Periodic Precipitation in a Confined Liquid Layer

Pattern formation is a ubiquitous phenomenon in animate and inanimate systems generated by mass transport and reaction of chemical species. The Liesegang phenomenon is a self-organized periodic precipitation pattern always studied in porous media such as hydrogels and aerogels for over a century. The primary consideration of applying the porous media is to prevent the disintegration of the precipitation structures due to the sedimentation of the precipitate and induced fluid flow. Here, we show that the periodic precipitation patterns can be engineered using a Hele–Shaw cell in a confined liquid phase, restricting hydrodynamic instability. The patterns generated in several precipitation reaction systems exhibit spatiotemporal properties consistent with patterns obtained in solid hydrogels. Furthermore, analysis considering the Rayleigh–Darcy number emphasizes the crucial role of fluidity in generating periodic precipitation structures in a thin liquid film. This exploration promises breakthroughs at the intersection of fundamental understanding and practical applications.

The paper by Itatani et al. reports an experiment study of Liesegang patterns in Hele-Shaw cells.HS cells are frequently used for analysis of viscous fingering and its relevance to oil exploration and via Darcy's law flow in porous media.Liesegang patterns (LPs) are usually produced in gels, which defines the main novelty of this paper.The authors avoid undesired fluid motion not by polymers but rather by vertical confinement to a very thin solution layer (typically 70 um) bound by two glass plates.The importance of this study lies at least partially in the possibility of characterizing the role of the polymer/gel on the observed patterns, ultimately providing a better understanding of this classic and interesting phenomenon.I think the study is of high quality and originality.It is also wellwritten and of interest to physical chemists.I recommend the paper for publication in JPC Letters with minor revisions.I specifically ask the authors to consider the following questions and comments.
1) Title: The current title made me think of small pores and similar three-dimensional spatial confinement.Perhaps (and I do not insist) change "phase" to "layer".
2) P. 18: What is meant by "Otherwise (d = 260 and 680 μm), a tubularlike pattern appeared no longer classified as periodic structures as LPs"?Could this be a chemical garden?
3) Figures 5a,b are not needed and could be moved to the SI.They simply describe how the authors varied the gap height.4) Eq. 5: I am confused that the cell length L enters the analysis.Would the patterns (onset of LP) be different if cells of say 1 vs 10 cm are used?Also ε^2Ra depends on L! Moreover, microgravity conditions would yield Ra=0, but I doubt that LPs would form in thick layers for g = 0.
5) While I am not requesting these data, it might be interesting to repeat the experiment in a wedgeshaped HS cell with heights between say 20 and 200 um.6) In the Science paper https://doi.org/10.1126/science.aah6350, the Aizenberg group shows some indication of Liesegang patterns formed by biomorphs in microfluidic devices (see Fig. 1G, see also the SI).The results are not fully convincing, but might be worth mentioning (or at least of interest to the authors).7) Can the authors provide more information on the size of the crystals formed in their different systems.Is the crystal height comparable to the layer height?Should we expect differences between a full and partial blockage of diffusion by these crystals?8) Does the surface (what is the material?) of the HS cell affect crystal growth?Would different materials potentially create different outcomes or do the crystals nucleate in solution?9) P. 21: rephrase ">empirical stereotypes< of experiments in use".10) Check for missing subscripts (e.g.caption 5, SI text at the end of the main paper).

Reviewer: 2
Comments to the Author This manuscript describes the periodic precipitation patterns using a Hele-Shaw cell in a confined liquid phase.I think that the authors have reported a really interesting phenomenon.In addition to the interesting phenomenon, the authors have also investigated the phenomenon in good amount of depth, both experimentally and theoretically.The conclusion arrived based on the investigations is thus strong.The demonstration of pattern formation in a gel-free environment is indeed interesting.I think this manuscript is very suitable for the Journal of Physical Chemistry Letters.
Author's Response to Peer Review Comments: "4) Graphics: One or more of your figures and tables includes a reference citation.Please confirm that this pertains only to data and not the figure itself.If it pertains to the use of a published image, permissions must be secured for any graphics NOT originally published by ACS or for Open Access content which permits reuse with credit only.Permission is needed if you are using another publisher's or copyright owner's figures/tables verbatim, adapting/modifying them, or using them in part.If the images are from an Open Access publisher that does not require permission for reuse, please confirm." Author reply: We confirm that Table S1 contains two citations that pertain only to data.

Reviewer #1 "The paper by Itatani et al. reports an experiment study of Liesegang patterns in Hele-Shaw cells. HS cells are frequently used for analysis of viscous fingering and its relevance to oil exploration and via Darcy's law flow in porous media. Liesegang patterns (LPs) are usually produced in gels, which defines the main novelty of this paper. The authors avoid undesired fluid motion not by polymers but rather by vertical confinement to a very thin solution layer (typically 70 um) bound by two glass plates. The importance of this study lies at least partially in the possibility of characterizing the role of the polymer/gel on the observed patterns, ultimately providing a better understanding of this classic and interesting phenomenon. I think the study is of high quality and originality. It is also well-written and of interest to physical chemists. I recommend the paper for publication in JPC
Letters with minor revisions.I specifically ask the authors to consider the following questions and comments." Author reply: We thank the Reviewer for his/her comments and the positive assessment of our work.
"1) Title: The current title made me think of small pores and similar three-dimensional spatial confinement.Perhaps (and I do not insist) change "phase" to "layer"." Changes: Page 18. Otherwise (d = 260 and 680 µm), the homogeneous distribution of the precipitate could be observed, and it can no longer be classified as a periodic structure.
"3) Figures 5a,b are not needed and could be moved to the SI.They simply describe how the authors varied the gap height." Author reply: We thank the Reviewer for this comment.We moved Figure 5 a,b to the SI (now Figure S3 in the SI).

Changes:
(i) Page 20.  "4) Eq. 5: I am confused that the cell length L enters the analysis.Would the patterns (onset of LP) be different if cells of say 1 vs 10 cm are used?Also ε^2Ra depends on L! Moreover, microgravity conditions would yield Ra=0, but I doubt that LPs would form in thick layers for g = 0."

Budapest University of Technology and Economics Faculty of Natural Sciences
Author reply: We thank the Reviewer for this valid issue.The Reviewer is right; the cell length does not affect the pattern formation in our study.We revised the calculations and updated the results.In the case of g = 0, the product ε 2 Ra is zero, indicating Darcy flow in the system, generating the periodic precipitation in this condition.

Changes:
We rewrote the corresponding part and updated Figure 5b and Table S1.
where g,  * ,  * , µ, and D are the standard acceleration of gravity, the smallest length of the cell, the length of the cell parallel with the gravity acceleration vector, the dynamic viscosity of the medium, and the molecular diffusion coefficient, respectively.Also, Δρs is the density difference between solute-saturated fluids and solute-free fluids.In our setup, both  * and  * equal d and equations 5 and 6 can be rewritten in the form: (iv) Table S1 in the SI.
Table S1.Parameters and results for calculating the Rayleigh-Darcy number (Ra) and the geometry parameter (ε).
7.0 × 10 -5 9.8 27 1.0 × 10 -9 9.9 × 10 -4 7.6 6.4 × 10 -1 1.2 × 10 -4 3.9 × 10 1 3.2 × 10 0 2.6 × 10 -4 3.9 × 10 2 3.2 × 10 1 6.8 × 10 -4 7.0 × 10 3 5.8 × 10 2 1.4 × 10 -3 6.1 × 10 4 5.1 × 10 3 "5) While I am not requesting these data, it might be interesting to repeat the experiment in a wedgeshaped HS cell with heights between say 20 and 200 um." Author reply: We thank the Reviewer for this suggestion.Investigating periodic precipitation in a wedge-shaped HS cell could be the next step in exploring this phenomenon in a liquid layer, but this type of investigation is outside the scope of this study.Pattern formation in a wedge-shaped HS provides complexity due to the cell's shape.The height increase can influence the stability of the colloids formed in the liquid layer (due to the changing of the Rayleigh-Darcy number) and the diffusion flux of the chemical species.
Changes: Page 22.We added a short discussion to highlight this issue in the conclusion.
A more complex approach could be the investigation of the formation of periodic precipitation in a wedge-shaped HS.This setup affects not only the fluidity of the medium but the diffusion flux of the chemical species, contributing to the creation of more complex chemical structures in reactiondiffusion systems.
"6) In the Science paper https://doi.org/10.1126/science.aah6350, the Aizenberg group shows some indication of Liesegang patterns formed by biomorphs in microfluidic devices (see Fig. 1G, see also the SI).The results are not fully convincing, but might be worth mentioning (or at least of interest to the authors)."

Figure 5 .
Figure 5.Effect of vertical gap distance (d) on the pattern formation of CuCrO4 system in both the HS cell and a hand-crafted HS cell.(a) Pattern formation in the hand-crafted HS cells with different d values for [CuCl2]0 = 1.0 M and [K2CrO4]0 = 0.2 M: d = 0.12 mm, 0.26 mm, and 0.68 mm.(b) Variation of the Rayleigh-Darcy number with the cell anisotropy ratio (ε 2 Ra) as a function of d.The purple square indicates the region where the periodic pattern was formed.The dotted line corresponds to the 2D HS and 3D flow regimes boundary. 68,69Error bars were obtained from 3 replicates calculated using p-values < 0.05.

BudapestFigure 5 .
Figure 5.Effect of vertical gap distance (d) on the pattern formation of CuCrO4 system in both the HS cell and a hand-crafted HS cell.(a) Pattern formation in the hand-crafted HS cells with different