1. Introduction

Good mixing is a major challenge in many areas of chemistry and chemical engineering, ranging from continuously stirred tank reactors (CSTRs) [1] to industrial level processes [2], where imperfect mixing can adversely affect product quality and yield. Many of us are familiar with mixing problems in making bread, stirring paint, mixing gasoline and air in internal combustion engines, and mixing driven by density differences (salt water-fresh water convection) but even mixing of ordinary liquids can pose challenges. (The density of pure water is 1000 kg/m³. Ocean water is more dense because of the salt in it. Density of ocean water at the sea surface is about 1027 kg/m³. There are two main factors that make ocean water more or less dense than about 1027 kg/m³. From http://www.csgnetwork.com/h2odenscalc.html we derived Table 1 (accessed 1/2009)).

Table 1: Densities of salt water and pure water at various temperatures

The effects of imperfect mixing are perhaps most dramatically evident in excitable chemical systems [1, 3] such as the BZ reaction [4–9] and the chlorite-iodide reaction [10–12]. An excitable chemical reaction is a reaction in which suitable small perturbations from steady state generate large excursions (excitations) before the system returns to steady state. Many excitable reactions display auto-oscillatory behavior in which sustained oscillations about an unstable steady state are observed. The effects of imperfect mixing can be seen readily in the unstirred, ferroin-catalyzed Beloushov-Zhabotinsky (BZ) reaction in a Petri disk, a quasi-two-dimensional (2D) system [3–8]. The catalyst in this reaction, ferroin/ferriin, also serves as an indicator; blue/oxidized at high [ferriin] and red/reduced at low [ferriin]. After an induction period of several minutes, one sees the “spontaneous” formation of target patterns of blue/oxidized rings moving outward from oscillatory (red/blue) centers into a red/reduced background.

Menzinger and Dutt [1] first described the effects of imperfect mixing upon reactions involving excitable media. We have explicitly demonstrated nonrandom spatiotemporal order in target formation in the unstirred BZ reaction, with targets preferentially occurring in excitable regions near existing targets, but too far from pre-existing targets to be generated by wave propagation or chemical diffusion [3]. We explained the observed spatiotemporal order in terms of mixing heterogeneities in preparing the 2D reaction medium likely because of the sensitivity of the dynamics to very small perturbations at the onset of oscillation [13, 14]. In conditions of almost complete mixing, striped patterns reminiscent of convection rolls with spacing approximately equal to the depth of the medium were often observed (Figure 1, panel (a)). Here we describe an even simpler model system which displays analogous pattern formation driven by density differences, c.f. [15–17].

Figure 1: (a) Excitable BZ reaction medium, stirred after initial pattern formation, 2.1 mm depth. From [9, Figure 2c]. Scale bar: alternating 2.1 mm lines and spaces. (b) Chemical model: Fe³⁺ and , 2 mm depth. Scale bar: alternating 2 mm lines and spaces. Striped patterns formed outside the visible droplet boundary (lower left of figure). The color corresponds to the concentration of FeSCN²⁺. Resolution 30 m in photos (b) through (f). Time: about two minutes after addition of one drop of Fe³⁺ solution. (c), (d) Evolution of panel (b). Same experimental run, 45 and 30 second earlier. Note pattern initiation 2 mm behind visible wavefront (of diffusion wave) and effect of diffusion in blurring visible boundaries as concentration gradients are reduced. (e) Same model, but 3 mm depth, early stage of pattern formation. Note visible droplet boundary in center of figure, and larger scale that corresponds to greater depth as in (b) through (d). (f) Same model, 1 mm depth, early stage of pattern formation. Scale bar: 1 mm. Note fine structure (1 mm) transverse to a spatial filament of the reaction, and initial excitation of 1 mm scale.

2. Pattern Visualization with Fe(SCN)²⁺

The present experiment arose from trying to understand how slight density differences might drive this pattern formation. Since BZ dynamics are relatively complex, we sought a simpler, minimal example to better visualize and understand the effects of mixing heterogeneity. We used a simple chemical reaction, in which two nearly colorless solutions, one containing Fe³⁺ (pale yellow) and the second containing (clear), are mixed, producing a deep reddish color where Fe³⁺ and combine to form the complex ion FeSCN²⁺ in an essentially irreversible reaction. We made a simple modification—instead of mixing the two solutions, we added a small drop (1 L) of 0.1 M Fe³⁺ (as Fe(NO₃)₃) to a 1–4 mm deep solution of 0.02 M (as KSCN) optionally containing a very small amount of surfactant (which had no apparent effect). The deep reddish color where Fe³⁺ and combine to form FeSCN²⁺ clearly labels interfaces between Fe³⁺-dominated and -dominated regions. We found patterns reminiscent of convective rolls with spacing equal to the depth of the corresponding reaction medium (Figure 1, panels (b)–(e), below, similar to patterns seen under conditions of almost complete mixing in the BZ reaction, panel (a)). Note that the Fe(NO₃)₃ solution is denser than the KSCN solution: 1.010(4) versus 1.001(2) g/mL.

The resulting patterns demonstrate behavior expected in the Rayleigh-Taylor instability [18, 19], which occurs when a heavier fluid (here the Fe(NO₃)₃ solution) is placed on top of a lighter fluid (here the KSCN solution). Gravitational forces drive the heavier solution downward through the lighter solution in an unstable, turbulent manner, until concentration (and thus density) differences are destroyed by diffusion. In our model system, one sees initial small scale instabilities (panel (f)) which are gradually destroyed by diffusion as longer length scales are excited, with one added twist—selection of a length scale corresponding to the depth of the reaction medium after a time delay a few multiples of the time it takes a drop to fall the depth of the solution. From Stokes law, the net force on a small drop depends upon its volume, velocity, and density difference (). In our model a drop of radius 50 m rapidly reaches terminal velocity of 50 m/s, making the natural time scale (time to fall 2 mm) 2 mm/ (50 m/s) = 40 second, consistent with the formation of relatively stable patterns over 3 minutes (4 times this natural time scale).

3. Conclusion

This simple minimal model, in a nonexcitable system, demonstrates excitation of the most unstable mode (the only one that survives over minutes). It visually demonstrates the persistence of spatial heterogeneities in liquids. The problem of mixing is important in real-world situations often encountered by chemical engineers (industrial chemical reactions), process/flow engineers, and even fields such as hydrology and geoscience–temperature and salinity differences can cause density differences in water similar in magnitude to the density differences in this experimental model. This simple model provides a mechanism to experimentally visualize and study the patterns formed by density differences.

Acknowledgments

This material is based upon work supported by the US Department of Energy under Award no. DE-FG02-08ER64623 for the Hofstra University Center for Condensed Matter Research and by the US National Science Foundation Grant CHE-0515691. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.