Turing instability under centrifugal forces

Abstract

Self-organized patterns are sensitive to microscopic external perturbations that modify the diffusion process. We find that Turing instability formed in a compartmented medium, a Belousov–Zhabotinski–aerosol-OT micelle reaction, responds sensitively to a change in the diffusion process. In order to modify the diffusion mechanism, we apply a centrifugal force that generates a perturbation with an anisotropic character. We find experimentally and numerically that the perturbation is able to modify the pattern and even force its disappearance. For different values of the perturbation significant changes can be seen in both the pattern wavelength and its morphology. Furthermore, for strong perturbations, the orientation of the patterns couples with the symmetry of the perturbation.

Full text

Turing instability under centrifugal forces
Jacobo Guiu-Souto,
*
a
Lisa Michaels,
b
Alexandra von Kameke,
a
Jorge Carballido-
Landeira†
a
and Alberto P. Mu
~
nuzuri
a
Self-organized patterns are sensitive to microscopic external perturbations that modify the diffusion
process. We find that Turing instability formed in a compartmented medium, a Belousov–Zhabotinski–
aerosol-OT micelle reaction, responds sensitively to a change in the diffusion process. In order to modify
the diffusion mechanism, we apply a centrifugal force that generates a perturbation with an anisotropic
character. We find experimentally and numerically that the perturbation is able to modify the pattern
and even force its disappearance. For different values of the perturbation significant changes can be
seen in both the pattern wavelength and its morphology. Furthermore, for strong perturbations, the
orientation of the patterns couples with the symmetry of the perturbation.
Introduction
Self-organization is a characteristic of many living systems that
involves three fundamental properties: order, pattern and
form.
1
Thereby the involved physico-chemical mechanisms
allow these systems to evolve from an initially disordered stage
into a pattern-organized conguration with well-dened
macroscopic properties. In the last few decades, this phenom-
enon has been studied from a theoretical point of view
2,3
concluding that many nonlinear systems can lead to macro-
scopic self-organization.
One of the most important mechanisms capable of gener-
ating organization through pattern formation is the Turing
instability in reaction–diffusion systems.
4,5
It is thought to be
responsible for morphogenesis
6,7
in living organisms, for
example, patterning on the sh skin
8
or seashells.
9
In Nature,
pattern formation does not occur in isolated systems but usually
in the presence of different external perturbations, as for
example, changes in the temperature, density gradients,
10–12
and
even certain types of periodical forces such as modulated gravity
or differential ows.
13–15
Furthermore, pattern formation in the
presence of a centrifugal force has been studied in different
systems such as Rayleigh–B
´
enard cells or active media.
16–18
Here we considered the Belousov–Zhabotinsky reaction
19,20
encapsulated into AOT micelles
19
(BZ–AOT system). Experi-
mentally, this system is shown to exhibit Turing patterns for the
appropriate concentrations of the reactants. We impose a
centrifugal force in such a way that any uid ow is suppressed,
and the only transport process is the microscopic diffusion of
micelles and reactants due to the physical constraints of the
system. The experimental observations are explained by theo-
retical analysis. Furthermore, numerical simulations conrm
and complement the experiments.
Methods
Experimental description
Two stock microemulsions (MEs), ME
1
and ME
2
, with the same
molar ratio, w ¼ [H
2
O]/[AOT] (¼18), and the same droplet
volume fraction, F
d
(¼0.72), were prepared at room temperature
by mixing aqueous solutions of Belosov–Zhabotinsky (BZ) reac-
tants and an oily 1.5 M solution of aerosol-OT (ref. 21) in octane.
ME
1
was composed of solutions of malonic acid (0.5 M) and
sulphuric acid (0.3 M), while for ME
2
we used bromate (0.32 M)
and ferroin (8.3 mM) as described in ref. 22. The reactive
microemulsion, BZ–AOT reaction system,
23–25
was obtained by
mixing equal volumes of the two MEs and diluting the mixture
with octane to the desired droplet fraction (F
d
¼ 0.48).
The behaviour of the reaction is governed by the two main
species, also known as the activator (bromous acid) and the
inhibitor (bromine) due to their chemical competition
processes.
23
The activator molecules remain on the inside of the
AOT-micelles in the aqueous phase, due to their polar nature
and ionic dissociation, and only the non-polar molecules
(inhibitor) can go through the micelle membrane and move into
the oil phase. The polar compound, the activator, is conned to
the aqueous phase within the micelle and its diffusion is linked
to that of the micelle. Thus, in the following, we will consider
the activator mass as that of the micelle containing it. The
micelle mass can be estimated
22
given the hydrodynamic radius
of our micelles
23
(R  1 nm) and we obtain a value of 2500 amu.
As a conclusion, the activator diffusion coefficient is much
smaller than the inhibitor diffusion coefficient (approximately,
10
9
m
2
s
1
and 10
7
m
2
s
1
, respectively
23
).
a
Group of Nonlinear Physics, Universidade de Santiago de Compostela, E-15782
Santiago de Compostela, Spain. E-mail: jacobo.guiu@usc.es
b
Max-Planck-Institut fuer Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
† Present address: Planetary Sciences, Harvard University, 100 Edwin H. Land
Boulevard Cambridge, MA 02142-1204
Cite this: DOI: 10.1039/c3sm27624d
Received 14th November 2012
Accepted 22nd February 2013
DOI: 10.1039/c3sm27624d
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A small amount of the reactive ME was sandwiched between
two optical glasses separated by a Teon gasket (thickness
80 mm, inner diameter 25 mm, and outer diameter 50 mm, see
Fig. 1). This reactor was placed at different distances, R, from
0 up to 20 cm from the rotation center of the rotor (Geared
Brushless 24V DC), performing an uniform circular motion on
the horizontal plane, with a maximum peak angular velocity of
u ¼ 40 rad s
1
. The system is such that no convective ow may
occur and this will be discussed below.
The rotating BZ–AOT system was analyzed with the following
acquisition image set. In order to achieve optimal contrast, the
sample was illuminated with a blue high-intensity diode array
LED. The light passes through a series of light diffusers until
reaching the reactor in order to guarantee uniform background
illumination. The image of the reactor was magnied with an
achromatic objective (DIN 4 Edmund Optics), recorded by a
camera (Guppy AVT 64 fps) and was nally processed in the
computer. In order to quantify the effect of the rotational force,
we compare the differences between two identical reaction
samples. One of them is subjected to a rotational motion and
the other is le at rest, it is subsequently referred to as the
standard pattern. This procedure is repeated for different
angular velocities and radii, obtaining a range of force from
uR ¼ 0ms
1
to uR ¼ 8ms
1
. The required time for the pattern
to appear and its evolution into the steady state is 40 min. The
room temperature was kept constant at 20  2

C. All experi-
ments lying on the same iso-velocity line (v ¼ uR) exhibit the
same behaviour and thus can be averaged for statistical
purposes.
BZ–AOT numerical model
The kinetics of the BZ–AOT reaction can be modelled by the
Oregonator equations revised in ref. 26:
F
1
¼
dc
1
dt
¼
1
3

fc
2
q  c
1
q þ c
1
þ c
1
1  mc
2
1  mc
2
þ 3
1
 c
1
2

(1)
F
2
¼
dc
2
dt
¼ c
1
1  mc
2
1  mc
2
þ 3
1
 c
1
(2)
where the subindex i ¼ 1, 2 labels the activator and inhibitor
species, and c
i
are their respective dimensionless concentra-
tions. The parameters f, m, q, 3
1
and 3 depend on the reaction
rates and they appear as a result of applying the mass action law
on the chemical equations.
27
The suitable values to model the
BZ–AOT system
28
are f ¼ 1.2, m ¼ 190, q ¼ 0.001, 3
1
¼ 0.01 and
3 ¼ 0.8, as used throughout this study. Numerical simulations
of eqn (7) are performed by applying an explicit three-level Du
Fort–Frankel scheme
29
with a spatial step of 0.2 s.u. and zero
ux boundary conditions. To trigger instability we use random
noise as an initial condition.
Experimental results
Fig. 2a shows exper imental results in the absence of force.
Turing patterns app ear as a mixed state that is composed of
stripes and spots with a labyrinthine congurat ion. However,
when we increase the force to 4.7 m s
1
(Fig. 2b) two different
phenomena can be distin guished: the dominance of stripes
over spots and the so orientation of patterns along the
direction of the centripetal force. Higher force values intensify
these phenomena an d stripe-lik e patterns completely orient
along the direction of the for ce (Fig. 2c). The phase diagram
can be divided into three represe ntative regions, as shown in
Fig. 2g.
The rst region, from uR ¼ 0ms
1
up to uR ¼ 2ms
1
,is
characterized by patterns that maintain a mixed conguration
(labyrinth patterns). The second region corresponds to uR
values ranging from 2 m s
1
up to 5 m s
1
(see discussion of
Fig. 6). Here, the system exhibits an intermediate state where
the orientation phenomenon begins to occur (transition stage).
Finally above 5 m s
1
all patterns present are well-oriented
stripes in the direction given by the centrifugal force (oriented
patterns).
Moreover, the pattern wavelength augments slightly
according to the centrifugal force as will be discussed below
(Fig. 5). It is interesting to note that the organization mecha-
nisms studied here are similar to those studied for sh skin
pattern formation by using the CDIMA reaction–diffusion
model,
30
where the velocity of a growing boundary determines
the nal structure of the pattern.
Theoretical approach
In order to understand the phenomena induced by a centrifugal
force, we propose a theoretical model based on a statistical
mechanics approach. The characteristics of our encapsulated
reactor allow us to neglect any macroscopic ow,
22
and there-
fore, the only transport mechanism is due to molecular diffu-
sion.
31,32
This approximation is possible because the thickness
(80 mm) of our system is much smaller than the estimated
boundary layer thickness
33
(3 cm). Given that no convective
transport may exist in this system, the only effect of the external
force is an alteration of the diffusion mechanism. According to
Fig. 1 Experimental setup. The rotor is controlled by a voltage source (VS) that
generates a uniform angular velocity u. BL is the blue LED array used for illumi-
nation and DF is the diffuser. The CCD camera on top of the reactor center is
connected to a PC. The reactor (RE), where the BZ–AOT reaction takes place, is
composed of a Teflon separator (TS) and two optical glasses (GL). R is the distance
from the rotation axis to the reactor center.
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this, canonical ensemble considerations can be applied in order
to evaluate the inuence of the force on the microscopic diffu-
sion processes.
34
We establish an expression for the position of the activator
and the inhibitor inside of the reactor by coupling the thermal
uctuations with the rotational motion:
r
i
¼ x
i
^
x + y
i
ˆ
y + z
i
^
z ¼ (x
0
i
+ x
rot
)
^
x +(y
0
i
+ y
rot
)
ˆ
y + z
0
i
^
z (3)
where x
rot
¼ Rcos(ut), y
rot
¼ Rsin(ut) and the subindex i ¼ 1, 2
represents the activator and inhibitor species, respectively. The
coordinates (x
0
i
, y
0
i
, z
0
i
) indicate the position of the particles due to
thermal motion. Thus the resulting Hamiltonian associated
with the kinetic contribution is:
H
i
¼
p
0
i
2
2m
i
þ
1
2
m
i
u
2
R
2
þ uðp
0
y;i
x
rot
 p
0
x;i
y
rot
Þ (4)
where p
0
i
¼ (p
0
x,i
, p
0
y,i
, p
0
z,i
) and m
i
denote the momentum and the
masses of the particles. The rst and the second term of the
Hamiltonian represent the kinetic and rotational energy,
respectively. The last term represents the angular moment of
the particles and it accounts for the coupling between the
rotational motion and thermal uctuations.
In order to estimate the changes in the diffusion processes
due to the applied force, we consider a diluted system approx-
imation
34
and assume that the microscopic collisions occur only
between identical particles, i.e. 1–1or2–2. The 1–2 collisions are
considered negligible because in the BZ–AOT reaction the
inhibitor molecules can diffuse through the micelle membrane
due their non-polar character.
23
One can also estimate easily the
frequency of collisions and the results indicate that collisions of
type 2–2 occur 100 times more frequently that those of type 1–1.
All of these arguments allow us to consider each species
independently.
Under these considerations we obtain the following expres-
sion for the diffusion coefficient of each species:
31,34
D
1;2
ðb; m
1;2
; u; RÞ¼
D
0
1;2
ðb; m
1;2
Þ
3
ffiffiffi
3
p

3 þ bm
1;2
v
2

3=2
(5)
D
0
1;2
ðb; m
1;2
Þ¼
1
8Ns
tot
ffiffiffiffiffiffiffiffiffiffiffi
3
bm
1;2
s
(6)
where v ¼ uR is the characteristic force velocity, N is the volu-
metric density of the particles, s
tot
is the total collision cross-
section, D
0
1,2
(b,m
1,2
) is the diffusion coefficient in the absence of
force and b ¼ 1/k
B
T. This analytical expression, eqn (5), was
obtained by considering that bm
1,2
u
2
R
2
 1 which is in agree-
ment with the experimental conditions. For example, for the
activator bm
1
u
2
R
2
 0.095 at the experimental force threshold.
It is also important to note that in the absence of force, i.e.
uR ¼ 0ms
1
, we recover the expressions for free diffusion.
34
In accordance with eqn (5) the diffusion process is aniso-
tropic, since it depends on the direction and magnitude of the
centrifugal force. Moreover, as x
rot
, y
rot
[ x
0
, y
0
, we consider the
following approximations x  x
rot
and y  y
rot
. Taking into
account all these considerations the resulting reaction–diffu-
sion equation for our rotating system is:
vc
i
vt
¼ F
i
þ 2D
0
i
bm
i
u
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
1
3
F
i
ðx; yÞ
r

x
vc
i
vx
þ y
vc
i
vy

þ
D
0
i
3
ffiffiffi
3
p
ð3 þ F
i
ðx; yÞÞ
3=2

v
2
c
i
vx
2
þ
v
2
c
i
vy
2

(7)
where c
i
with i ¼ 1, 2 represents the activator and inhibitor
species, respectively. F
i
(x,y) ¼ bm
i
u
2
(x
2
+ y
2
) is related to the
magnitude of the centrifugal forces. F
i
is the reaction term from
eqn (1) and (2). The second term on the right corresponds to a
microscopic ow obtained from VD
i
Vc
i
. Finally, the last term
represents the diffusive transport given by D
i
V
2
c
i
.
By performing a linear stability analysis
7
of eqn (7), we nd
the following eigenvalues L

:
Fig. 2 Turing pattern under centrifugal force (direction represented by the red arrow) far from the rotation center. The patterns orient in the direction of the force.
Experimental (top row) and numerical (bottom row) Turing patterns obtained at different force values: (a and d) uR ¼ 0ms
1
, (b and e) uR ¼ 4.7 m s
1
and (c and f)
uR ¼ 7ms
1
. The experimental images have a radius of 1.53 mm and the computational domain consists of a two-dimensional mesh of 300  300 grid points, with a
spatial step of 0.2 s.u.; (g) is the experimental (u, R)-phase diagram with the corresponding characteristic velocities uR. The iso-velocity curves uR ¼ 2ms
1
and uR ¼
5ms
1
divide the diagram into three regions; labyrinth patterns, transition stage and oriented patterns.
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2L

¼

k
2
trðDÞjktrðZ ÞtrðJÞ


ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½k
2
trðDÞjktrðZÞtrðJÞ
2
 4Pðk; D; J; ZÞ
q
(8)
with,
P(k,D,J,Z) ¼ k
4
det(D)  jk
3
tr(D*Z)  k
2
[tr(D*J) + det(Z)]
+ jktr(JZ*) + det(J) (9)
where k is the wavenumber, j the imaginary unit and J the
Jacobian matrix corresponding to the linearization of the reac-
tion terms, F
i
. D is the diagonal diffusion matrix (d
ii
¼ D
i
), and Z
is an auxiliary diagonal matrix (z
ii
¼ 3D
0
i
bm
i
u(x + y)) that
accounts for the anisotropy of the diffusion processes. The
matrices D* and Z* are dened by D
1
det(D) and Z
1
det(Z),
respectively. Once again, in the absence of force we recover the
standard expressions for the reaction–diffusion system.
35
Fig. 3 shows the real eigenvalues, Re(L
+
) versus the wave-
number for different forces. Positive values of Re(L
+
) are
responsible for generating the Turing instability. Negative
values of the eigenvalue indicate the region where the Turing
structures are unstable. Thus for each uR-isovelocity-curve of
Fig. 3 there are a range of wavenumbers denoted by (k
min
, k
max
),
where Turing patterns can be generated. Moreover, the most
unstable wavenumber
7
(k
C
) inside that range determines the
nal observed stable conguration of the pattern, i.e. its critical
wavelength given by the relationship l
C
¼ 2p/k
C
. As the force is
increased k
max
and k
C
decrease and k
min
remains almost
invariable. It points to the fact that the pattern wavelength, l
C
,
increases with the force. Note that for values of the force above
uR ¼ 30 m s
1
Turing instability is inhibited.
Numerical results
Eqn (7) was numerically integrated and the results are shown in
Fig. 2d–f. Note that they show a good agreement with the
experimental observations; the radial force gradually orients the
previously labyrinthine patterns in the direction of the centrif-
ugal force. Additionally, we again obtain a structure that evolves
from a mixed state composed of stripes and spots to a well-
dened stripes con guration. Analogous to the experimental
results, the numerical wavelength undergoes an increment that
will be discussed in Fig. 5.
Fig. 4 shows the effect of a strong force close to the rotation
center. In the absence of any force (Fig. 4a) we obtain a laby-
rinth-like pattern with a uniform wavelength and without a
Fig. 3 The real part of the eigenvalues obtained by a linear stability analysis of
eqn (7). The curves were calculated for the following values of force uR ¼ 0, 5, 15,
25 and 40 m s
1
. The most probable Turing wavenumber k
C
, is the maximum of
each Re(L
+
) curve and the Turing region (k
min
, k
max
) is given by the zero-crossing,
i.e. Re(L
+
(k
min,max
)) ¼0. For forces above uR ¼30 m s
1
Turing patterns disappear.
Fig. 4 Numerical study of Turing patterns under rotation. The center of rotation lies in the middle of the numerical domain. The radius scale is around the sizeof
patterns (R ranges from 0 to 2 mm). (a) Labyrinth pattern in the absence of force. (b) Labyrinth pattern and transition stage at u ¼ 6  10
3
rad s
1
. (c) Developed radial
modifications (oriented stage and black spots), at u ¼ 35  10
3
rad s
1
. (d) Pattern inhibition at u ¼ 57  10
3
rad s
1
. A summary of all these results is plotted in the
(u, R)-phase diagram in (e). The iso-velocity curves uR ¼ 2ms
1
, uR ¼ 5ms
1
, uR ¼ 20 m s
1
and uR ¼ 30 m s
1
delimit five regions: I, II, III, IV and V related to the
effects of force over the patterns. Lines (a) to (d) indicate the force situations of panel (a) to (d) on the left.
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preferred direction. Similar to the experimental results, when
we increase the angular velocity up to u ¼ 6  10
3
rad s
1
, the
pattern shows an increment in the wavelength with the radius,
and also a growing dominance of stripes (Fig. 4b). At higher
angular velocities, the patterns re-align along the direction of
the force (see Fig. 4c and d) but also new phenomena are
observed. On the one hand, black spots appear at the boundary
(see for example Fig. 4c), and on the other hand, for values
higher than uR ¼ 30 m s
1
, Turing patterns are strongly
inhibited (see Fig. 4d).
These different states are summarized in the numerical
phase diagram shown in Fig. 4e. The iso-velocity curves (uR ¼
2ms
1
, uR ¼5ms
1
, uR ¼20 m s
1
and uR ¼30 m s
1
) divide
the phase diagram into ve regions denoted as I, II, III, IV and V.
Region I includes the labyrinth patterns and region II the
transition state shown in Fig. 2g. Region III corresponds to the
experimental oriented regime. The black spots appear in region
IV and the inhibition phenomenon occurs in region V. When
this last region is reached the conditions for the Turing insta-
bility are not satised
7
as we also observed from the analytical
considerations. Lines marked with (a) to (d) in Fig. 4e show the
values of R and u in the corresponding Fig. 4a–d. Each one of
these lines determines the transitions along the different
regions (related to the type of pattern organization) in the phase
diagram as seen in the corresponding Fig. 4a–d.
Discussion
Fig. 5 compares the differences in the wavelength of the Turing
pattern for the experimental results, the numerical results and
the theoretical predictions from Fig. 2 and 3. This comparison
is feasible due to the dimensionless force term (bm
1,2
u
2
R
2
) and
the use of the normalized wavelength increment, Dl. This
quantity is dened as Dl ¼ (l  l
0
)/l
0
where l
0
stands for the
wavelength of the pattern in the absence of external force, i.e.,
the standard pattern. Note that the normalized wavelength
Fig. 5 Normalized wavelength (Dl ¼ (l  l
0
)/l
0
, with l
0
as the standard pattern
wavelength) versus the force applied. The theoretical critical wavelength incre-
ment, Dl
C
(‘red continuous line’) from stability analysis, numerical estimations,
Dl
num
(‘blue circles’), and experimental measures, Dl
exp
(‘black squares’)reflect a
monotonous growth with the force. The theoretical region for Turing patterns lies
between the maximum and the minimum wavelength increments, i.e. Dl
max
and
Dl
min
(‘dashed lines’). The uncertainties in the Dl were estimat ed from the width
of the wavenumber ring obtained by a Fourier transform of the Turing patterns
images. Uncertainties associated with uR are estimated by the sensitivity of the
experimental devices.
Fig. 6 Representation of the L and T-length mode increment depending on the force. The increments Dl
L,T
are calculated with respect to the wavelength of the
standard pattern (l
0
), analogous to Fig. 5. Circles (squares) and red (blue) color stand for transversal (longitudinal) mode. Filled (empty) markers correspond to
experimental (numerical) results. (a) Characterization of different orientation regimes versus applied force. Point P
1
marks the first divergence between the L-mode and
T-mode and corresponds to the transition state (transition between region I and II). This can be more clearly seen in the inset of the figure. Point P
2
marks the transition
between regions II and III. (b) Semilog plot of the same results (log(Dl
L,T
) ¼ a
L,T
uR with a
L
¼ 1.02  0.16 and a
T
¼ 0.42  0.09).
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augments with the force. Experimentally, increments of up to
18% were reported while theoretically and numerically any
increment can be achieved up to the inhibition of the Turing
pattern (Fig. 3). In this sense, the experimentally encapsulated
BZ–AOT reactions are quite robust to a rotational force, and
therefore Turing patterns are only moderately modied. Similar
effects on the Turing patterns were observed in systems where
temperature or concentrations modify the diffusivity.
36,37
In addition to the changes in the wavelength, we quantify the
orientation of the pattern with respect to the centrifugal forces.
From the numerical and experimental data shown in Fig. 2 we
calculate the mean separation between stripes in the direction
of the force (Dl
L
), and in the transversal direction (Dl
T
), deno-
ted as the longitudinal (L) and transversal (T) wavelength
modes, respectively (the denition of these normalized quan-
tities is equivalent to that in Fig. 5). The T-modes are related to
pattern wavelength and L-modes take into account the pattern
orientation. According to this, patterns in the absence of force
(Fig. 2a and d) have similar values for both L and T-modes since
there is no preferential direction. The relative wavelength
increments, Dl
L,T
, for the two modes are shown in Fig. 6. Note
that both experimental data and numerically data t reasonably
well onto the same curve.
Fig. 6a shows that above uR ¼5ms
1
(P
2
in the gure) the L-
modes experience an important increment because stripes
become largely oriented. At the bifurcation point P
1
(uR ¼ 2m
s
1
), the L and T-modes rst separate from each other. Thus,
the interval of force between P
1
and P
2
(from 2 to 5 m s
1
)
denes a transition zone where the orientation phenomenon
begins to evolve. P
1
and P
2
were calculated by intersecting the
best correlated linear ts of both modes (see Fig. 6). Note that
the points P
1
and P
2
coincide with the transition from stage I to
II and from II to III, respectively (see Fig. 2 and 4).
The growth of the L and T-modes can be characterized by a
semilog function (log( Dl
L,T
) ¼ a
L,T
uR) (see Fig. 6b). We nd that
the growth of Dl
L,T
follows exponential laws whose exponents
for the L and T modes are a
L
¼ 1.02 0.16 and a
T
¼ 0.42 0.09.
According to this, the longitudinal modes have a rate of growth
twice as high as the translational mode. In order to maximize
the correlation coefficient of the t, low force points were
neglected due to their logarithmic dispersion.
Conclusions
We studied the effect of an external rotational force on the
Turing instability. The force modies the diffusion dynamics
at the microscopic scale due to coupling of the pattern forma-
tion and the centrifugal force. An estimation of the external
force states that it is about a 10% of the thermal energy
(m
1
u
2
R
2
 0.1 k
B
T).
Nevertheless, their effects have a crucial role in the pattern
formation mechanism. This may seem counterintuitive at rst
sight. Nevertheless, we have to consider that this is a highly
non-linear phenomenon where small disturbances may be
reected in a dramatic change in the macroscopic properties of
the system. In our present case, this small perturbation destroys
the isotropy of the system that it is translated into a non-
isotropic diffusion coefficient that induces the reported
changes. A theoretical model for the anisotropic diffusion
process was developed, neglecting convective ows due to
boundary constraints. We show, both experimentally and
numerically, that the anisotropic (inhomogeneous) force has
various important effects on the pattern organization. The force
changes the pattern from a labyrinth con guration to stripes
oriented in the direction of the centrifugal forces. Moreover, a
numerical study reveals that for increasing angular velocity, the
patterns exhibit a clear radial orientation before passing
through an intermediate black-spots conguration and, nally,
are inhibited. In order to clarify these transitions, phase
diagrams for the experimental and the numerical case were
calculated as shown in Fig. 2e and 4e, respectively. A posterior
stability analysis of the reaction–diffusion equations indicates
that the force is also able to augment the pattern wavelength
and completely inhibit the formation of Turing structures. We
also observed an increment in the wavelength with the force
experimentally and numerically, see Fig. 5. The numerical and
the experimental results are in good agreement. Furthermore,
the effect of the centrifugal force on the directionality of the
patterns has been quantied by comparing the transversal and
longitudinal wavelength modes (see Fig. 6).
Previous work
22
showed that it was possible to modify the
diffusive regime homogenously; here we present a mechanism
capable of introducing anisotropies in the system. The coupling
between the perturbations and the pattern development allows
us to control the type of nal structure. In this regard, the
authors want to remark that the present work contributes to
obtaining a better understanding of the effects that microscopic
ows produce on Turing pattern formation. These mechanisms
might be extended to other systems that involve self-organiza-
tion phenomena. Further, this study also constitutes a
straightforward method to introduce anisotropy in a system by
simply applying an external eld without actually altering the
nature of the system.
Acknowledgements
The authors thank Dr. P. Taboada for helpful discussions. This
work was supported by the Ministerio de Educaci
´
on y Ciencia
and Xunta de Galicia under Research Grants no. FIS2010-21023
and no. CN2012/315. J.C.-L. and A.v.K. were supported by the
MICINN under a FPI and FPU (AP-2009-0713), respectively and
J.G.-S. by the Xunta de Galicia under a Predoctoral Grant.
Notes and references
1 J. Tabony, Biol. Cell, 2006, 98, 589.
2 N. Rashevsky, Bull. Math. Biophys., 1940, 2, 15.
3 I. Prigogine and G. Nicolis, Q. Rev. Biophys., 1971, 4, 107.
4 A. M. Turing, Philos. Trans. R. Soc., B, 1952, 237, 37.
5 S. Setayeshgar and M. C. Cross, Phys. Rev. E: Stat. Phys.,
Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58(4), 4485.
6 J. M. W. Slack, From Egg to Embryo. Determinative Events in
Early Development, Cambridge University Press, 1983.
Soft Matter This journal is ª The Royal Society of Chemistry 2013
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7 J. D. Murray, Mathematical Biology, Springer, Berling, 2nd
edn, 1993; P. A. Lawrence, The Making of a Fly: The Genetics
of Animal Design, Blackwell Scientic Publications, Boston,
1992.
8 S. Kondo and R. Asai, Nature, 1995, 376, 765.
9 H. Meinhardt, The Algorithmic Beauty of Sea Shells, Springer,
Berlin, 1995.
10 D. A. Vasquez and C. Lengacher, Phys. Rev. E: Stat. Phys.,
Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58, 6865.
11 I. P. Nagy and J. A. Pojman, J. Phys. Chem., 1993, 97, 3443.
12 J. Carballido-Landeira, V. K. Vanag and I. R. Epstein, Phys.
Chem. Chem. Phys., 2010, 12, 3656.
13 S. R
¨
udiger, D. G. M
´
ıguez, A. P. Mu
~
nuzuri, F. Sagu
´
es and
J. Casademunt, Phys. Rev. Lett., 2003, 90, 128301.
14 G. Fern
´
andez-Garc
´
ıa, D. I. Roncaglia, V. P
´
erez-Villar,
A. P. Mu
~
nuzuri and V. P
´
erez Mu
~
nuzuri, Phys. Rev. E: Stat.
Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2008, 77,
026204.
15 A. von Kameke, F. Huhn, G. Fern
´
andez-Garc
´
ıa,
A. P. Mu
~
nuzuri and V. P
´
erez-Mu
~
nuzuri, Phys. Rev. E: Stat.
Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2010, 81,
066211.
16 V. P
´
erez-Villar, J. L. F. Porteiro and A. P. Mu
~
nuzuri, Phys. Rev.
E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2006,
74, 046203.
17 M. Fantz, R. Friedrich, M. Bestehorn and H. Haken, Phys. D,
1992, 61, 147.
18 A. P. J. Breu, C. A. Kruelle and I. Rehberg, Europhys. Lett.,
2003, 62, 491.
19 V. K. Vanag and D. V. Boulanov, J. Phys. Chem., 1994, 98,
1449.
20 A. N. Zaikin and A. M. Zhabotinsky, Nature, 1970, 225, 535.
21 T. K. De and A. Maitra, Adv. Colloid Interface Sci., 1995,
59, 95.
22 J. Guiu-Souto, J. Carballido-Landeira, V. P
´
erez-Villar and
A. P. Mu
~
nuzuri, Phys. Rev. E: Stat. Phys., Plasmas, Fluids,
Relat. Interdiscip. Top., 2010, 82, 066209.
23 V. K. Vanag, Phys.-Usp., 2004, 47, 923; V. K. Vanag and
I. R. Epstein, Phys. Rev. Lett., 2001, 87, 228301.
24 V. K. Vanag and I. R. Epstein, Phys. Rev. Lett., 2003, 90,
098301.
25 I. R. Epstein and V. K. Vanag, Chaos, 2005, 15, 047510.
26 A. Kaminaga, V. K. Vanag and I. R. Epstein, J. Chem. Phys.,
2005, 122, 174706.
27 L. Gyorgyi, T. Turanyi and R. J. Field, J. Phys. Chem., 1990, 94,
7162.
28 A. Kaminaga, V. K. Vanag and I. R. Epstein, Angew. Chem.,
Int. Ed., 2006, 118, 3159.
29 F. A. William, Numerical Methods for Partial Differential
Equations, Academic Press, New York, 1977.
30 D. G. M
´
ıguez and A. P. Mu
~
nuzuri, Biophys. Chem., 2006,
124, 161.
31 J. Guiu-Souto, J. Carballido-Landeira and A. P. Mu
~
nuzuri,
J. Phys.: Conf. Ser., 2011, 296, 012016.
32 J. Guiu-Souto, J. Carballido-Landeira and A. P. Mu
~
nuzuri,
Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip.
Top., 2012, 85, 056205.
33 I. M. Cohen and P. K. Kundu, Fluid Mechanics, Academic
Press, San Diego, 1990.
34 K. Huang, Statistical Mechanics, John Wiley & Sons,
Massachusetts, 1987.
35 S. H. Strogatz, Nonlinear Dynamics and Chaos: With
Applications to Physics, Biology, Chemistry, and Engineering,
Westview Press, 1994.
36 O. Ouyang, R. Li, G. Li and H. L. Swinney, J. Chem. Phys.,
1995, 102, 2551.
37 R. McIlwaine, V. K. Vanag and I. R. Epstein, Phys. Chem.
Chem. Phys., 2009, 11, 1581.
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