stripe profiles (onyx-like profiles) via the external introduction of pulsating ionic fluxes into a reactant-depleted medium.

In summary, the two-dimensional RTR system provides a compelling framework for studying agate banding. It preserves the high controllability characteristic of hydrogel platforms while enabling the simulation of intricate geological environments distinct from the Liesegang phenomenon. Specifically, the following investigative applications can be envisioned. In the genesis of natural agate, hydrothermal fluids frequently infiltrate the silica matrix not merely from a unidirectional front but through multiple surrounding structural gaps. Systems that facilitate simultaneous influx from multiple positions—such as the two-dimensional RTR architecture—can replicate these multi-directional infiltration environments to more faithfully mimic natural conditions. Additionally, since the discharge of hydrothermal fluids from subterranean reservoirs is inherently episodic, geological factors such as tectonic movements and seasonal variations heavily influence agate banding morphogenesis. Within the two-dimensional RTR setup, the temporal profile and flux rate of ionic introduction can be precisely modulated via voltage parameters, making it a highly effective platform to systematically unravel the impact of these environmental fluctuations.

The two-dimensional (2D) RTR patterning technique enables the controlled fabrication of intricate, periodic metal hydroxide precipitation profiles within agarose hydrogels, with multi-directional morphology dynamically modulated by the configuration and abundance of the electrodes. The emerging patterns closely resemble the morphological hallmarks of natural agate and onyx—architectures that resist replication through the canonical Liesegang phenomenon alone. Thus, as a physicochemical model platform highly complementary to the Liesegang phenomenon, this 2D RTR architecture warrants systematic exploration to decode the non-equilibrium mechanisms governing complex geological morphogenesis.