Modification of the Matalon–Packter Law for Self-Organized Periodic Precipitation Patterns by Incorporating Time-Dependent Diffusion Flux

Abstract

Spontaneous pattern formation is common in both inanimate and living systems. Although the Liesegang pattern (LP) is a well-studied chemical model for precipitation patterns, various recent LP systems based on artificial control could not be easily evaluated using classical tools. The Matalon–Packter (MP) law describes the effect of the initial electrolyte concentration, which governs the diffusion flux (F diff), on the spatial distribution of LP. Note that the classical MP law only considers F diff through the initial concentration of electrolytes, even though it should also depend on the volume of the reservoir used for the outer electrolyte because of the temporal change in the concentration therein due to diffusion. However, there has been no report on the relationship between the MP law, the reservoir volume, and F diff. Here, we experimentally demonstrated and evaluated the effect of the reservoir volume on LP periodicity according to the classical MP law. Numerical simulations revealed that the reservoir volume affects the temporal modulation of F diff. By expressing the MP law as a function of estimated F diff after a certain period of time, we provide a uniform description of the changes in periodicity for both small and large reservoir volumes. Such modification should make the MP law a more robust tool for studying LP systems.

Effect of the Polarity of Solvents on Periodic Precipitation: Formation of Hierarchical Revert Liesegang Patterns

“… 9 , 14 The regular-type LP generation can be observed in the CuCrO 4 system and many other precipitation systems. 9 , 14 − 20 However, when Pb 2+ is used instead of Cu 2+ as an outer electrolyte, exceptional LP can be formed, in which Δ x n decreases in contrast to the regular-type pattern, which is the so-called revert (invert) LP. 21 − 23 This difference was discussed from the perspective of the property of colloidal stability.…”

“…The periodicity of the typical LPs shows a general property, namely the distances of bands measured from the gel interface are the members of a geometric progression. This empirical rule is known as the spacing law 12 − 14 where n is the band number, x n and x n +1 , are the distance of the n th and ( n + 1)th bands measured from the interface between media containing M + and X – , and p is the spacing coefficient, which is constant at large values of n . Furthermore, the inter-band spacing (Δ x n ) deduced by the spacing law is given by …”

“…Generally, Δx n increases with increasing n, and this type of pattern is called regular LP. 9,14 The regular-type LP generation can be observed in the CuCrO 4 system and many other precipitation systems. 9,14−20 However, when Pb 2+ is used instead of Cu 2+ as an outer electrolyte, exceptional LP can be formed, in which Δx n decreases in contrast to the regular-type pattern, which is the so-called revert (invert) LP.…”

Tunability of Self-Organized Structures Based on Thermodynamic Flux

“…The formation of the periodic zone was apparent at [KI] above 0.1 M on the line profile. A further increase in [KI] moved the position of the periodic band farther away from the interface, which is a general feature of Liesegang pattern formation. − Two empirical rules, namely, a spacing law and the Matalon–Packter law, can be used to confirm that the observed periodic band may be classified as a Liesegang band. The spacing law implies that the distance, x n , from the interface to the n th band obeys the following relationship (eq ): where n and p are the band number and spacing coefficient, respectively.…”

Periodic Precipitation of Zeolitic Imidazolate Frameworks in a Gelled Medium

“…One can notice that the distance between the two consecutive bands in both systems is greater once the concentration of the cations increases (Figure ). This is the manifestation of the Matalon–Packter law of the Liesegang system, namely, the spacing coefficient (defined as the ratio of distances of two consecutive bands measured from the gel–liquid interface) is linearly proportional to the concentration of the inner electrolyte (in this setup, the concentration of cations). − …”