## 4 Discussion  
The position-dependent size and shape of crystallites formed in the proposed RDR setup (Figure 13) suggest the potential applications of the observed periodic banding phenomenon for selective preparation of PBA, as mentioned in the introduction. Unfortunately, the current periodic banding only occurred stochastically (having a probability of \(\sim\)50%), and some byproducts were also present. To establish RDR banding systems for practical applications in material science and engineering, better control over the periodic bands is required, especially concerning (1) the amounts, sizes, and shapes of crystallites, (2) the amounts of formed byproducts, (3) the spacing between bands, and (4) reproducibility. First, achieving the above objectives requires a deeper understanding of the observed periodic banding, in which the \(d_{\mathrm{n}}\) values are randomly distributed around an average (Figure 10). Currently, no mathematical model could explain such precipitation banding. As a starting point towards a theoretical framework, the models of Liesegang bands may be useful. Modern theoretical models for Liesegang banding can be approximately classified into pre- or post-nucleation models, depending on the sequence of elementary events [1, 2, 3, 5, 33, 34, 35, 36]. Pre-nucleation models assume that the precipitation bands are formed as a result of repeated cycles of the supersaturation, nucleation, and depletion of the reaction products. In these models, the crystallites are formed by rapid nucleation and crystallization triggered by supersaturation, and hence they should be relatively small in size and irregular in shape [23]. In contrast, post-nucleation models assume the competitive growth of small particles of reaction products (including crystallites). According to these models, the crystallites in Liesegang bands must be relatively large in size because most of them have undergone Ostwald ripening [23]. Overall, the sizes and shapes of the crystallites in Liesegang bands can be used to identify the dominant formation processes: small and irregular crystallites are likely formed according to the pre-nucleation models, whereas relatively large crystallites indicate the dominance of the post-nucleation processes. Although Liesegang banding (via RD processes) and the currently studied banding (via RDR processes) have different mechanisms, the correlation between the dominant processes in precipitation banding and the sizes/shapes of the resultant crystallites should remain valid. The SEM image in Figure 13b is more consistent with pre-nucleation models, i.e., rapid nucleation and crystallization triggered by supersaturation were dominant in forming the main bands. Nevertheless, this suggestion is qualitative and tentative, and more quantitative and comprehensive models are required. Second, the optimal experimental conditions to produce well-controlled periodic bands should be further explored. Conventional precipitation banding of PBA in gels (using the RD setup in Figure 1a) is known to be dependent on the experimental conditions, including the gel type [29], gel density [29], and presence of magnetic fields [37]. The precipitation patterns formed using the proposed RDR setup should be examined while varying these factors (as well as the initial concentration of [Fe(CN)\({}_{6}\)]\({}^{3-}\) and the gel length). The optimal external potential, including the voltage, period, and number of cycles for cyclic alternating voltage application, should also be explored further. Additionally, variation in the sizes, materials, and shapes of the electrodes may also have interesting effects on the banding. Third, the reaction products formed in the periodic bands, the blue region in the gel, and the anode and cathode regions should be more comprehensively characterized. For instance, the defect concentration in the cubic (Cu-Fe PBA) crystallites formed in these regions should be measured quantitatively and separately.