When confined to a Hele-Shaw cell, chemical gardens can grow as filaments, narrow structures with an erratic and tortuous trajectory. In this work, the methodology applied to studies with horizontal Hele-Shaw cells is adapted to a vertical configuration, thus introducing the effect of buoyancy into the system. The motion of a single filament tip is modeled by taking into account its internal pressure and the variation of the concentration of precipitate that constitutes the chemical garden membrane. While the model shows good agreement with the results, it also suggests that the concentration of the host solution of sodium silicate also plays a role in the growth of the structures despite being in stoichiometric excess.
A vast and fascinating array of different patterns can be observed when performing chemical garden experiments, which occur when metal salts come into contact with solutions, such as sodium silicate. These structures are shaped by forces, such as osmosis and buoyancy, that can be controlled by changing the concentration of the reactants. Recently, these experiments have also been performed in Hele-Shaw cells, quasi-two-dimensional micro reactors that allow the removal of the effect of osmosis when performed with injection, and buoyancy if placed horizontally; the results are thus only dependent on the relationship between the flow and chemical reaction. Among the various observable regimes, filaments are some of the most prominent and intriguing ones: thin tubes that periodically change direction after short rectilinear paths. In this study, the structures are grown in vertical Hele-Shaw cells, introducing the effect of gravity, and a physical model is developed by building upon previous work on horizontal filaments to attempt to explain their erratic trajectory.
I. INTRODUCTION
Chemical gardens are remarkable self-assembling structures formed by the precipitation reaction between metal salts and an aqueous solution with anions, such as silicate, phosphate, carbonate, oxalate, or sulfide.1 The classic method of growth of such structures involves simply dropping a metal salt seed into such a solution. As the seed dissolves, the metal ions precipitate with the anions in the outer solution, forming a semipermeable membrane around the seed. The steep concentration gradient leads to a greater osmotic pressure within the membrane, causing it to inflate and eventually rupture. This allows the release of a buoyant jet of a metal salt solution, around which a new precipitate membrane can form. As the process repeats, new tubes can branch out of the original structure, leading to growths resembling plants, hence the name “chemical garden.” The formation of these chemical gardens is thus controlled by osmotic and buoyant forces, together with the precipitation reaction between the two reactants. Many applications exist for these structures, such as a better understanding of corrosion tubes,2 cement hydration,3–6 and even the origin of life.7,8
Since the first experiment of this kind, described by Johann Glauber in the 17th century,9 new experimental methods have been established that allow for greater control over the process. Injection growth10 involves pumping one of the solutions into a reservoir containing the other one. This method still leads to the formation of a membrane at the interface of the two solutions and the growth of a tube, while making it possible to determine the concentration of the injected solution and the internal fluid pressure. Thouvenel-Romans and Steinbock10 reported three different growth regimes in experiments with this method: “jetting,” “popping,” and “budding.” These are due to the difference in buoyancy between the injected fluid and the denser silicate solution that surrounds it, which decreases with increasing metal salt concentration in the injected fluid. Air bubbles can also be placed at the tip of the growing tube to force it to grow vertically as the bubble rises through buoyancy. In addition, the width of the tube can be changed by changing the radius of the air bubbles.11 The effect of osmosis can be isolated by performing the experiments under microgravity, thus removing the contribution of buoyancy on the growth of these structures. Chemical gardens formed in space are found to grow at a much slower rate and to exhibit new morphologies.12,13
One other way of simplifying the system is to grow chemical gardens in Hele-Shaw cells.14–17 These consist of a quasi-two-dimensional system made by two horizontal parallel plates separated by a small gap. By injecting a metal salt solution into a host solution of sodium silicate, a surprising wealth of different patterns may be observed by just changing the concentrations of the two solutions. Such patterns include “spirals,” “flowers,”“worms,” and “filaments.”14–16,18 For fixed concentrations of the reactants, different patterns are observed by changing the flow rate of injection. Sumino and co-workers16 report “shells” for low flow rates, eventually leading to increasing numbers of filaments as the flow rate is raised. Given the horizontal orientation of the cell and the very small gap, buoyant effects are negligible and the resulting patterns are only the result of the combination of fluid flow and the precipitation reaction. Hele-Shaw cells can also be used to grow chemical gardens from a seed rather than injection. Ding et al.17 described the crucial role of the osmotic pressure in this case, with experiments with a solid cobalt chloride pellet placed at the center of a cell filled with a sodium silicate solution. For increasing concentrations of the solution in the cell, the resulting precipitate membrane is shown to grow in a stable, oscillatory, or unstable manner. The unstable regime eventually causes the explosion of the structure after some time, drawing a parallel between chemical gardens and clock reactions.
In this work, experimental data are presented for confined chemical gardens grown in a vertical Hele-Shaw cell, thus introducing the effect of buoyancy into the system. The effect of buoyancy on the internal pressure of chemical garden tubes has been modeled;19 however, the dynamics of formation of confined filaments still requires further study. Cobalt chloride and sodium silicate are used as the two reactants for four different pairs of concentrations and a range of injection flow rates. The filament regime, one of the main regimes identified in the literature, is examined more closely, with the main characteristics of the filaments determined through the means of image analysis. The active tip of these filaments exhibits an erratic, zig-zag motion; the main purpose of this work is to better understand this behavior. This paper builds upon a previous study conducted for the dynamics of chemical garden filaments in a horizontal Hele-Shaw cell.20
II. EXPERIMENTAL METHODS
Experiments were performed in a vertical Hele-Shaw cell, consisting of two square Perspex plates with a side length of 30 cm. A 0.5 mm rubber spacer was placed between the plates, around the edge of the cell. The top was left open to allow air to escape. Sodium silicate solutions with concentrations of 6.25 and 3.13 M were prepared by dilution of a commercial solution (27 wt. % with respect to silica SiO, which corresponds to a concentration of 6.25 M; Honeywell), and cobalt chloride solutions with concentrations of 0.63 and 1 M were prepared by dissolution of the powder in water [cobalt (II) chloride hexahydrate, ClCo6HO; 98%–102% (manufacturer’s description); ACROS Organics]. Before each experiment, the cell was filled with the host solution of sodium silicate, and the cobalt chloride solution was then pumped into the cell with a syringe pump (NE-1000, New Era Pump Systems, Inc.). A light pad was placed behind the cell for illumination, and the experiment was recorded with a Nikon D300s single lens digital camera (DSLR, pixels) with a Hoya circular polarizing lens filter. A photograph was taken every 2 s, covering an area of mm of the cell. The experimental conditions tested are summarized in Table I.
Regimes observed for the different chemical systems studied in this work. Blank squares, filled squares, and blank triangles represent the regimes with no filaments, one filament, and multiple filaments, respectively. At least three experiments were performed for each symbol in this table. The last column presents a condensed reference for each experiment. Figures present data as a function of flow rate; therefore, the experiment reference will have an “X” as shown in this table. When the data in a figure refer to an experiment performed at a specific flow rate, the respective legend will indicate the flow rate value after “Q.”
. | Q (mL min−1) . | . | |||||
---|---|---|---|---|---|---|---|
Chemical system . | 0.1 . | 0.5 . | 1.0 . | 1.5 . | 2.0 . | 3.0 . | Experiment name . |
Na2SiO3 3.13 M CoCl2 0.63 M | ■ | ■ | ■ | SS3.13 CC0.63 QX H0.5 | |||
Na2SiO3 6.25 M CoCl2 0.63 M | ■ | ■ | ■ | ■ | SS6.25 CC0.63 QX H0.5 | ||
Na2SiO3 3.13 M CoCl2 1.00 M | ■ | ■ | ■ | ■ | SS3.13 CC1.00 QX H0.5 | ||
Na2SiO3 6.25 M CoCl2 1.00 M | ■ | ■ | ■ | ■ | SS6.25 CC1.00 QX H0.5 |
. | Q (mL min−1) . | . | |||||
---|---|---|---|---|---|---|---|
Chemical system . | 0.1 . | 0.5 . | 1.0 . | 1.5 . | 2.0 . | 3.0 . | Experiment name . |
Na2SiO3 3.13 M CoCl2 0.63 M | ■ | ■ | ■ | SS3.13 CC0.63 QX H0.5 | |||
Na2SiO3 6.25 M CoCl2 0.63 M | ■ | ■ | ■ | ■ | SS6.25 CC0.63 QX H0.5 | ||
Na2SiO3 3.13 M CoCl2 1.00 M | ■ | ■ | ■ | ■ | SS3.13 CC1.00 QX H0.5 | ||
Na2SiO3 6.25 M CoCl2 1.00 M | ■ | ■ | ■ | ■ | SS6.25 CC1.00 QX H0.5 |
The photographs were analyzed with MATLAB in order to extract the main properties of the filaments. The image of each filament was first binarized and then skeletonized, which corresponds to thinning the width of the filament to a single pixel. The Euclidean length of the filaments (straight line distance between the two extremities of the filament) and their arc length (length of the filament measured along its path) were determined from the skeletonized image. The width of each filament was calculated from the binarized image. From the coordinates of the points of the thinned filament, it is possible to determine the parametric curves that define the path of the filament and with those calculate the curvature of the filament along its arc length.2 The average distance between turns was thus defined as the ratio between the total length of the filament and its number of turns. All these properties were averaged for the entire length of the filaments under the same experimental conditions. The speed of an active filament tip, , was taken by dividing the length of each filament by the time elapsed during its growth, determined by the photographs taken every two seconds.
III. EXPERIMENTAL OBSERVATIONS
Three main regimes are observed, in the range of injection flow rates tested, as presented in Fig. 1. No filaments are observed for low flow rates, with the precipitate growing in a flame shape instead. For intermediate flow rates, a single filament is formed, and eventually, multiple filaments grow simultaneously for high flow rates, in accordance with observations in horizontal Hele-Shaw cells.16,20 Haudin et al.15 described different types of structures found in the filament regime. F1 type filaments correspond to the narrow, fast growing tubes observed in all cases. F2 type filaments were also observed; these correspond to much wider structures with thicker walls than the regular F1 filaments. These wider structures were only observed in experiments with the higher concentration of sodium silicate (6.25 M) and were more common for higher injection flow rates. For lower flow rates, F2 structures are not observed at all, with small wider regions appearing as the flow rate is increased. For the higher flow rates tested, practically, only F2 filaments are observed. An example of these two structures in a single experiment is shown in Fig. 2. The results and modeling presented in this work only concern the F1 type filaments.
Time sequence of photographs taken during experiments with cobalt chloride and sodium silicate. Three regimes can be observed: no filaments in the top row, one filament in the middle row, and multiple filaments in the bottom row. It is possible to observe from these pictures that the cobalt chloride solution starts to leak from the membrane after some time, with the resulting precipitate spreading as shell-like structures. This leaking causes a decrease in and may eventually halt the growth of the filaments.
Time sequence of photographs taken during experiments with cobalt chloride and sodium silicate. Three regimes can be observed: no filaments in the top row, one filament in the middle row, and multiple filaments in the bottom row. It is possible to observe from these pictures that the cobalt chloride solution starts to leak from the membrane after some time, with the resulting precipitate spreading as shell-like structures. This leaking causes a decrease in and may eventually halt the growth of the filaments.
F1 and F2 type filaments in a vertical Hele-Shaw cell. Wide “lakes” with thick walls sporadically appear during the growth of the filament. Experimental conditions: SS6.25 CC0.63 Q1.0 H0.5. Photo taken 40 s after the onset of the filament.
F1 and F2 type filaments in a vertical Hele-Shaw cell. Wide “lakes” with thick walls sporadically appear during the growth of the filament. Experimental conditions: SS6.25 CC0.63 Q1.0 H0.5. Photo taken 40 s after the onset of the filament.
In order to obtain a better understanding of the effect of flow rate on the growth of a chemical garden filament, the range of flow rates where only one filament appears was studied more closely. Under these conditions, the flow of cobalt chloride inside the filament is known to be the same as the one imposed by the syringe pump. The main properties of the solutions are presented in Table II, together with the Reynolds number of the fluid inside the filament and its aspect ratio. The speed of the tip of the filaments and their width were both found to increase with increasing flow rate, as shown in Fig. 3. In addition, the filaments exhibit a straighter, less tortuous trajectory as their flow rate increases. Indeed, chemical garden filaments exhibit an erratic motion, changing direction after short rectilinear paths with typical length . The only moving part of this chemical garden is the active filament tip, as the filament stays in place during the experiment, without changing its width or position (the sides of the filament do eventually start leaking). This shows that the walls of the filament are rigid. As presented in Fig. 4, the average value of does tend to increase with the flow rate, in spite of the large experimental error. In agreement with this observation, the average angle between turns decreases with the flow rate. This parameter is directly connected to the tortuosity of the filament’s path since ,2 where is the Euclidean length of the filament and is the arc length.
Results obtained from the analysis of the filament images. (a) Average speed of the tip of the filaments. (b) Average width of the filaments. Error bars represent standard deviation.
Results obtained from the analysis of the filament images. (a) Average speed of the tip of the filaments. (b) Average width of the filaments. Error bars represent standard deviation.
Results obtained from the analysis of the filament images. (a) Average distance between turns of the filaments. (b) Average angle between the filament-motion and the direction with regard to the injection site. Error bars represent a standard deviation.
Results obtained from the analysis of the filament images. (a) Average distance between turns of the filaments. (b) Average angle between the filament-motion and the direction with regard to the injection site. Error bars represent a standard deviation.
Density and viscosity of the injected solutions of cobalt chloride, as well as the density differences and viscosity ratios relative to the host solutions of sodium silicate. Data taken from the literature.21 The range of Reynolds numbers for the fluid inside the filament is also shown for each chemical system, together with the range of aspect ratios, defined as AR = 2Rf/H.
. | SS 3.13 CC 0.63 . | SS 6.25 CC 0.63 . | SS 3.13 CC 1.00 . | SS 6.25 CC 1.00 . |
---|---|---|---|---|
ρCC (kg m−3) | 1070 | 1070 | 1110 | 1110 |
Δρ (kg m−3) | 140 | 350 | 100 | 310 |
μCC (mPa s) | 1.4 | 1.4 | 1.6 | 1.6 |
μSS/μCC | 2.6 | 28.6 | 2.3 | 25 |
Re | 12.1–33.4 | 10.7–47.9 | 10.6–44.0 | 10.4–42.6 |
AR | 2.8–5.7 | 4.5–7.9 | 2.9–4.5 | 4.9–7.1 |
. | SS 3.13 CC 0.63 . | SS 6.25 CC 0.63 . | SS 3.13 CC 1.00 . | SS 6.25 CC 1.00 . |
---|---|---|---|---|
ρCC (kg m−3) | 1070 | 1070 | 1110 | 1110 |
Δρ (kg m−3) | 140 | 350 | 100 | 310 |
μCC (mPa s) | 1.4 | 1.4 | 1.6 | 1.6 |
μSS/μCC | 2.6 | 28.6 | 2.3 | 25 |
Re | 12.1–33.4 | 10.7–47.9 | 10.6–44.0 | 10.4–42.6 |
AR | 2.8–5.7 | 4.5–7.9 | 2.9–4.5 | 4.9–7.1 |
One relevant experimental observation concerns the behavior of the filaments when an air bubble is attached to its tip. As shown in Fig. 5, rather than the usual erratic trajectory, the filament grows in a straight manner, which is consistent with results reported in horizontal Hele-Shaw cells.20 This observation shows that the changes in cannot be explained by the growing inertia of the injected fluid as the flow rates are increased; on the contrary, something must be changing periodically at the tip for its direction to be constantly changing. Furthermore, the Dean number for flow in these filaments is very low [; , where is the equivalent diameter], and as a result, the flow can be assumed to be completely unidirectional; the effect of the curves along the filament on the friction factor can thus also be neglected.22
Chemical garden filament with an air bubble attached to its tip (left) and without a bubble (right) for the same experimental conditions (SS6.25 CC1.00 Q2.0 H0.5). The Reynolds number is identical in both cases (, see Table II); however, the result is completely different: the bubble guided filament shows no turns, while the regular filament is continually changing direction with an almost “zig-zag” path. This strongly suggests that the precipitation reaction occurring at the tip is the main factor controlling the trajectory of the filament. The approximately constant distance between turns further suggests that some properties of the filament tip may be changing periodically. The filaments with a bubble occurred accidentally due to an incorrect connection of the feeding tube to the Hele-Shaw cell. It is possible to purposefully generate these bubble guided filaments by not completely filling the feeding tube before connecting it to the cell.
Chemical garden filament with an air bubble attached to its tip (left) and without a bubble (right) for the same experimental conditions (SS6.25 CC1.00 Q2.0 H0.5). The Reynolds number is identical in both cases (, see Table II); however, the result is completely different: the bubble guided filament shows no turns, while the regular filament is continually changing direction with an almost “zig-zag” path. This strongly suggests that the precipitation reaction occurring at the tip is the main factor controlling the trajectory of the filament. The approximately constant distance between turns further suggests that some properties of the filament tip may be changing periodically. The filaments with a bubble occurred accidentally due to an incorrect connection of the feeding tube to the Hele-Shaw cell. It is possible to purposefully generate these bubble guided filaments by not completely filling the feeding tube before connecting it to the cell.
Since for any given filament, the speed of the tip and the typical distance between turns are constant (until the walls of the structure start leaking), these can be combined into a single parameter, the frequency with which the filament tip changes direction,
This parameter is referred to as the frequency of oscillation. Chemical gardens have been found to oscillate before: when grown in a tank with a pellet, water flows into the membrane by osmosis, causing the internal pressure to increase and the chemical garden tube to rupture and the top, ejecting the metal salt solution flowing within. A new membrane forms around the ejected solution, and the cycle repeats. This mechanism causes the internal pressure of the structure to oscillate.23,24 A similar behavior is found when pellets are placed in the confined environment of a Hele-Shaw cell; these oscillations may even lead to the explosion of the chemical garden.17 Oscillations can also occur in chemical gardens grown via injection, in the “popping” and “budding” regimes.10,25 Under the co-flow of the reactants in a microfluidic reactor, in the budding regime, the droplets rupture periodically, imprinting orderly spaced membranes on the precipitate surface.26 The spacing between these membranes is dependent on the velocity of the injected metal salt solution.26 All these processes involve the rupturing of the precipitate membrane. Here, chemical garden filaments are found to oscillate without ruptures, under the mechanism proposed in Sec. IV.
It is clear from the experimental observations that something must be changing periodically at the tip—oscillating—to constantly throw the active filament tip in new directions. The method for estimating the frequency of oscillation, by measuring the speed of the tip and the distance between turns, is simple and follows directly from the main observation that motivates this work. Observing the oscillations with a camera would also be interesting, but any reliable measurement would require a camera with both a very high resolution and a very high frame rate of capture of images since the oscillations happen quickly and over a very small area. The pressure oscillations are also expected to be small, in the order of 5 mPa.20 The mathematical model presented below thus aims to explain this parameter , which can be determined experimentally and then compared with the theoretical predictions.
IV. MATHEMATICAL MODELING
The modeling of a chemical garden filament in a vertical Hele-Shaw cell follows the same principles applied to the horizontal configuration.20 The dynamics of a single active filament tip are described according to the variation of the concentration of a precipitate product between the two fluids, , and the pressure difference across the membrane, . Assuming the pressure outside the membrane is steady, is thus the rate of change of the internal pressure at the tip. For the fluid outside the filament, the Reynolds number is low (); therefore, quasi-steady flow of the silicate can be assumed. The concentration of the product is governed by the balance between its formation as a result of the precipitation reaction and its dilution due to the spreading of the outflow. The gauge pressure at the tip depends on the change in the volume of the fluid inside the filament and the deformation of the membrane and buoyancy. Hence, the system evolves with time according to
The first term of Eq. (2a) represents the diffusive supply of cobalt ions with coefficient and across a length scale , to the outer surface of the membrane, where precipitation occurs. Indeed, as cobalt ions are consumed by the precipitation reaction, a concentration boundary layer of thickness is formed at the tip of the filament. The precipitate particles form a membrane of thickness . The concentration of product is, therefore, the concentration of these particles across the perimeter of the membrane at the tip of the filament. The concentration of cobalt ions, , is assumed to be identical to that delivered at the source. While it is likely that it is in reality lower than assumed due to consumption by a chemical reaction and leaking through the membrane, this difference is considered negligible. The membrane is initially extremely thin (in the order of micrometers), and its growth with time is negligible in 10–20 s of an experiment.27,28 In a similar way, leaking of cobalt chloride on the sides of the membrane only starts after considerable growth of the filament, and once it starts, the results are not considered from that point forward due to the change in the flow rate of the filament. The assumption of constant cobalt chloride concentration during each experiment is supported by the fact that the color of the filaments does not change during the duration of the experiment. As a result, is one of the parameters of each experiment, controlled by the concentration of the solution of cobalt chloride injected into the cell.
Since is the rate of change of a precipitate along the perimeter of the membrane, if it is nonzero, then the concentration of the product is oscillating at a fixed location: in that sense, the membrane can be said to be “opening and closing,” not due to ruptures but due to a variation in the precipitate concentration. Meanwhile, since the walls of the filament are parallel to the direction of advection of fluid, they have time to grow thicker and are thus much more rigid.
The second term of the equation refers to the spreading of the product across the curvature of the membrane, . As shown in the schematic of Fig. 6, the flow of the host solution is expected to spread the newly formed precipitate particles away from the tip to the sides of the filament, thus decreasing their concentration at the tip. This continuous spreading of the product will keep the membrane at the tip very thin, and its center is likely to always be void of any precipitate. The radius of curvature of this membrane will thus be a fraction of the width of the filament. Since the exact radius of curvature is unknown and is likely to change for different experimental conditions, the curvature is calculated as , where is the gap of the Hele-Shaw cell. is always smaller than , as shown in Table II; this allows for a simplification of the model without introducing a significant error. The Heaviside step function indicates that the precipitate is only formed when there is outflow of cobalt ions. is the volumetric flow rate of a cobalt chloride solution injected into the filament, and is the speed of the tip of the filament. The elastic deformation of the precipitate membrane is given by , where is the radius of the filament, is Young’s modulus, and is the Poisson ratio for the precipitate product that constitutes the membrane material. This deformation is considered analogous to the upward buckling of a thin plate, embedded on both ends, due to a uniform horizontal load,29 and assumes that there is no significant friction between the membrane and the Hele-Shaw cell plates. This mechanism is illustrated in Fig. 6. is the cross-sectional area for outflow, assumed constant as . The coefficient represents the pressure drop due to the viscous flow along the length of the filament, calculated with the following equation for rectangular ducts:30
where is the dynamic viscosity of the fluids without a product. The term represents the effect of buoyancy,19 where corresponds to the density difference between the two fluids and is the acceleration due to gravity. The cobalt chloride solution flows across the precipitate membrane of thickness and permeability , a porous medium. The speed of the tip of the filament is thus determined by Darcy’s law,
The Heaviside step function ensures that and . Thus, as the concentration of the product increases, the speed decreases and ceases when the critical product concentration is reached. The dependence of the outflow speed on the concentration of the product was first proposed by Wagatsuma et al.16
Schematic of a two-dimensional filament. The frame of reference is the filament, with the black arrows representing the flow of the host solution (sodium silicate) relative to the tip of the filament. The red wavy arrows represent the diffusive supply of cobalt ions to the outer surface of the membrane, leading to the precipitation reaction and the formation of more product. The relative flow of sodium silicate spreads the product away from the center of the tip of the filament to its sides. The parameter represents the concentration of the product over the perimeter of the membrane at the tip, the dashed line in the figure. These two effects increase and decrease , respectively, ultimately causing its value to oscillate. Experimental observations show that the side walls of the filament are rigid and do not change their position. On the other hand, the membrane at the tip is thin and is continually deformed due to pressure oscillations (the three dashed lines shown here represent the various positions the membrane may have as it is deformed). This deformation is modeled as analogous to the buckling of a thin 2D plate embedded on both ends; assuming no friction with the Hele-Shaw cell plates, these points are fixed, as shown in the schematic. The actual chemical garden membrane is not actually two-dimensional; however, the 2D approximation is considered valid since the width of the filament is always at least twice as large as the Hele-Shaw cell gap (see Table II).
Schematic of a two-dimensional filament. The frame of reference is the filament, with the black arrows representing the flow of the host solution (sodium silicate) relative to the tip of the filament. The red wavy arrows represent the diffusive supply of cobalt ions to the outer surface of the membrane, leading to the precipitation reaction and the formation of more product. The relative flow of sodium silicate spreads the product away from the center of the tip of the filament to its sides. The parameter represents the concentration of the product over the perimeter of the membrane at the tip, the dashed line in the figure. These two effects increase and decrease , respectively, ultimately causing its value to oscillate. Experimental observations show that the side walls of the filament are rigid and do not change their position. On the other hand, the membrane at the tip is thin and is continually deformed due to pressure oscillations (the three dashed lines shown here represent the various positions the membrane may have as it is deformed). This deformation is modeled as analogous to the buckling of a thin 2D plate embedded on both ends; assuming no friction with the Hele-Shaw cell plates, these points are fixed, as shown in the schematic. The actual chemical garden membrane is not actually two-dimensional; however, the 2D approximation is considered valid since the width of the filament is always at least twice as large as the Hele-Shaw cell gap (see Table II).
While the contribution of buoyancy into the internal pressure of the filament is taken into account, no buoyant convection effects are assumed to occur at the tip of a filament in these experiments. Convection occurs when the Rayleigh number is above a critical value: , where is the characteristic length scale.31–33 For the system studied here, will at most reach a value of approximately 1.1, far below the threshold for convection to occur. Additionally, even if the Rayleigh number was high enough, the characteristic time scale for buoyant convection to start over a length scale is expected to follow , which simplifies as .31–33 Considering the length scale to be the width of the precipitate membrane, approximately 5 , then s. On the other hand, the time scale for the filament tip to cross that same distance, considering the slowest moving filament tips shown in Fig. 3(a), is around s, almost ten times lower than . The filaments thus move too quickly to allow buoyant instabilities to affect the precipitate membrane.
Substituting Eq. (4) into Eq. (2) and non-dimensionalizing using the scales for time , pressure , and concentration , we obtain the following simplified coupled equations:
The system is described by three non-dimensional groups:
is a non-dimensional rate of accumulation of a precipitate;
is a non-dimensional volumetric injection rate of a metal ion;
measures the pressure drop along the filament; and
measures the buoyancy effect on the system. Linear stability analysis of the governing equations (5a) and (5b) around the steady state solution , reveals that the system can be unstable, oscillatory, and stable. The unstable regime occurs when the critical product concentration is reached, and the filament growth is blocked. Experimentally, this occurs for low injection flow rates (low ) and a slow radial spreading of precipitate is observed, with no filaments being formed.
The filaments appear under the oscillatory regime; and are lower than in the previous case and oscillate. As a result, the membrane is constantly being deformed, with the concentration of the precipitate changing periodically across its perimeter; this causes the filament to grow in “bursts” and to adopt the zig-zag trajectory usually observed. It is important to note that this mechanism does not involve the rupturing of the membrane. One possible reason behind this zig-zag pattern is the pressure difference between the interior and exterior of the filament being highest at the edges of the membrane (the fixed points shown in Fig. 6; these edges have been found to be the areas of highest stress in a chemical garden24). The external pressure should be the lowest at these points since the external fluid speeds up from the stagnation point at the vertex to the edges. Assuming the internal pressure is uniform, a pressure gradient would be established that would preferentially drive fluid to the edges, changing the direction of the evolution of the filament.
The stable regime corresponds to injection rates of a metal salt solution above a certain threshold, and the internal pressure and concentration of the product remain constant. In theory, this would correspond to a filament growing in a straight line. In practice, the injection flow rates required are extremely high, and on the contrary, multiple filaments are observed in the Hele-Shaw cell. The stable regime is thus not observable experimentally, at least with the setup used in this work. From stability analysis, the frequency of oscillation of a growing filament is calculated by
where
These expressions were determined with the Wolfram Alpha software. With the method presented in Sec. II, the theoretical predictions can be compared with the experimental results. Among the various variables that define the groups , , , and , the only unknown parameters are , , and . As a result, these are the fitting parameters of the model to the experimental measurements of the frequency of oscillation.
V. RESULTS AND DISCUSSION
The pressure and product of reaction concentration oscillations at the tip of the filament predicted by the model are expected to lead to the changes of direction of the filaments observed in the experiments. The frequency of oscillation can be obtained from the results with Eq. (1) and compared with the theoretical prediction by fitting the data to the model, shown in Fig. 7. Commercial software MATLAB was used to determine the best fitting values of the non-dimensional groups; the variable was then obtained from the group , was calculated from the value of , and finally, was solved from group . can also be estimated from , leading to very similar results. Thus, the properties of the membrane are m, m, and mol m. Young’s modulus and Poisson’s ratio for the membrane were assumed to be Pa and 0.5, respectively, based on previous studies of chemical garden membranes of a similar size.24 For the Hele-Shaw cell gap of 0.5 mm used here, the curvature is 4000 m.
Comparison between the experimental results (symbols) and the model predictions (lines). These represent the fitting of Eqs. (6)–(8) to the experimental results of shown in Sec. III. (a) Frequency of oscillation as a function of flow rate in the single filament regime. Shaded regions represent the transition regions between regimes; at lower flow rates, no filaments are present; above a certain threshold flow rate, multiple filament tips are active. (b) Same data as shown in (a), zoomed in on the single filament regime area for easier analysis. The experimental results are in good agreement with the model, with the exception of the SS6.25 CC1.00 QX H0.5 case. The model only takes into account the concentration of cobalt chloride due to the host solution of sodium silicate being in stoichiometric excess. These results suggest that the concentration of sodium silicate also influences the dynamics of the system and the frequency of oscillation.
Comparison between the experimental results (symbols) and the model predictions (lines). These represent the fitting of Eqs. (6)–(8) to the experimental results of shown in Sec. III. (a) Frequency of oscillation as a function of flow rate in the single filament regime. Shaded regions represent the transition regions between regimes; at lower flow rates, no filaments are present; above a certain threshold flow rate, multiple filament tips are active. (b) Same data as shown in (a), zoomed in on the single filament regime area for easier analysis. The experimental results are in good agreement with the model, with the exception of the SS6.25 CC1.00 QX H0.5 case. The model only takes into account the concentration of cobalt chloride due to the host solution of sodium silicate being in stoichiometric excess. These results suggest that the concentration of sodium silicate also influences the dynamics of the system and the frequency of oscillation.
The model is in good agreement with the experimental results for all chemical systems tested except for the sodium silicate 6.25 M/cobalt chloride 1.00 M pair. In fact, the frequency of oscillation for the experiments with high concentration of sodium silicate is identical, as the results are within the error bars. Qualitative differences can also be observed in the experiments when changing the concentration of sodium silicate, such as the appearance of F2 structures. In addition, during the experiment run with low flow rates and no filaments, the progressive growth of precipitate can be seen to adopt a different color when the concentration of sodium silicate is changed, as shown in Fig. 8.
Results for the “no filament” regime for two chemical systems with the same cobalt chloride concentration and the injection flow rate, but different sodium silicate concentrations. (a) SS3.13 CC0.63 Q0.1 H0.5. (b) SS6.25 CC0.63 Q0.1 H0.5. In this regime, the concentration of product reaches its critical value , blocking the filament growth. Instead, internal pressure leads to a slow leak across the entire area of the membrane, leading to the accumulation of the product and the spreading of the precipitate structure. Even though the only difference between the two experiments shown in the figure is the concentration of the excess reactant, the structure formed with a higher concentration of sodium silicate exhibits a darker color, possibly indicating a more concentrated precipitate. The mechanism of the formation of a precipitate is likely to be influenced by the concentration of both reactants, independently of which is the limiting reagent. As a result, the properties of the membrane at the tip of a growing filament are probably not only dependent on the metal salt used.
Results for the “no filament” regime for two chemical systems with the same cobalt chloride concentration and the injection flow rate, but different sodium silicate concentrations. (a) SS3.13 CC0.63 Q0.1 H0.5. (b) SS6.25 CC0.63 Q0.1 H0.5. In this regime, the concentration of product reaches its critical value , blocking the filament growth. Instead, internal pressure leads to a slow leak across the entire area of the membrane, leading to the accumulation of the product and the spreading of the precipitate structure. Even though the only difference between the two experiments shown in the figure is the concentration of the excess reactant, the structure formed with a higher concentration of sodium silicate exhibits a darker color, possibly indicating a more concentrated precipitate. The mechanism of the formation of a precipitate is likely to be influenced by the concentration of both reactants, independently of which is the limiting reagent. As a result, the properties of the membrane at the tip of a growing filament are probably not only dependent on the metal salt used.
This suggests that in spite of being in stoichiometric excess in all cases, the concentration of sodium silicate still influences the properties of the precipitate membrane at the tip of the filament, which affects the dynamics of the system. In the model, the term in Eq. (4) can be seen as an “effective” permeability of the membrane since the concentration of the product will affect the outflow speed. It is possible that changes in the concentration of the host solution affect the value of , initially assumed to be equal across all experimental conditions. Another possibility that might contribute to the discrepancies between the model and the experiments is the large increase in viscosity for concentrated sodium silicate solutions. For a concentration of 3.13 M, these solutions have a viscosity of 3.6 mPa s; however, this property increases more than ten times to approximately 40 mPa s when the concentration is doubled.21 The model, as shown in Eq. (4), assumes the viscosity of the liquid as it flows through the porous medium of the membrane to be equal to that of the cobalt chloride solution; however, if the membrane includes entrapments of sodium silicate as the filament progresses, this viscosity will surely be different. It will, therefore, depend on the viscosity of both solutions and the size of these potential entrapments.
Given the previous work on horizontal confined filaments,20 it is relevant to compare the results for the two Hele-Shaw orientations used in this investigation. For the same experimental conditions, the filaments behave slightly differently when grown in a horizontal or vertical orientation; the frequency of oscillation is not the same in both cases. This slight variation is attributed to the effect of gravity on the pressure difference across the membrane. When comparing the non-dimensionalized model equations for the two cases, the new group is merely added to the pressure equation for the horizontal direction. Indeed, the term can be simplified into a single group, , making all the model equations mathematically identical to the ones for the horizontal case. This means that for a couple of horizontal and vertical experiments with the same values of , and or , the model predicts the same behavior for the filaments; the non-dimensionalized frequency of oscillation, should, therefore, be identical. Table III compares these values for a couple of different cases. These data suggest that the model accurately captures the oscillatory behavior of the system, and the growth of confined filaments vertically rather than horizontally simply adds the effect of gravity to the pressure balance. A change in the metal salt solution injected into the cell yielded minor changes in the dynamics of horizontal filaments, and the different membrane properties are estimated to have practically identical properties. It is very likely that the same similarity exists in the vertical orientation as well.
Comparison of the system’s behavior in the horizontal and vertical cases. The non-dimensional groups and f.ts were all calculated with the experimentally measured variables. While the values of M, N and W or Wv are not exactly the same in all three examples, they are as close as could be obtained from the available experimental results; we consider that they are close enough to allow for a comparison. As expected, the non-dimensional frequency of oscillation is practically the same, which is consistent with the non-dimensional groups being identical.
. | . | M . | N . | W or Wv . | f (s−1) . | f.ts . |
---|---|---|---|---|---|---|
Horizontal | SS6.25 CC0.63 Q0.5 H0.5 | 0.082 | 0.23 | 0.12 | 2.65 | 0.012 |
Horizontal | SS3.13 CC1.00 Q1 H0.5 | 0.066 | 0.28 | 0.16 | 5.95 | 0.014 |
Vertical | SS 3.13 CC1.00 Q1 H0.5 | 0.080 | 0.32 | 0.12 | 5.16 | 0.014 |
. | . | M . | N . | W or Wv . | f (s−1) . | f.ts . |
---|---|---|---|---|---|---|
Horizontal | SS6.25 CC0.63 Q0.5 H0.5 | 0.082 | 0.23 | 0.12 | 2.65 | 0.012 |
Horizontal | SS3.13 CC1.00 Q1 H0.5 | 0.066 | 0.28 | 0.16 | 5.95 | 0.014 |
Vertical | SS 3.13 CC1.00 Q1 H0.5 | 0.080 | 0.32 | 0.12 | 5.16 | 0.014 |
VI. CONCLUSIONS
In this work, novel experimental data are presented for the growth of chemical garden filaments in a vertical Hele-Shaw cell, with the sodium silicate/cobalt chloride chemical system. Four different pairs of concentrations were tested, and the experimental photographs were analyzed to obtain the main characteristics of the filaments. A single filament is formed under a particular range of injection flow rates, exhibiting increasing speed, width, and typical distance between turns as the flow rate is increased. The filaments are essentially found to adopt a less tortuous trajectory as the flow rate is increased. Above a certain threshold flow rate, multiple active filament tips are observed. For higher sodium silicate concentrations, F2 structures with thick walls are seen together with the generic F1 filaments. These structures become more prevalent at higher flow rates, eventually replacing the thinner filaments entirely. A mathematical model was compared with the experimental results, with the estimation of the properties of the precipitate membrane at the tip of the filament. Practical observations and an analysis of the model results suggest that the concentration of the host solution of sodium silicate still influences the reaction in spite of being in excess. Further experimental studies are necessary to determine whether this is due to the formation of precipitate structures with different properties or the possibility of silicate entrapments in the membrane. This will allow the development of a more accurate model.
ACKNOWLEDGMENTS
L.A.M.R. gratefully acknowledges funding from the Fundação para a Ciência e Tecnologia (FCT), Portugal (Grant No. SFRH/BD/130401/2017). The authors acknowledge the contribution of the COST Action chemobrionics CA17120.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available within the article.