Pattern selection by material aging: Modeling chemical gardens in two and three dimensions

Significance

While living systems rely on the growth of diverse macroscopic structures, the range of patterns in inorganic chemistry is significantly limited. Nonetheless, certain abiotic processes far from equilibrium can self-organize into life-like shapes. A prime example is the formation of tubular precipitates in chemical gardens. Despite these structures forming readily in various reaction systems, modeling efforts have mostly fallen short. In this study, we present numerical simulations that effectively reproduce these structures, including the distinctive precipitate tubes. Our results highlight the crucial role of gradual changes in the precipitate, which influence the likelihood of local size expansion through dynamic self-healing processes. This form of pattern generation indicates potential for designing self-growing devices and might also be present in biological systems.

Abstract

Chemical gardens are complex, often macroscopic, structures formed by precipitation reactions. Their thin walls compartmentalize the system and adjust in size and shape if the volume of the interior reactant solution is increased by osmosis or active injection. Spatial confinement to a thin layer is known to result in various patterns including self-extending filaments and flower-like patterns organized around a continuous, expanding front. Here, we describe a cellular automaton model for this type of self-organization, in which each lattice site is occupied by one of the two reactants or the precipitate. Reactant injection causes the random replacement of precipitate and generates an expanding near-circular precipitate front. If this process includes an age bias favoring the replacement of fresh precipitate, thin-walled filaments arise and grow—like in the experiments—at the leading tip. In addition, the inclusion of a buoyancy effect allows the model to capture various branched and unbranched chemical garden shapes in two and three dimensions. Our results provide a model of chemical garden structures and highlight the importance of temporal changes in the self-healing membrane material.

Self-organizing macroscopic patterns can unveil surprising universalities among vastly different systems (1–3). Examples of such patterns include rotating spiral waves observed in autocatalytic reactions, living cells, organs, and groups of people (4–9) Similarly, fractal-like patterns emerge in dendritic crystals, dielectric breakdown figures, and river deltas (10–13). This spatiotemporal convergence often stems from simple mechanisms that function at large scales, controlling microscopic processes in a top-down manner. These mechanisms rely on a minimal set of rules and act on a restricted number of system-level variables or states. For instance, wave patterns in the previously mentioned examples arise from local coupling, such as diffusion or voltage gradients, which trigger a transition between an excitable and excited state, followed by a refractory phase (3, 8).

Cellular automata (CA) are a class of models that effectively capture system-level rules and states. The most well-known example is Conway’s “Game of Life”, which generates a variety of patterns including oscillators, spaceships, and replicators (14). This type of CA has roots in the work of von Neumann and Ulam, who in the 1940s used simple lattice models to explore crystal growth and self-replicating machines. In the same decade, Wiener and Rosenblueth proposed a CA model capable of describing reentrant waves and fibrillation in the human heart as well as the rotating wave examples mentioned above. In the 1990s, Wolfram and others expanded on this work, introducing new CA classes that could reproduce pigmentation patterns on mollusk shells and three-dimensional concentration patterns in the self-organizing Belousov–Zhabotinsky reaction (15–18). Today, probabilistic CAs are commonly used in the fields of condensed matter science and engineering to study phase transitions, fluid dynamics, corrosion, and active matter (19). Although more detailed models, based on reaction-transport equations, have replaced simple CAs for some applications, CAs remain a valuable tool for gaining mechanistic insights into complex experimental systems.

In this paper, we describe a CA model for the complex phenomenon known as chemical gardens. Chemical gardens form during precipitation reactions and are macroscopic structures, such as colorful thin-walled tubes, that grow in the upward direction (20). This quintessential example involves the placement of mm-sized salt particles into a silicate solution. The dissolving salt quickly reacts in the alkaline solution to form a thin layer of metal hydroxide around the salt seed. This semipermeable membrane subsequently ruptures under the osmotically driven influx of water and ejects an upward-rising jet of salt solution (21–23). Continued precipitation around this jet creates life-like tubes and branched structures. Their several μm-thick walls tend to consist of an outer silica layer covering the dominant metal hydroxide (or oxide). Numerous chemical variants of chemical gardens exist including metal sulfides, phosphates, borates, and carbonates as well as polyoxometalates and hydrogels (24–31). The range of usable metal ions spans the entire periodic table including the group-1 element lithium (32). This chemical diversity combined with the existence of characteristic shapes and patterns signals the presence of underlying universalities that to date are only poorly understood.

Much of the progress toward a better understanding of chemical gardens has come from both physical and chemical simplifications of the classical salt-seed experiment. Our group replaced the salt seed with a salt solution, which is steadily injected into a larger volume of silicate solution (33–35). This method showed the presence of distinct growth modes and resulting morphologies, specifically “jetting” (slender tubes with open tips), “popping” (closed tubes that release membrane segments), and “budding” (closed tubes consisting of chains of membrane balloons). The latter two regimes qualitatively revealed that the fresh precipitate membrane is stretchable whereas older membranes are rigid and tend to break. This seeming elasticity of the fresh material is not a traditional elasticity but the result of microscopic failures of the material followed by rapid self-healing as the internal and external reactants come into contact. While evidence for this intriguing process is mainly indirect, it might also explain rim-like damage lines on the inorganic shell of reactant-loaded polymer beads undergoing precipitation reactions (36).

Another step toward more controlled experimental conditions was the introduction of spatial confinement (37). Haudin et al. investigated precipitate patterns that formed in thin, horizontal layers of silicate solution when the CoCl₂ solution was injected radially from a central port (38). They found a variety of patterns for different reactant concentrations and grouped them according to categories coined filaments (highest concentrations), worms, flowers, hairs, spirals, and lobes (lowest concentrations). Subsequent studies reported additional types of patterns and expanded the range of chemical reactions to other product materials such as metal hydroxides, phosphates, carbonates, and oxalates (39–53). A small number of examples is shown in Fig. 1.

Fig. 1.

Experimental examples of precipitation patterns in thin horizontal layers. The patterns form when one reactant solution (B) is steadily injected into another one (A). Using this B/A notation, the reactants are (A) CoCl₂/Na₂SiO₃, (B) NiCl₂/NaOH, (C and D) CoCl₂/Na₂SiO₃, (E) Na₂CO₃/BaCl₂, (F) CaCl₂/Na₂CO₃, (G) Na₂CO₃/CaCl₂, (H) NiCl₂/Na₂(COO)₂, and (I) CuCl₂/Na₂(COO)₂. All scale bars: 2 cm. Reprinted with permission from refs. 41, 44, 45, 47, 50, 51, and 53.

The primary motivation for this study is to elucidate the origin of filament-like precipitation structures (Fig. 1 A–C) and their transitions to more dense patterns (D–F) and circular fronts (G–I). For this, we formulate and characterize a CA model that assumes time-dependent changes in the precipitate. Considering the broad applicability of the underlying rules, the simulated patterns constitute a universality class that might find unexpected realizations in other systems such as acid-induced “wormholes” in oil exploration and stiffening roots in plant growth (54, 55).

Results

Basic Model – Formulation.

We first describe and characterize a bare-bones version of our model. The model domain consists of a regular grid of N × N cells. Each cell is occupied by one of three possible chemical species. Species A and B are the reaction partners that create the precipitate C. Reactant A initially occupies most of the domain while reactant B is injected during the simulation. In the following, we define the two main rules of the model. These rules are applied for each timestep and update the entire domain. An example is shown in Fig. 2.

Fig. 2.

Example illustrating the operation of the basic model. Gray and blue regions represent the reactants A and B, respectively. The precipitate C is shown as a range of colors between gold (fresh) and black (old). The initial condition is shown in the leftmost panel (t = 0) and includes no C cells. At t = 1, Rule 1 has created a ring of C (black) and the area of A and B decreased. For t = 2, the C cell (x, y) = (8,7) was selected randomly for reset to B and the nearby A cell (10, 8) received the old C element (Rule 2a). At this point, all but one C cell are of age 2 (Rule 2b) and only (10, 8) is age 1 (gold). For t = 3 and later, this process repeats. At t = 5, a total of four B elements have been “injected”, and the C cells have five different ages with black now indicating C = 5.

Rule 1 aims to capture the reaction of A and B to the precipitate species C. We demand that a cell is updated to C if its 3 × 3 neighborhood contains at least one A and one B.

This rule of C production does not alter the states around the central cell and therefore A or B are consumed under this rule only if the central cell is in the A or B state. Notice that this rule is inconsequential if the central cell is already in the C state.

Rule 2a aims to model the steady injection of B. We require that during each timestep, a randomly chosen C cell with at least a specified number (B_min) of B-type cells in its 3x3 neighborhood transforms into a B-type cell. In addition, one A cell closest to this site changes to C.

This rule expresses that an injected B volume can only be accommodated by pushing out the preexisting membrane and draining A from the system. Accordingly, we remove one C cell from the inside (where it is in contact with preexisting B) and add one C cell to the outside of the membrane (where A is present). Notice that if more than one target site has the same shortest distance, one of them is randomly selected.

The value of B_min is a small, positive integer. Its effect on the resulting patterns is small and mainly alters the thickness of the precipitates (SI Appendix, Fig. S1). In the following, we use B_min = 2 for all two-dimensional simulations and B_min = 4 for all three-dimensional simulations.

Rule 2b is a nonessential feature of the basic model but insightful and the basis for a later modification. We demand that the value of each C cell increases by +1 per timestep and that a newly formed C cell has the value 1. This simple counter uses the C value to store the age of the local precipitate and implies that shifting a C cell resets its age.

Last, we define the domain’s initial condition and the criterion for terminating the simulation. For time t = 0, we set all cells to A with the exception of cells within a small central disk that are set to B. The radius of this disk equals 10 cells (smaller only in Fig. 2). Larger or smaller disks show qualitatively identical results. Simulations are terminated when the first C cell reaches a distance of N/2-1 from the central cell (Figs. 2–5) or come close to the top or side boundaries of the domain (Figs. 6 and 7).

Fig. 3.

Characterization of the basic model. (A and B) Pattern after t = 15,000 and 30,000 timesteps, respectively. Gray and blue regions represent the reactants A and B, respectively. The precipitate C is shown as a range of colors between gold (fresh) and black (old). The Inset in A magnifies a short front segment. (C and D) Temporal evolution of the total B and C area, respectively. The black curve in D is a fitted square root function. (E) Histogram of C values (i.e. ages) at t = 30,000. (F) Sorted C values at various times with the rightmost curve showing C at t = 30,000. The dashed lines in C and F are the identity line. Grid size: 250 × 250.

Fig. 4.

Phase diagram of precipitation patterns in terms of the model parameters k and δ, which are the aging rate constant and the spatial slack, respectively. Color scheme as in prior figures. Simulations are run until the precipitate reaches a fixed distance from the domain center. Grid size: 150 × 150.

Fig. 5.

(A and B) Total number of C and B cells as a function of the rate constant k for δ = 0. Gray circles show the results of 384 simulations (12 per k value) that were terminated when the precipitate reached a distance of N/2-2 from the domain center (N = 150). The red markers and error bars in A are the median values and their respective 67% percentiles; in B, they represent means and SDs. The dashed line at log(k) = −2.5 approximates the data’s maximum and inflection point. The Inset shows the ratio of total B and C cells, which is the area-to-edge ratio of the patterns.

Fig. 6.

(A) Phase diagram of gravity-dependent, two-dimensional precipitation patterns in terms of the model parameters k and β, which are the aging rate constant and the buoyancy strength, respectively. Color scheme as in prior figures. Simulations were run until the precipitate reaches the top or side boundaries of the domain. Grid size: 150 × 150. (B) Precipitate patterns observed in vertical Hele-Shaw experiments by Rocha et al. (reprinted with permission from ref. 42). (C) Calcium alginate balloons reported by Zahorán et al. (reprinted with permission from ref. 31). (D–F) Three-dimensional chemical garden structures (D and E) reprinted with permission from refs. 25 and 26). Seed crystals in (F): CoCl₂, NiCl₂, FeCl₃, and MnCl₂. (Black scale bars: 1 cm; white scale bars: 1 mm.) Gravity direction in all panels: downward.

Fig. 7.

(A) Simulated three-dimensional precipitation patterns as a function of the aging rate constant k. (B) Front and side views of a cut 3D structure. Blue represents the reactant B while the precipitate C is shown using the same color scheme as in previous figures. (C) Two examples of chemical garden tubes, featuring nodules formed when the C increment alternates between the usual value of 1 and 10⁻⁶. Height of the structures: (A and B) 100 cells (80 cells for the shell-like structure on the Left), and (C) 200 (Left) and 250 cells (Right). In all panels, gravity acts in the downward direction and the buoyancy factor is β = 10.

Basic Model – Characterization.

This section describes the performance of the basic model and characterizes the emerging precipitation front. Fig. 3 A and B shows representative patterns at timesteps t = 15,000 and 30,000, respectively. The local states are encoded as different colors with gray and blue indicating A and B, respectively. C is rendered as colors that encode the age of the precipitate: The freshest material is golden (low values of the C counter under Rule 2b), and the oldest material is black (highest C values in the given pattern). We reemphasize that, in the basic model, the C counter and Rule 2b have no impact on the pattern evolution.

In Fig. 3 A and B, the precipitate is located along the edge of an expanding B disk and steadily expels the surrounding A from the system. The width of the precipitate front varies between 1 and 5 cells, with 1 and 5 being rare. Most of the precipitate is part of the front and only very few isolated C cells exist. These smaller islands are all within the interior B region, whereas no unconnected C cells are found in the surrounding A region.

We traced the execution frequency of Rule 1 for non-C cells (SI Appendix, Fig. S2). This frequency is high during the first timestep and very low during the following iterations (about one execution every 17 timesteps). Accordingly, the movement of the precipitate front is driven by the injection of B under Rule 2 while the growth and the integrity of the membrane are assured by Rule 1. Fig. 3C shows the progression of the total B area, which is found to be linear with a slope of 0.98. The small mismatch to the mandated injection rate of one B cell per timestep is again due to reactions between nearby A and B cells (Rule 1) and the resulting conversion of B to C. The evolution of the total C area obeys a square root law as shown in Fig. 3D, which is a consequence of the linearly increasing area of the B disk, for which C acts as an edge of nearly constant thickness.

The expanding precipitate ring in our basic model consists of C elements of different ages. Fig. 3E shows the distribution of the log(C) values at the late time t = 30,000. The C values spread over a wide range, between 1 and about 6,000, with a distribution that widens over time and a standard deviation scaling like $\sqrt{t}$ (SI Appendix, Fig. S3).

Fig. 3F provides additional details by graphing the sorted C values for 15 select time points. As the youngest C cells tend to have unique C values, we find one occurrence of C = 1 (as per Rules 2a, b), typically one occurrence of C = 2, and so on. Accordingly, the sorted C values trace initially a straight line of slope one. Larger C values become rarer, and the graph sharply slopes upward. The C values are bound by t but the maximal C is typically much smaller. The analytical nature of the C distribution is unclear to us. However, we point out that the behavior of our basic model is related to a simpler statistical problem that is illustrated in SI Appendix, Fig. S4.

Last, we compare the precipitation patterns of our basic model to the experimental examples in Fig. 1. We find good qualitative agreement between the thin circular front in the model and the example shown in Fig. 1H. Similar fronts were also reported for systems injecting CoCl₂, NiCl₂, CuCl₂, and ZnCl₂ into oxalate solutions (53). All of these patterns appear to consist of loose particles rather than a connected membrane. Interestingly, Schuszter et al. observed a redissolution of the precipitates, which is related to the formation of metal complexes and argued that radially expanding patterns occur when the time scales of reaction and flow are comparable (53).

Model with Aging.

We now modify the basic model by including a simple age bias to Rule 2a. The underlying assumption is that freshly formed precipitate is more susceptible to deformation or replacement than older rigid material. Experimental evidence for slow changes of the precipitate membrane has been reported in several studies but those changes are typically deduced from color changes that indicate variations in hydration or redox states (20, 33, 35, 53). In addition, other studies of three-dimensional chemical garden tubes revealed closed “budding” structures that grow via alternating phases of self-healing stretches and breach-like ruptures (33).

For the age bias, we first transform all C values to multinomial probabilities using a kinetic rate law f(C) and normalization $\sum \mathit{f}\left(\mathit{C},\right)=1$ . For most of our simulations, the rate law is the function $\mathit{f}\left(\mathit{C}\right)=\mathit{exp}\left[-\mathit{k}\left(\mathit{C}-1\right)\right]$ , where k is a rate constant and key model parameter. Notice that the basic model effectively uses the function f(C) = const. After the resulting vector is normalized, we randomly select one C cell from all C cells according to their respective probabilities. Once selected, the cell is—as before—replaced by B, and then C is transferred as C = 1 to the closest available A site. A sample MATLAB script can be found in SI Appendix.

The lowest row of Fig. 4 (see also Movie S1) shows representative patterns obtained in this model for different k values (see SI Appendix, Fig. S5 for additional examples). For the smallest value shown (k = 10⁻⁴), the model generates a nearly circular precipitate front that is very similar to the front in the basic model (Fig. 3 A and B). This similarity indicates a mild and nearly negligible influence of aging, which is expected if we compare the corresponding half-life of t_1/2 = ln(2)/k = 6,932 timesteps to the typical termination time of about 13,000 timesteps.

Moving rightward in Fig. 4, we find that an increase in k leads to more wrinkled fronts and a larger amount of interior precipitate (black inclusions). Slightly above log(k) = −2.5, these irregular patterns give way to filamentous structures that at −1.5 and above have a clear conduit-like character. These conduits consist of a narrow passageway for B with thin C-based walls. The filaments extend their length via expansion near their tips, which show the freshest precipitates (golden cells). This growth preference is in accord with the model’s preference to move fresh C cells and the very short half-life of t_1/2 = 2.2 timesteps at log(k) = −0.5.

The differences between the patterns directly result from local “hot spots” of continuing C change. For filaments, there are a few, often only one, of these hotspots. For intermediate, disordered patterns, this number increases but the individual hotspots exist only intermittently. Last, for circular fronts, there are no hotspots, and the site selection is essentially random. These three scenarios are illustrated in SI Appendix, Fig. S6.

The filament patterns in Fig. 4 share striking similarities to the experimentally observed filaments in Fig. 1 A–C. For instance, both the simulated and the experimental filaments have the tendency to meander and form regions of layered and winding filaments. This process can occasionally result in residual A islands surrounded by large areas of B and C. Such an “empty” spot can be discerned in the lower right panel of Fig. 4 and in Fig. 1B.

In addition, the simulated patterns show self-trapping events that also exist in experiments (39, 40, 45). In these situations, the growth tip gets surrounded by its own tail and—as injection of new solution continues—forces the rupture of membrane material at an older site. Such an event can be seen in the lower right panel in Fig. 4 as it created a small B inclusion (blue island). The phenomenon is further illustrated in SI Appendix, Fig. S7. If the self-entrapped region is large, the aging system loses its preferred growth region and must select target cells along older C deposits with more similar f(C) values. This effect leads to a broad leakage phenomenon (SI Appendix, Fig. S7) that is reminiscent of experimental observations (39, 40, 45).

We now discuss the upper rows in Fig. 4, which test the model behavior under a greater variability in the placement of C under Rule 2b. The main motivation for this modification is to allow membrane segments and precipitate particles to be moved to more distant positions as it would likely be the case for loosely aggregated and nonaggregated particles as well as for systems in which fluid motion plays a larger role. The original rule demands that C cells (that were removed to accommodate new B) are placed onto the nearest available A site. Now we allow for some spatial “slack” δ. Accordingly, we first locate the nearest A cell, compute its distance d from the original C site, and then randomly select the target cell from all A cells within a distance of d+δ from the original site. The resulting patterns tend to have thicker precipitate walls and appear fuzzier. Notice that no attachment to preexisting C cells is required. The resulting appearance of scattered C cells ahead of the main structure can be understood as the result of precipitate transport via convection which is known to exist in certain experimental systems (51, 56). We also investigated precipitate patterns with required attachment of new C cells to preexisting precipitate. The resulting patterns have a hair-like front that is vaguely reminiscent of hair-like patterns that were reported for silicate-cobalt experiments (see SI Appendix, Fig. S8 for both numerical and experimental examples) (38).

Values of δ ≈ 6 promote the simultaneous growth of multiple filaments, which is frequently observed in experiments, and cause larger C inclusions for intermediate rate constant around log(k) = −2.5. The parameter δ is therefore a possible way to account for denser precipitate patterns including those shown in Fig. 1 D and E. Moreover, δ affects the density gradient of the circular precipitation fronts. For large δ and small k values, we find density profiles that increase in the direction opposite to the propagation direction (i.e., fuzzy front and steep wake). This feature is reminiscent of the experimental pattern shown in Fig. 1G while smaller δ values cause patterns similar to Fig. 1H.

So far, we have discussed qualitative aspects of the simulated precipitate patterns as well as possible similarities to their experimental counterparts. Fig. 5 provides a quantitative view by plotting the total number of B and C cells as a function of the rate constant k for δ = 0. The C count in Fig. 5A reveals a pronounced maximum slightly above log(k) = −2.5 that corresponds to the demarcation line between fronts and filaments in Fig. 4. Notice that around this maximum the simulated patterns are disordered and show a large amount of C inclusions. The C values for the filaments show a greater variability because they can reach the termination distance swiftly or undergo longer phases of meandering and layering. The fronts, however, have reliable and longer termination times.

The existence of a maximum in the C(k) curve can be explained as follows. At high k values, filaments exist and tend to leave the domain quickly, thus, creating only small amounts of precipitate (see Fig. 4). At low k values, fronts exist and, due to their nearly circular and thin edge also produce relatively small amounts of C. Only for intermediate k values, patterns evolve for sufficiently long times and show wrinkled and folded fronts as well as left-behind C inclusions. Accordingly, the C(k) maximum constitutes a disordered transition state between two well-defined pattern classes: circular fronts and elongated thin filaments.

These two states can also be discerned from the total number of B cells in Fig. 5B. Here, we find a sigmoidal curve with wings indicating relatively pure manifestations of the two pattern types. The approximate transition point agrees well with the inflection point of the B curve. Finally, the inset in Fig. 5B presents an additional order parameter: the ratio between the total number of B cells and C cells. This number reflects the area-to-edge ratio of the patterns. They vary from about 1 to 2 for filaments to 10 for circular fronts and show a steep decline at the transition point. The value of 10 can be readily understood as the area-to-edge ratio πr²/2πrw (r ≫ w) of a circle of radius r and an edge thickness w. For the maximal radius of about 70 and w = 3.5, this expression yields 70/7 and is in good agreement with the measured value of 10.

We also investigated nonexponential dependencies for the aging kinetics of the precipitate material such as sigmoidal aging functions f(C). Several examples are shown in SI Appendix, Fig. S9 and are qualitatively similar to those found for exponential decay. Moreover, we tested various modified rule sets to increase the amount of interior C. One interesting modification amends Rule 2a, which in its original form requires that the newly injected B cell must replace a C cell with at least B_min = 2 B cells in its 3 × 3 neighborhood. If we further demand that a larger m×m neighborhood shall also contain at least A_min A cells, we find patterns with much larger C densities that are reminiscent of the experimental example shown in Fig. 1E (SI Appendix, Fig. S10). Last, we also investigated a spatially one-dimensional version of our model, which shows bidirectional growth at low aging rates and unidirectional growth at high rates (SI Appendix, Fig. S11).

Model with Buoyancy.

We now return to the model with exponential aging defined by the original Rules 1 and 2a, b with the goal to include a buoyancy-like feature. Such gravity effects are responsible for the upward-directed growth in most chemical garden systems (20, 21). Our first step is to model chemical gardens confined to vertical solution layers that were first studied by Rocha et al. in 2022 (42, 43). Notice that the chemical gardens in these pseudo-two-dimensional experiments were filled with CoCl₂ solutions (0.6 to 1 mol/L) that are lighter than the surrounding Na₂SiO₃ solution (3.1 to 6.3 mol/L). They included wide and narrow filamentous structures that grew upward along straight or erratic paths (Fig. 6B).

To include buoyancy, we modify the probability weight function f(C) by a gravity-dependent factor, so that buoyancy affects horizontal membrane segments more strongly than tilted or vertical segments. In addition, we only consider buoyancy effects on C segments separating an upper A region from a lower B volume (i.e. the “top” of the structure). Utilizing the dot product of the unit surface normal n and the vertical unit vector e_y, we explore the function f(C) = exp[−k(C − 1)] × (1 + βθn·e_y). Here, β denotes the effective buoyancy strength, which qualitatively captures the density difference between inner and outer solutions; θ equals 0 for local B-over-C-over-A cases and 1 otherwise. Notice that this function f(C) leaves the aging kinetics unchanged for vertical membrane segments but introduces a preference for enhanced growth at and near the horizontal top. Computationally, this expression is evaluated in local neighborhoods of 3 × 3 cells.

Fig. 6A shows precipitate patterns formed for different values of k and β (see also SI Appendix, Fig. S12). The examples in the lowest row are obtained for a buoyancy strength of β = 0 and correspond to the earlier results (lowest row in Fig. 4) with the exception of a modified boundary condition for the lower edge of the domain. This boundary condition does not terminate the growth but rather excludes the lowest lines of cells (y ≤ 5) as targets for replacement with B.

The examples in Fig. 6A obtained for β ≥ 1 have a clear preference for growth in the upward direction. For large k values, they are filament-like and can show branching. With increasing β, they become less erratic and more erect. Some structures at intermediate k values look brush-like and low k values create precipitate envelopes of various shapes. Comparison to the experimental results by Rocha et al. yields good agreement with the right row of our phase diagram (42). The remaining structures might have experimental counterparts in vertical Hele-Shaw cells, but the currently available data are too limited to allow a meaningful discussion. We also note that a higher placement of the initial B disk allows for some downward growth.

We now conclude our study by expanding the buoyancy-aging model to three dimensions. The main modification is that the neighborhood definition is changed to a cube of 3 × 3 × 3 cells and B_min is set to 4. The latter accounts for the change in the relevant neighborhood from nine cells (two dimensions) to 27 (three dimensions) but does not strongly affect the model behavior. Fig. 7A shows a collection of precipitate structures obtained for different values of k at β = 10. The structures are rendered based on the C values only and, with increasing k values, vary from shell-like precipitates to thick columns and thin tubes. All structures grow in the upward direction but thicken to various degrees. Tubes and columns show additional variations in the growth direction that qualitatively agree with the erratic appearance of chemical gardens in experiments (Fig. 6 D–F). In addition, all C structures are filled with the reactant B and some detached C cells (Fig. 7B). We also investigated buoyant tube growth in thin horizontal layers and found a strong preference for the attachment to the upper boundary (SI Appendix, Fig. S13).

An extension of our 3D approach induces nodules along the tube (Fig. 7C and Movie S2) that are very similar to structures observed in experimental chemical gardens (Fig. 6F). The effect is triggered by adjusting the C increment from the usual value of +1 to a much smaller value per timestep. This change causes the initially slender tube to widen into a nodular segment that is reminiscent of the shell-like structures in Fig. 7A (lowest k value). Notice however that decreasing k causes a different behavior that allows the already-formed tube stalk to undergo active changes. We interpret the effect of decreasing the C increment as an effective increase in the injection rate of B; in other words, a significantly larger reactant volume is delivered into the structure cavity during the length of the original timestep. In the context of chemical garden experiments, this faster delivery likely results from sudden changes in the osmotically driven inflow of water or other connected tubes becoming inactive.

Discussion

Despite being one of chemistry’s oldest and most iconic reactions, chemical gardens have remained enigmatic and resisted modeling efforts. While certain quantitative aspects were captured by theoretical analyses (34), the most striking features—the actual shapes—have been simulated here. As a common theme, our 2D model variants revealed three types of patterns: filaments/tubes, irregular structures, and front-like envelopes. The primary model parameter selecting these different types is the rate constant k that regulates the rate of precipitate aging or more precisely its likelihood of being moved outward as more reactant solution is injected. Our simulations show that these changes are responsible for major characteristics of the pattern.

The origins of this age dependence are diverse and system-dependent. One extreme case is the absence of a mechanically continuous membrane. In this scenario, individual precipitate particles move outward driven by fluid flow and the dynamics are therefore age-independent. The resulting patterns tend to be circular precipitate bands and are well captured by our simulations for very low k values. Very similar fronts are also found in experiments with very soft membranes such as hydrogel systems (30). The other extreme is the rapid hardening of a membrane that remains deformable and expandable only in the most recently formed locations. These conditions give rise to filaments and are reproduced by our model for large k values. We reemphasize that these shape changes and elongations of the fresh membrane are unlike other elastic materials that would return to the original shape and size. They also differ from conventional plasticity for which the material would remain deformed but not increase in size and volume. Both the fresh membranes of chemical gardens and our model expand in response to microbreaches and thus constitute an interesting example of self-healing materials that can lead to enormous increases in mass and volume. The physicochemical nature of these processes remains poorly understood but—depending on the reaction system—likely involves changes in the hydration level, redox changes, and possibly secondary transformations such as the transition of aggregated microparticles to more compact amorphous materials. In systems with silicate ions, additional hardening might occur due to the precipitation of silica on the outside of the metal salt membrane.

In addition to the morphological similarities between simulated and experimental chemical gardens, we also find insightful quantitative agreement. For the Hele-Shaw experiments by Haudin et al. (38) the aging rate constant k in our simulations can be directly mapped to the employed sodium silicate concentration c using a simple linear relation between log k and c (SI Appendix, Fig. S14). This connection does not only quantitatively recover the measured perimeter-to-area ratios but also provides strong support for our interpretation of silica as a key species in the aging process of the precipitate. As a second example, we aimed to reproduce a scaling law reported by Brau et al. for precipitate filaments (57). The scaling law relates the shortest distance of the active growth point from the injection site to the reaction time and is convincingly recovered by our simulations (SI Appendix, Fig. S15). This agreement between simulations and experiments illustrates the model’s ability to capture quantitative physical features such as the intricate balance of Brownian and ballistic motion.

A feature of pseudo-two-dimensional chemical gardens that is not captured by our model is the friction between the membrane and the confining glass or plastic plates of the Hele-Shaw cell. We believe that this friction strongly affects or controls the formation of both lobes and spiral shapes that were reported by Haudin et al. (38). In addition, our model does not account for the increasing pressure head of elongating filaments. This length-dependent resistance to fluid flow through the thin membrane channel likely limits the maximal filament length and ultimately induces new breach sites closer to the injection site. Last, the model currently does not allow for open tube growth (“jetting”) and the detachment of larger sections of the structure (“popping”) (33).

Future work could aim to match our model parameters and aging kinetics more closely to specific reaction systems and also explore different injection protocols including decreasing rates of B delivery that can be expected for slowly diminishing osmosis. It will also be interesting to see whether our cellular automaton approach can be transferred to continuous time models such as nonlinear reaction-diffusion-advection equations. The complexity of the continuously moving boundaries and stochastic effects in the chemical garden system, however, make this task a formidable challenge.

Materials and Methods

To explore the behavior of the different models described here, we wrote computer scripts for MATLAB (version R2020b). Script examples for 2D and 3D simulations are included in SI Appendix, respectively. Although the basic model is in essence a three-state system, the chemical species A, B, and C are represented as independent arrays. Furthermore, we made use of the MATLAB command mnrnd to select age-biased precipitate cells via multinomial random numbers. The scripts were not optimized for speed.

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (TXT)

Dataset S02 (TXT)

Movie S1.

Video illustrating the growth dynamics of filaments as well as irregular and front-like patterns. The five video sequences were obtained for aging rate constants of log k = -0.5, -1, -2, -3, and -4. The last two video sequences are shown 6x faster than the first three. Gray and blue regions represent the reactants A and B, respectively. The precipitate C is shown as a range of colors between gold (fresh) and black (old). The scaling of the C colors varies between simulations. Simulation parameters: N = 200, δ = 0, Bmin = 2. Filename: movie1.mp4; duration: 76 seconds.

Movie S2.

Video illustrating the simulated growth dynamics of a realistic chemical garden. The alternating slender and nodular segments occur for C increments of 1 and 10^-6 respectively. The color of the precipitate varies from light gray to black as it ages. Filename: movie2.mp4.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information. Previously published data were used for this work [Fig. 1: refs. 41, 44, 45, 47, 50, 51, and 53; Fig, 6 D and E: refs. 25 and 26; Fig, 6 C: ref. 31; SI Appendix, Fig. S8B and S14a (38)].