# Spatial Heterogeneity and Imperfect Mixing in Chemical Reactions: Visualization of Density-Driven Pattern Formation

Authors: Sabrina G. Sobel; Harold M. Hastings; Matthew Testa
Journal: Research Letters in Physical Chemistry
 (2009)

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## Abstract

Imperfect mixing is a concern in industrial processes, everyday processes (mixing paint, bread machines), and in understanding salt water-fresh water mixing in ecosystems. The effects of imperfect mixing become evident in the unstirred ferroin-catalyzed Belousov-Zhabotinsky reaction, the prototype for chemical pattern formation. Over time, waves of oxidation (high ferriin concentration, blue) propagate into a background of low ferriin concentration (red); their structure reflects in part the history of mixing in the reaction vessel. However, it may be difficult to separate mixing effects from reaction effects. We describe a simpler model system for visualizing density-driven pattern formation in an essentially unmixed chemical system: the reaction of pale yellowFe3+ with colorless SCN− to form the blood-red Fe(SCN)2+ complex ion in aqueous solution. Careful addition of one drop of Fe(NO3)3 to KSCN yields striped patterns after several minutes. The patterns appear reminiscent of Rayleigh-Taylor instabilities and convection rolls, arguing that pattern formation is caused by density-driven mixing.


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## Body

## 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
Temp (C°)Ionic strengthCalc. density (g/cm3)2335001.00021500.9975682600.9968142000.9982343000.995678The 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 SCN−,
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 μm
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 SCN− (clear), are mixed, producing a deep reddish color where Fe3+ and
SCN− 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 μL)
of 0.1 M Fe3+ (as Fe(NO3)3) to a 1–4 mm deep
solution of 0.02 M SCN− (as KSCN) optionally containing a very small amount of surfactant (which had no
apparent effect).  The deep reddish color
where Fe3+ and SCN− combine to form FeSCN2+ clearly labels interfaces between Fe3+-dominated
and SCN−-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
(ρ1−ρ2).  In our model a drop of radius 50 μm
rapidly reaches terminal velocity of ~50 μm/s,
making the natural time scale (time to fall 2 mm) ~2 mm/ (50 μm/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.



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*Source: 350424-2009-05-06.xml*