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Simple, simpler, simplest: Spontaneous pattern formation
in a commonplace system
Evelyn H. Strombom
Swarthmore College, Swarthmore, Pennsylvania 19081
Carlos E. Caicedo-Carvajal
3D Biotek, North Brunswick, New Jersey and Rutgers University, Piscataway, New Jersey 08902
N. Nirmal Thyagu, Daniel Palumbo, and Troy Shinbrota)
Rutgers University, Piscataway, New Jersey 08854
(Received 21 June 2011; accepted 14 April 2012)
In 1855, Lord Kelvin’s brother, James Thomson, wrote a paper describing “certain curious
motions” on liquid surfaces. In the present paper, we describe several curious motions produced in
the simplest possible manner: by introducing a droplet of food coloring into a shallow dish of
water. These motions include the spontaneous formation of labyrinthine stripes, the periodic
pulsation leading to chaotic stretching and folding, and the formation of migrating slugs of
coloring. We use this simple experiment to demonstrate that the formation of ordered macroscopic
patterns is consistent with the requirement of the second law of Thermodynamics that microscopic
disorder must increase. This system is suitable for undergraduate experimentation and can be
modeled by advanced students in a straightforward finite difference simulation that reproduces the
labyrinths and other patterns. VC 2012 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4709384]
I. INTRODUCTION
The second law of Thermodynamics dictates that disorder
must increase: when we add dye to water, we expect it to
mix and not to separate. Yet we see ordered patterns wherever we care to look. In Biology, we see stripes on zebras
and the intricate regularity of peacock feathers. In Chemistry, we see perfectly regular crystals that lead to the iridescence of opals and other precious gems. And in Physics, we
see periodic ripples of sand on riverbeds and perfectly hexagonal convection patterns in a pan of heated oil. Understanding how these patterns can form in concert with
Thermodynamics has occupied many giants of scientific inquiry. Erwin Schrödinger described how life achieves order
in the presence of the second law,1 and many Nobel Prizes
have been awarded for deepening our understanding of patterns in Nature; examples include de Gennes for revealing
how patterns are formed in liquid crystals, and Lewis,
Nüsslein-Volhard, and Wieschaus for elucidating how patterning is controlled during embryonic development.
Perhaps the deepest and most influential work in the field
was provided by Turing,2 who in 1952 described a mechanism in which diffusion—the mixing engine by which the
second law operates—when combined with reaction can lead
to the spontaneous formation of a rich variety of different
patterns, including stripes, spots, hexagons, and other more
elaborate structures.3 The intrinsic beauty of these patterns
has stimulated research in a broad spectrum of scientific subspecialties, and this research has been facilitated by descriptions of simple archetypal systems that can clarify how
complex patterns are formed in the presence of the second
law.4
In the present paper, we describe a commonplace experiment that involves precisely and literally the example that
we began our discussions with: adding dye to water. As we
will describe, although it seems self-evident that the dye
should mix, in fact it separates and counter-intuitively forms
intricate and complex patterns similar to those seen in much
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Am. J. Phys. 80 (7), July 2012
http://aapt.org/ajp
more complicated systems. Some of the patterns observed
are displayed in Fig. 1 and include labyrinthine stripes [Fig.
1(a)], chaotic stretching and folding [Fig. 1(b)], and an intriguing new state in which a “slug” spontaneously migrates
across a container [Fig. 1(c)]. Through experiments, we
show that the pattern-formation mechanism in this system
can be understood using the most elementary of methods,
and through simple simulations, we show that—as revealed
by Turing—these patterns actually depend on diffusion,
which tends to increase entropy in complete accord with the
second law.
The experimental system consists of dropping an aliquot of food coloring into a thin layer of deionized water.
Existing work is plentiful on convective and other patterns
in deep fluid containers5–7 and several authors have produced stunning photographs of surface-tension-mediated
flows;8 however, comparatively little work has been presented on patterns in thin layers of this kind. The literature
also contains a few studies that consider doubly diffusive
convection (of heat and density) in shallow layers,9 and
patterns termed vermiculated rolls10 have been described
in the literature as being associated with evaporative convection of volatile fluids such as acetone or benzene
beneath thin layers of non-volatile materials. Few such
studies are suitable for teaching laboratories or simple
classroom demonstrations, and few systems exhibit as
wide a variety of very different patterns as are shown in
Fig. 1. The system described here provides an attractive
and exceedingly simple illustration of complex patterning
behaviors, is amenable to straightforward simulation, and
is comprehensible enough for secondary school while containing unanswered questions appropriate for undergraduate study.
This paper is organized as follows. In Sec. II, we describe
the materials and methods used, which we follow in Sec. III
with experimental results. Then in Sec. IV, we describe a
computational model that reproduces many of the features
seen in experiment, and in Sec. V we conclude.
C 2012 American Association of Physics Teachers
V
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578
ments, a small quantity of powdered fluorescein was mixed
with automotive antifreeze.11
C. Water layer and dye mixing
Deionized water was added to a 9-cm polystyrene Petri
dish to make a 4-mm water layer. This depth was chosen
after several trials with other depths ranging from 3-mm to
8-mm. Depths were established by metering fixed volumes
of water by pipeter, and dividing by the known area of the
dish. Depending on the experiment, a 37–50-ll drop of food
coloring or fluorescent dye solution was released at the center of the dish from a height of about 2 mm above the water
surface; the height was kept small to minimize splashing or
other transient effects.
D. Surface tension measurement
Fig. 1. (Color online) Three patterns from a dye droplet in a shallow dish of
water. (a) Successive enlargements of a fully developed labyrinthine pattern;
(b) chaotic stretching and folding (false colored); (c) pulsating and migrating
slug of dye moving in the direction of the arrow. Panels (a) and (b) show
commercial food coloring added to a 4-mm deep layer of water, and panel
(c) shows commercial antifreeze mixed with fluorescein added to a 4-mm
deep water layer.
II. METHODS
A. Materials and reagents
We use commercially available materials and reagents
including deionized water, McCormick food coloring (propylene glycol (density:1.04 g/ml) in water), sodium dodecyl
sulphate (SDS) detergent, 95% ethanol, automotive antifreeze (ethylene glycol (density:1.11 g/ml) and fluorescein),
CorningTM 9-cm polystyrene Petri dishes, a regulated pipeter
(Drummond Pipet-Aid XP), virgin glass micro-hematocrit
tubes (VWR International, LLC: 0.55 6 0.05 mm ID), a goniometer (Ramé-Hart Instrument Co.) to measure surface
tension, and an ultraviolet lamp to induce fluorescence.
B. Dye solutions
For non-fluorescent experiments, McCormick food coloring was mixed with deionized water in concentrations of
20%, 40%, 60%, and 80% (v/v). For fluorescent experi-
Surface tension measurements were performed using the
capillary tube and goniometer methods. Capillary height
measurements were taken for food coloring solutions at
20%, 40%, 60%, and 80% (v/v) in deionized water, and goniometer measurements were taken at 2%, 4%, 6%, 8% (v/v)
in deionized water. For the capillary measurements, we prepared ten identical, virgin glass micro-hematocrit tubes by
rinsing them in ethanol, removing excess liquid and then
allowing them to dry for an hour. We then rinsed the tubes
with the solution being tested, removed excess liquid again,
and then inserted the tubes into ten identical microcentrifuge
containers, each filled with 200 ll of solution. Finally, the
capillary rise height was measured by photographing the
tubes at high resolution using a macro lens (Nikon D90 camera with AF Micro Nikon 60-mm lens) alongside calibrated
gratings. Three independent experiments were performed at
each concentration using the capillary height method.
III. EXPERIMENTAL RESULTS
A. Effects of surface tension
We begin our discussion with an examination of the labyrinthine patterns shown in Fig. 1(a). As shown in the time series of Fig. 2, these patterns emerge spontaneously from a
nearly uniform initial state obtained after a droplet of food
coloring is introduced into a shallow dish of water. In the
top-left panel of this figure, we show a state immediately
Fig. 2. (Color online) Time series of dye droplet on surface of shallow dish of water. Notice that as time progresses the dye does not diffuse. Rather, slight variations
in dye concentration intensify. An emerging localization of dye is identified by black arrows, and enlarged globular “feet” (see text) are highlighted in dotted circles.
Note that the scale bar, bottom left corner, coincides with the depth of the water layer. A movie of the evolution shown is available as supplementary material (Ref.
27) (enhanced online) [URL: http://dx.doi.org/10.1119/1.4709384.1] [URL: http://dx.doi.org/10.1119/1.4709384.2] [URL: http://dx.doi.org/10.1119/1.4709384.3].
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Am. J. Phys., Vol. 80, No. 7, July 2012
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579
after introduction of the dye droplet, visible as a dark spot at
the center of the Petri dish. The dark spot contains the bulk
of the dye, which has sunk to the bottom of the dish, while
the remainder of the dye spreads rapidly over the surface.
This spreading is consistent with the behavior one would
expect if the dye reduces the surface tension of the water. In
this case, the higher-tension clear water would pulls the
lower-tension dye outward toward the container walls.12 In
separate experiments, described in Sec. II.C we find that if a
small amount of surfactant is added to the water, none of the
patterns shown here appear and the dye uneventfully falls to
the bottom of the dish. Conversely, adding surfactant to the
dye causes the dye to cover the entire water surface and
thereafter to rapidly mix throughout the water volume. Other
experiments performed using different surfactants, temperatures, and water depths are described in Sec. C.
For the time being, we note that added surfactant destroys
the effects described, and hence surface tension must play a
significant role—at a minimum in causing the initial aliquot
of dye to spread across the surface (upper left panel of Fig.
2). By the same token, because surface tension causes the
dye to spread initially, it seems paradoxical that dye isn’t
thereafter spread by surface tension, but instead concentrates
into localized patterns as shown after 15, 30, and 60 s in the
panels to the right of Fig. 2. The concentrated dye is highlighted in enlarged views in the lower panels. The localized
patterns become fully developed as shown in Fig. 1(a) within
about 2 min, and within 5 min the pattern fades away as surface dye diffuses into the water beneath.
This very simple experiment presents contradictory
results. On the one hand, surface tension appears to spread
dye across the water surface. On the other hand, an apparently nearly uniform initial surface state becomes increasingly heterogeneous over time, an effect that should be
opposed by surface tension. In fact, several authors have
described surface patterning as the result of variations in
surface tension (e.g., Refs. 10, 13, and 14). We will present
evidence that this is not the mechanism at work in the present experiment. Nevertheless, it is conceivable that the dye
used could increase surface tension, causing local regions
of higher dye concentration to contract, thus leading to the
behavior shown in Fig. 2. In this regard, although most
additives to water interrupt hydrogen bonds and thereby
cause the surface tension to decrease,15 some reports indicate that certain inorganic salts can increase the surface tension of water.16 We, therefore, seek to establish whether
surface tension might increase with dye concentration in
our system. For this reason, we measure surface tension vs
food coloring concentration using two standard techniques
(see Sec. II).
Figure 3 shows surface tension values resulting from capillary height17 and goniometer measurements. The capillary
height method has limited accuracy, as it is affected by
unavoidable surface imperfections and variations in capillary
diameter, and it relies on visual estimation of the height of a
curved meniscus.18 We, therefore, produced finer measurements at low dye concentrations using a goniometer,19 as
shown in the inset to Fig. 3. With the possible exception of a
single data point at 80% dye concentration—statistically
inconclusive in view of measurement uncertainties (and in
any event outside of the likely concentration range of
interest)—both capillary and goniometer data are consistent
with a monotonic decrease in surface tension with dye
concentration.
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Am. J. Phys., Vol. 80, No. 7, July 2012
Fig. 3. Measurements of surface tension as a function of concentration of
food coloring in deionized water. Data in the main plot are obtained by
measuring the height of fluid in a capillary tube, while data in the inset are
obtained using a goniometer. Results shown are for McCormick blue food
coloring, although similar data have been obtained using other colors. Error
bars are standard errors based on accuracy estimates of the base height of
the capillary tubes and uncertainties in the height measurement due to the
fluid meniscus.
Thus, we find both qualitatively and quantitatively that
surface tension decreases with dye concentration: qualitatively, dye is rapidly pulled outward as soon as the droplet is
introduced; and quantitatively, direct measurements show
that surface tension decreases monotonically as dye concentration increases. In view of this evidence, it remains counterintuitive that slight inhomogeneities grow, rather than
diminish, as shown in Fig. 2. Increased dye concentration
ought to produce lower surface tension, causing the high
concentration region to be pulled away by the lowerconcentration (higher-tension) surrounding fluid.
B. Mechanisms
In view of these results, we performed several experiments
to isolate what the dye is doing near incipient inhomogeneities. Beginning with the enlargements shown in Fig. 2, we
see that initially, apparently uniform regions spontaneously
develop inhomogeneous stripes—the emergence of such an
inhomogeneity shaped like twin sickles is indicated by black
arrows in the enlargements. After an inhomogeneity
becomes clearly defined, it invariably develops small spreading “feet” (dotted circle, enlarged in bottom-right panel of
Fig. 2).
If we examine the feet more closely, we find that they are
produced by submerging dye tendrils beneath the concentrated stripes. We recall that the dye—consisting largely of
propylene glycol for food coloring and ethylene glycol for
antifreeze—is heavier than water, and so dye on the surface
unavoidably produces a density inversion. The consequent
submersion of dye is shown in Fig. 4. Figure 4(a) shows a
side view of food coloring in water in a rectangular container; the top view here looks similar to the 60-s image
from Fig. 2. As before, the water depth is 4 mm. Figure 4(b)
shows a well-developed striped pattern using fluorescein in
antifreeze; here, the fluorescence causes the dye beneath the
surface to be clearly visible. As shown in the enlargement
(circled), feet appear beneath the sharply defined curves of
Strombom et al.
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580
Fig. 4. (Color online) Details of dye “feet” falling in water. (a) Time series of a
foot submerging beneath surface from a side view, using blue food coloring in
water. (b) Top view of fluorescein in antifreeze illuminated under UV light 30 s
after dropping an aliquot into water, alongside an enlarged view with “feet”
highlighted in dotted circles. (c) Isometric view of stripe and falling “foot” of
green food coloring in water. Water depth is approximately 4 mm in all experiments. A movie of this foot formation is available as supplementary material
(Ref. 27) (enhanced online) [URL: http://dx.doi.org/10.1119/1.4709384.4].
dye. Most clearly, Fig. 4(c) shows an isometric view of a
curved line of food coloring at the surface of the water along
with feet that have reached the container bottom, as well as a
“curtain” of dye falling from the high-concentration curve.
The feet that form appear to be similar to those reported long
ago20 of fluid droplets falling in deep containers.
Apparently, high concentration regions of dye fall from
the surface and by mass conservation, surface fluid must be
drawn in toward these regions to replace the submerging
fluid. Since the dye is on the surface, this enhances the local
concentration of dye in downwelling regions. In other contexts, increases in concentration of surface contaminants due
to related mechanisms have been reported. For example,
pond scum driven by convective downwelling has been
found to be attracted to fractal regions;21 likewise, in the socalled “Cheerios effect,”22,23 floating bodies have been found
to attract one another by deforming the elastic surface downward. Additionally, in the presence of temperature or evaporation gradients, elaborate Bénard-Marangoni convection
patterns have been described,24,25 and a large body of work
deals with buoyancy-driven flows, as appears, for example,
in salt fingers in the ocean.26
We emphasize that in our system, as shown in Fig. 4(c),
foot formation and associated surface dye downwelling
occur in the absence of a surface layer of dye—only a thin
stripe of dye is present at the surface. Thus, the downwelling
does not appear to be associated with previously reported
evaporative or thermal convection mechanisms, which
involve variations in interfacial tension across a fluid surface
(see, e.g., Refs. 10, 13, and 14). Conversely, in our system,
only a thin stripe of dye appears to be necessary to produce
downwelling and the resulting increased surface concentration and foot formation.
C. Additional trials
The observations reported exhibit considerable variability,
both with respect to details of dye and water movements,
and with respect to the occurrence of the reported effects
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Am. J. Phys., Vol. 80, No. 7, July 2012
themselves. For example, whenever dye is spread across the
surface (upper left of Fig. 2) the emergence of labyrinthine
stripes with feet invariably follows; however, dye does not
invariably spread across the surface. Sometimes the droplet
submerges uneventfully without spreading on the surface,
and other times dye spreads over only a part of the surface.
Similarly, under identical conditions, some trials produced a
motile submerged droplet, which we term a slug, while other
trials generated a stationary droplet that emitted pulsations, a
periodic expansion and contraction of the dye around the
connection of the slug to the water surface. These pulsations
led to chaotic mixing.
We performed numerous experiments under identical conditions, precisely metering amounts of dye, fixing the height
at which the dye is released, premixing our own dye from
known materials, varying temperatures and concentrations
both of the dye and of the water along with various additives
such as surfactants, glycerol, and alcohol. Additionally, we
varied the dish diameters and materials, and manufactured
customized Petri dishes of multiple diameters with hypodermics inserted from the bottom through drilled holes at carefully determined heights. In all of these experiments, over
multiple trials by different experimenters, we found variability in outcome to be the norm. From this, we conclude that
the processes involved are likely highly unstable. For
instance, a slug with a slightly off-center surface connection
might migrate in a direction led by the connection, while,
less frequently, a perfectly centered surface connection
might produce the stationary pulsations described. Similarly,
like Olympic divers, one droplet might fall cleanly to the
bottom, leaving scarcely a trace on the surface, while another
droplet might spread widely while submerging. Because of
this variability, producing a particular outcome in this
experiment requires repeated trials and patience. Nevertheless, the outcomes seen in any trial are invariably complex
and attractive, and the mechanisms behind these outcomes
are intriguing and unexplained. From this perspective, the
simple problem of dropping an aliquot of food coloring into
a shallow dish of water is an ideal platform for future
investigation.
To guide such investigations, we provide here results of
several attempts to isolate parameters that strongly affect the
outcomes. Most significantly, the phenomena reported here
depend strongly on water depth and are not reliably seen in
water shallower than 2 mm or deeper than 8 mm. Slug formation in particular has only been regularly observed over
numerous trials at depths of 2–4 mm, and since slug formation is the most common outcome at short times, we report
the effects of the following parametric changes on this outcome: water depth, temperature, addition of surfactant or
alcohol, and volume of injected dye. Mean pulse repetition
frequency, duration of migration, and lag times before movement started have been measured for trials resulting in slug
formation; these results are shown in Table I for depths from
1 to 7 mm in a 9-cm diameter Petri dish. As shown in the table, slug motion predominates between 3 and 4 mm of water
depth. At 2 mm of depth, the slug moves rapidly and produces little more than a subsurface streak.
We also examined the effect of water temperature on this
pattern formation process. For this purpose, room temperature dye was dropped into about 4 mm deep water at three
temperatures: 2 ! C, 22 ! C, and 48 ! C. As shown in Fig. 5,
labyrinths formed reliably and lasted several minutes at the
lowest temperature, while at the highest temperature, the dye
Strombom et al.
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581
Table I. Effect of water depth on slug behavior. Data are averages over
three trials and dashes indicate that slugs did not form. At 3- and 4-mm
depths, the movement duration ended when the slug reached an edge of the
dish.
Water depth (mm)
1
2
3
4
5
6
7
Duration of movement (s)
Pulses per second
Lag time before slug motion started (s)
—
—
—
1
0
0
32
1
3
64
0.5
16
—
—
—
—
—
—
—
—
—
spread slowly and gradually without producing either slugs
or labyrinths. At the intermediate temperature, both slugs
and labyrinthine patterns formed but were inconsistent from
trial to trial.
To explore the effects of surface tension on the observed
phenomena, we performed experiments adding either detergent or ethanol to the water. For the detergent experiments,
we varied the concentration of SDS in deionized water of
depth 4 mm. We found that above 0.5 g/l of detergent, slugs
never formed and the dye submerged and spread more rapidly at the bottom of the water with increasing detergent concentration. At detergent concentrations below 0.25 g/l, slugs
reliably formed and migrated. This supports the proposition
that slugs are driven by surface tension of the water. We performed additional experiments using varying concentrations
of anhydrous ethanol, which also reduces the surface tension
of water. In these experiments, no slugs formed above 9%
(v/v) alcohol concentration in the water, again supporting the
link between slugs and surface tension. For alcohol concentrations between 4% and 9%, we observed multiple tails (up
to 5) from the same drop of dye, often with different tails
and trailing slugs traveling in opposing directions. At
2%–4% alcohol concentration, we observed significantly
slower pulsations as well as slowed surface pattern
formation.
To probe effects of surface tension of the dye itself, we
also added ethanol to the dye. In this case, we performed two
separate sets of experiments, first dropping the dye into
deionized water from about 2 mm above the surface, and
second injecting the dye beneath the surface with a hypodermic. Under both conditions, 35 ll of dye was injected. Here,
we found that dye with 25% (v/v) or more added alcohol rapidly and completely mixed with the water, while concentrations between 6.25% and 12.5% resulted in the dye rising to
the surface, where it rapidly spread and mixed. In none of
these experiments did we see slugs or other patterns, a result
that we interpret to indicate that alcohol in the dye caused
the free surface of the water to expand too rapidly to sustain
these new effects.
We also varied the volume of injected dye in the range
from 10 ll to 2.5 ml. We performed these experiments by
rapidly introducing the dye just below the surface using a hypodermic, with volume controlled with a regulated pipettor.
Here, we found very slow slug migration for dye droplets
smaller than 20 ll, and increasing migration and pulsation
rate of slugs as the slug volume increased up to a volume of
1.5 ml. For larger droplets than this, we observed that the
slug started migrating before all of the dye was released
from the syringe.
IV. COMPUTATIONAL MODEL
A. Simulations
The mechanism by which local regions of increased dye
concentration cause downwelling and consequent contraction of surface dye toward these regions can be simulated in
a straightforward way. We consider a purely 2D model, representing the free surface, and define C(x,y,t) and W(x,y,t) to
be, respectively, the concentrations of colored dye and water
on the surface at location (x, y) and time t. We permit both
dye and water to diffuse and assume that the water and dye
interact nonlinearly with a term that depends more strongly
on dye than on water concentration, as defined by
@C
¼ Dc r2 C þ a Cn W;
@t
@W
¼ Dw r2 W $ a Cn W;
@t
(1)
where all terms are taken to be dimensionless. Here the parameters Dc and Dw define the diffusivity of dye and water,
respectively, a defines the strength of interaction between
water and dye, and n defines the nonlinearity of the interaction (i.e., the importance of dye vs water concentration). We
emphasize that the patterns produced by this model depend
on the presence of molecular diffusion—without the diffusive terms, no patterns would emerge. Yet diffusion proceeds
in accordance with the second law of Thermodynamics as it
causes a concentrated state to evolve into a disordered, more
uniform state. Thus, the patterns we describe not only agree
with the second law, they depend on it.
In our model, we impose the additional condition that the
sum of water and dye concentrations cannot exceed 100%.
We achieve this constraint by calculating S ¼ C þ W at every
gridpoint and timestep, and whenever S > 1 we normalize C
and W in the simplest possible way, by letting
C ! C=S
(2)
W ! W=S:
Fig. 5. (Color online) Examples of dye patterns at 2 ! C and 48 ! C after similar lengths of time (2–3 min).
582
Am. J. Phys., Vol. 80, No. 7, July 2012
Equation (1) is one of the simplest possible analytic descriptions of diffusing dye and water that interact and is quite
similar to the Gray–Scott model (described below),4 a
well-established reaction-diffusion system that leads to autocatalytic patterning. In Fig. 6, we show finite difference simulations using a 256 % 256 grid with timestep Dt ¼ 0.25,
Dc ¼ 0.8, Dw ¼ 1, a ¼ 20, and n ¼ 2. The initial state is 100%
dye with random noise of 1% concentration subtracted to initiate pattern formation, and the boundary conditions are periodic. Similar patterns are produced in simulations (not
shown) using different initial conditions and choices of parameters, provided Dc = Dw (it has been long established on
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582
Fig. 6. (Color online) Time sequence of finite difference simulations of Eq. (1) starting from a nearly uniform initial state. The plot is 3-dimensional with
higher (lower) concentration shown as “peaks” (“valleys”) and lighting used to provide “shadows.”
dimensional grounds that this is a necessary condition to produce a length-scale for patterns in reaction-diffusion equations).
These simulations capture two distinctive features of the
experiments: the first is that the patterns nucleate rapidly
starting from imperceptibly small initial concentration variations, and the second is that the concentrated regions are
sharply defined. We can evaluate this sharpness in the
experiment by fitting the color density of cross sections
through stripes (illustrated in Fig. 7) to a Gaussian and evaluating their standard deviations. Likewise, computationally
we can directly fit the concentration C from simulations with
a Gaussian and evaluate its standard deviation. In Fig. 7, we
plot the averages of these deviations for five stripes tracked
over time from the experiment and from the simulation of
Eq. (1). For comparison, we also perform a simulation of the
Gray–Scott equations
@U
¼ Du r2 U $ UV 2 þ Fð1 $ U Þ;
@t
@V
¼ Dv r2 V þ UV 2 $ ðF þ kÞV;
@t
(3)
on a 256 % 256 grid using Du ¼ 0.2, Dv ¼ 0.1, F ¼ 0.05, and
k ¼ 0.063, which generates labyrinthine stripes as shown in
Fig. 7. (Color online) Evolution of labyrinth widths in: (a) experiment shown in Fig. 2, (b) simulation of Eq. (1), and (c)–(d) simulation of Gray–Scott equations (3). Note in (a) and (b) that nearly uniform states rapidly produce global labyrinths that sharpen over time, as measured by standard deviations of horizontal or vertical line samples. In (c) and (d), however, the Gray–Scott equations take tens of thousands of timesteps to produce labyrinths that cover the
computational domain and then produce waveforms that do not sharpen over time and are significantly more rounded than seen using Eq. (1). Panel (d) shows
an intensity plot of U from Eq. (3) after 40,000 timesteps, overlaid with exemplars of horizontal and vertical cross-sectional locations at which concentrations
would be evaluated. Panel (e) shows actual intensity data through linescan #3 from (d) with a Gaussian fit. The standard deviations of such fits provide the
measures of width used in (a)–(c).
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583
Fig. 7(d).27 To minimize interpolation errors, we evaluated
the width of only horizontal or vertical stripes, as illustrated
in Fig. 7(d).
As shown in Figs. 7(a) and 7(b), stripes of concentrated
dye appear rapidly in both experiment and simulation using
Eq. (1) (within 2 min experimentally, or about 50 timesteps
computationally), and then sharpen, with widths monotonically decreasing by a factor of 2 or 3 over time. By comparison, the Gray–Scott simulation shown in Fig. 7(c) exhibits a
much longer transient period [typically 20,000 timesteps of
the same size as used to simulate Eq. (1)]. Additionally, the
Gray–Scott system requires a finite amplitude disturbance,4
unlike Eq. (1) that only requires small random noise to initiate the patterns shown. This indicates that the simplified
description of Eq. (1) accurately describes the qualitative
features of the dye patterning shown in Figs. 2 and 4. Further, we note that since the flow leading to labyrinthine patterns appears to be convective, driven by downwelling of the
heavier dye into the lighter water, the asymptotic wavelength
of the patterns seen in the experiment should be set by the
depth of the water layer. Indeed, comparison with the scale
bar on the bottom left of Fig. 2 indicates that the wavelength
approaches a value close to the water depth.
We remark that Eq. (1) is essentially the same model that
has been used previously to describe Turing patterns.4 However, in prior work, nonhomogeneous terms are typically
included to account for effects of feed or degradation of material (e.g., in chemical patterns when chemicals are added
and terminal reaction products are precipitated,28 or in biological patterns to model activation and inhibition29 or
growth and death of cells).30 We have examined modified
equations in which additional terms are added to Eq. (1) to
accommodate advection (terms of the form rð~
v CÞ and
rð~
v W Þ, where ~
v is a chaotic velocity field)31 and loss of submerged dye [by subtracting a term proportional to C in the first
equation of Eq. (1)]. In this case, the requirement that C þ W < 1
can be dropped and separate simulations (not shown) produce
patterns essentially indistinguishable from those shown here.
Other embellishments to the model are of course possible.
For example, the nonlinear interaction term (C2W) included
to produce Fig. 6 is arbitrary; the form we use has the merit
of being algebraically simple, but more complicated terms
are equally plausible. As examples, Fig. 8 shows the effect
Fig. 8. (Color online) Simulations using autocatalytic terms in Eq. (1) C2W,
C3W, and C4W. Notice that for C2W, the high concentration regions are
nearly uniform along the directions of the stripes, while for stronger nonlinearities, high concentration spots localize as indicated by dotted circles in
the enlarged views. Similar to Fig. 6, this is a 3-dimensional plot, with dye
concentration on the vertical axis.
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Am. J. Phys., Vol. 80, No. 7, July 2012
of increasing the order of the dye interaction from second
(C2W) to fourth (C4W) order. Each of these images is taken
after 200 timesteps using identical parameters and initial
states. As illustrated in these images, the uniform concentration curves seen for C2W give way to a transverse instability
leading to locally high concentration points, identified in the
enlarged snapshots by dotted circles. Of course, our 2D simulation has no capacity to model submersion of dye in the
third (vertical) dimension, but these qualitative results suggest that the formation of subsurface feet in the experiment
may simply be a result of a transverse instability that can be
modeled using higher-order nonlinearities for dye-water
interactions.
B. Slugs and chaotic mixing
Beyond the formation of labyrinthine stripes over times of
a minute or two, the experimental system investigated is rich
with unexplored phenomena over shorter time scales. Two
of these are shown in Figs. 1(b) and 1(c). In both these case
of chaotic stretching and folding and slug formation, the
droplet of submerged dye makes a thin connection to the surface that appears to periodically disgorge dye aliquots onto
the surface, which in turn causes the surface to periodically
expand and contract.27 At its simplest, the expansion and contraction amounts to a periodic forcing of the surface, leading
to stretching and folding of the surface fluid. While a flow
over a finite time interval cannot be rigorously demonstrated
to be chaotic, stretching and folding are the hallmarks of chaotic mixing, producing a cascade of successively finer tendrils
after each generation of folding. As shown in Fig. 9(a), these
tendrils are evident down to the limits of photographic resolution. For qualitative comparison, we plot the chaotic oscillatory “sine” flow31 in Fig. 9(b). This flow is defined by
x ¼ x þ 3 t sinð2pyÞ
y ¼ y þ 3 t sinð2pxÞ
for
for
n < t < ðn þ 1=2Þ;
ðn þ 1=2Þ < t < ðn þ 1Þ;
(4)
where n is an integer, and where we show the evolution of an
initially circular blob of marker particles after n ¼ 2 periods
of oscillation.
The periodic disgorgement of fluid from beneath is displayed in Fig. 10. In the top panels of this figure, we show
the evolution of a small round “slug” of antifreeze mixed
with fluorescein. This slug sits at the bottom of the water and
is connected to the surface with a thin tail that is visible to
the eye but is very fine and difficult to capture
Fig. 9. (Color online) (a) Enlargements of successively finer generations of
tendrils produced by repeated stretching and folding of dye and water on the
surface of a thin layer of water. Photograph is false colored to highlight the
surface features. (b) For comparison, we plot a portion of the chaotic “sine”
flow [Eq. (4)] ranging over x [ [0,p] and y [ [$2.55, $0.55].
Strombom et al.
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584
Fig. 10. (Color online) Periodic expansion and wander of dye “slug.” Top panels: one cycle of expansion and contraction of clear water and dye (delineated
by dashes) as slug periodically disgorges aliquots of dye through tail (not visible). Bottom panels: the submerged slug of dye wanders as its point of contact
with the surface (cross) is transported through interaction with surface stresses. The evolution shown is available as supplementary material (Ref. 27)
(enhanced online) [URL: http://dx.doi.org/10.1119/1.4709384.5].
photographically. Surrounding the slug in Fig. 10 is an irregularly shaped layer of dye on the surface of the water. We
identify the center of the slug with a cross, and around the
slug we delineate a clear region of fluid with a dashed curve.
Initially, the dashed curve is very close to the slug, but after
one second the clear fluid has expanded, and after 2.7 s the
clear region has contracted again. This sequence of events
repeats itself and coincides with a steady migration of the slug
toward the nearest edge of the Petri dish, as shown in the
lower panels of Fig. 10. In some trials, the slug changes direction and wanders haphazardly, but typically the motion continues until the slug reaches an edge of the container, at which
point the slug either adheres to the edge or hovers nearby.
The observed motion appears to be consistent with the
conjecture that the slug is driven by periodic disgorgements
of dye through the connection to the free surface. We surmise that as a small quantity of dye is disgorged, it causes
the water surface to contract (as occurred when the dye was
first introduced, cf. Fig. 2), and this surface contraction
appears to draw clear fluid from below the surface through a
mechanism that has yet to be determined. This clear fluid in
turn contributes to the motive force for subsequent pulsations
of the slug, although details of this process in particular, and
the mechanism in general, by which the slug produces and
responds to surface stresses remain speculative. Our best
assessment based on (a) measurements that dye concentration affects surface tension and (b) observations that dye
appears to be periodically disgorged to the surface through
the slug’s tail, is that the slug acquires its motility by locally
increasing the surface tension (by entraining clear fluid)
ahead of it, and decreasing surface tension (by trailing dye)
behind.
V. CONCLUSIONS
We have described an extremely simple experiment that
produces a variety of new complex behaviors as well as a
traditional labyrinthine patterned state. We have shown that
the labyrinthine state can be simulated as a Turing pattern
using a straightforward finite difference code. As we have
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described, this simulation depends on the presence of diffusion of dye and water. Without diffusion, the dye and water
would remain separated and would never produce regular
patterns. These patterns are thus a consequence of diffusion,
which tends to spread dye across the water surface, combined with a density inversion, which causes a nearly uniform state to concentrate into stripes. Other behaviors seen
appear to be associated with a coexistence of surface tension,
which provides the motive force for shorter-time, longer-distance states such as chaotic folding and slug migration, and
this density inversion, which leads to longer-time, shorterdistance states such as convective stripes and feet. This
experiment is rich with unexplored phenomena and shows
promise for pedagogical demonstrations of pattern formation
as well as providing new and visually appealing avenues for
research into fluids surface instabilities.
ACKNOWLEDGMENTS
The authors thank Professor Paul Takhistov for the use of
his laboratory facilities, and NSF REU supplement to
CBET-0827404 and Mathworks, Inc. for financial support.
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
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Information on a supplemental simulation for this article appears on the next page.
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586
The Gray-Scott Reaction Diffusion model displays the spatial concentration of chemical species U and V under the influence
of the reaction U þ 2V ! 3V and V ! P. The simulation models this reaction in an open system with a constant addition of U
and removal of V due to a flow rate f and with the removal of V by the reaction V ! P with reaction rate k. Combining this
autocatalytic process with diffusion results in pattern formation that depends on the f and k rates and on the U and V diffusivities DU and DV. This reaction diffusion system has a surprising variety of spatiotemporal patterns when starting in the initial
state U ¼ 1 and V ¼ 0 except for a square grid at the center where U ¼ 1/2 and V ¼ 1/4. The simulation can be found at
http://www.compadre.org/OSP/items/detail.cfm?ID=12017
The Gray-Scott Reaction Diffusion model is a supplemental simulation for the paper by Evelyn Strombom, Carlos E. CaicedoCarvajal, N. Nirmal Thyagu, Daniel Palumbo and Troy Shinbrot and has been approved by the authors and the AJP editor.
Partial funding for the development of these models was obtained through NSF grant DUE-0937731.
Wolfgang Christian
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587