scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Pattern-fluid interpretation of chemical turbulence.

16 Apr 2015-Physical Review E (Phys Rev E Stat Nonlin Soft Matter Phys)-Vol. 91, Iss: 4, pp 042907-042907
TL;DR: A pattern-fluid model is introduced as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction-diffusion system, where the nonlinearity forces one single mode to dominate the ensemble.
Abstract: The spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems, from biological tissue to evolutionary population dynamics. The standard model for pattern formation in general, and for Turing patterns in chemical reaction-diffusion systems in particular, are deterministic nonlinear partial differential equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern. However, these models fail to fully explain the experimental observation of turbulent patterns with spatio-temporal disorder in chemical systems. Here we introduce a pattern-fluid model as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction-diffusion system. The transition from turbulent to stationary patterns is then interpreted as a condensation phenomenon, where the nonlinearity forces one single mode to dominate the ensemble. This model leads to better reproduction of the experimental concentration profiles for the "stationary phases" and reproduces the turbulent chemical patterns observed by Q. Ouyang and H. L. Swinney [Chaos 1, 411 (1991)].

Summary (2 min read)

I. INTRODUCTION

  • Spatially heterogeneous concentration profiles that form spontaneously in chemical reaction-diffusion systems represent the epitomical example of non-linear pattern formation, with immediate analogies to many other nonlinear systems [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] .
  • Chemicals are continuously fed through a gel reactor and reactiondiffusion patterns are observed on the surface.
  • The formation of stationary patterns in this model is well understood [16] [17] [18] with three types of observed patterns: hexagonal honeycomb, hexagonal dot, and stripe patterns.
  • There are, however, two essential inconsistencies between the experimentally observed patterns and the numerical solutions of the nonlinear models: (1) A detailed comparison between the solutions to the LE model and the CIMA patterns obtained in [13] reveals significant morphological differences in the concentration profiles between experiment and simulation, which cannot be attributed to slight variations of the involved parameters.
  • Turing patterns exist in deterministic reaction-diffusion systems, such as chaotic oscillations [20] , self-replicating spots [21] , and defect mediated turbulence [17, 22, 23] , the experimental patterns appear to be different from the systems reported in the literature.

II. METHODS: SIMULATED LE PATTERNS AND EXPERIMENTAL CIMA DATA

  • The authors discussion of the pattern-fluid model is motivated and validated by comparison of numeric simulations of the LE model and experimental data for the CIMA reaction.
  • The initial conditions are given by the uniform steady state solution perturbed by additive uniform noise with a relative amplitude of 1%.
  • The simulation was iterated for T = 500 000 time steps until the system converged, and no significant change in the concentration profile was observed.
  • The experimental patterns of the CIMA reaction were provided as contrast-stretched gray-scale images with a resolution of 512 × 480 pixels by the original authors of [13] (a series of 590 images, six for every data point in Fig. 10 from [13] and a time-dependent series of the turbulent state was provided).

III. MORPHOLOGICAL DIFFERENCES BETWEEN LE MODEL AND CIMA PATTERNS

  • An established method suitable for the extraction of characteristic wavelengths of Turing patterns and instabilities is Fourier analysis.
  • To characterize gray-scale concentration fields such as u(x,y), these shape measures are evaluated for the isocontours, u(x,y) = ρ, and analyzed as a function of the isocontour threshold value ρ.
  • For each binary image the Minkowski functionals can be calculated directly by counting the number of pixels corresponding to each phase (area), the number of pixels at the border between the phases and the difference in the number of connected regions of each phase (Euler characteristic) [24] .
  • Application to experimental data of the stationary patterns in the CIMA reaction [Fig. 3(a) ], reveals substantial differences to the same phase in the LE patterns, as shown in Fig.
  • 3(c Additionally these differences persist for all experimental images of both hexagonal and stripe patterns compared to the LE model for various parameters (see section 1 in the Supplemental Material [19] ).

IV. PATTERN-FLUID MODEL

  • Note that this superposition could directly correspond to a projection of different layers onto a planar image, if the system had finite depth.
  • These mean squared differences are here used to establish quantitatively the parameter values for the pattern-fluid model.
  • The authors interpretation is that the kinetic energy of the pattern-fluid increases significantly at larger values of [MA] so that the amplitude of the pattern fluid decreases in favor of the stationary stripe state.
  • In qualitative agreement with previous experiments, their model reproduces the broad range of parameters with a turbulent phase observed in experiments.
  • The pattern-fluid model proposed here invokes an analogy between the treatment of nonequilibrium pattern formation and of condensation phenomena in equilibrium thermodynamics.

V. VARIATIONAL APPROACH

  • The success of the pattern fluid in this context motivates a chemical and mathematically more rigorous treatment which may elucidate more precisely the equivalents of the thermodynamic concepts of energy and temperature.
  • For nonpotential systems, such as the LE model a variational approach is not directly applicable [37, 38] .
  • For such an approach an energy functional F [u] from which time-evolution equations for the fundamental patterns can be derived is required.
  • This approach is not feasible since a function (u) whose partial derivatives with respect to the concentration fields u 1 and u 2 are the reaction rates f and g cannot be found.
  • For small perturbations, where the reaction rates can be expended around the homogeneous steady state, this energy functional can be properly defined [31] .

VI. CONCLUSION

  • The authors have presented a novel interpretation of dynamic pattern formation based on the concept of randomly overlapping fundamental patterns.
  • The random phases of all fundamental patterns can easily be made time dependent.
  • However an appropriate analysis to determine this time-dependence has yet to be developed and in the particular case of the CIMA reaction requires more experiments on the turbulent phase.
  • This could also reveal connections of their model and other types of spatiotemporal dynamics in reaction-diffusion systems that lead to chaotic behavior [20] .
  • Beyond its ability to quantitatively account for all observed phases of the chemical CIMA reaction, the pattern-fluid interpretation could also be relevant to different nonlinear systems with stable heterogeneous phases, e.g., in vibrated granular systems [39] or fluid convection [40] , focusing in particular on the search for systems with nonoscillatory temporally disordered phases [41, 42] .

Did you find this useful? Give us your feedback

Figures (6)

Content maybe subject to copyright    Report

PHYSICAL REVIEW E 91, 042907 (2015)
Pattern-fluid interpretation of chemical turbulence
Christian Scholz,
1,2
Gerd E. Schr
¨
oder-Turk,
3,2
and Klaus Mecke
2
1
Institute for Multiscale Simulation, Friedrich-Alexander-Universit
¨
at Erlangen-N
¨
urnberg, N
¨
agelsbachstraße 49b, 91052 Erlangen, Germany
2
Theoretische Physik I, Friedrich-Alexander-Universit
¨
at Erlangen-N
¨
urnberg, Staudtstraße 7b, 91058 Erlangen, Germany
3
Murdoch University, School of Engineering & IT, Mathematics & Statistics, Murdoch, Western Australia 6150, Australia
(Received 25 February 2015; published 16 April 2015)
The spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems, from biological
tissue to evolutionary population dynamics. The standard model for pattern formation in general, and for Turing
patterns in chemical reaction-diffusion systems in particular, are deterministic nonlinear partial differential
equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern. However, these
models fail to fully explain the experimental observation of turbulent patterns with spatio-temporal disorder in
chemical systems. Here we introduce a pattern-fluid model as a general concept where turbulence is interpreted
as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying
reaction-diffusion system. The transition from turbulent to stationary patterns is then interpreted as a condensation
phenomenon, where the nonlinearity forces one single mode to dominate the ensemble. This model leads to better
reproduction of the experimental concentration profiles for the “stationary phases” and reproduces the turbulent
chemical patterns observed by Q. Ouyang and H. L. Swinney [Chaos 1, 411 (1991)].
DOI: 10.1103/PhysRevE.91.042907 PACS number(s): 05.45.a, 82.40.Bj, 82.40.Ck
I. INTRODUCTION
Spatially heterogeneous concentration profiles that form
spontaneously in chemical reaction-diffusion systems repre-
sent the epitomical example of non-linear pattern formation,
with immediate analogies to many other nonlinear systems
[111]. While Turing predicted their spontaneous formation
by a diffusion-driven instability in driven two-component
systems in the 1950s [12], it took four decades to realize these
experimentally in a chemical system [2,13,14].
One of these systems is the chlorite-iodide-malonic acid
(CIMA) reaction [2]. In this nonequilibrium reaction, chemi-
cals are continuously fed through a gel reactor and reaction-
diffusion patterns are observed on the surface.
Mathematically this system can be treated as a two-
component reaction-diffusion model for the spatial concen-
tration fields u(x,y) of chlorite and v(x,y) of iodide. This
so-called Lengyel-Epstein (LE) model [15] is given by
∂u
∂t
= D
u
u + f (u,v), (1)
∂v
∂t
= D
v
v + g(u,v), (2)
with the Laplace operator =
2
∂x
2
+
2
∂y
2
, effective
1
diffusion
constants D
u
= 1 and D
v
= and nonlinear rate equations
f (u,v) = a u 4uv/(1 + u
2
), g(u,v) = [u uv/(1 +
u
2
)] with constants c, σ , a, and b.
The formation of stationary patterns in this model is well
understood [1618] with three types of observed patterns:
hexagonal honeycomb, hexagonal dot, and stripe patterns.
The parameter space for these phases is consistent with
1
D
v
= is obtained as an effective diffusion constant from the
CIMA reactions, while c = 1.07 is the ratio of the actual diffusion
coefficients of iodide and chlorite and σ a reaction constant that
enhances the diffusivity difference.
experiments in terms of qualitative pattern morphology and
quantitative characteristic wavelengths [16].
There are, however, two essential inconsistencies between
the experimentally observed patterns and the numerical solu-
tions of the nonlinear models:
(1) A detailed comparison between the solutions to the
LE model and the CIMA patterns obtained in [13] reveals
significant morphological differences in the concentration
profiles between experiment and simulation, which cannot be
attributed to slight variations of the involved parameters.
(2) Experiments have observed spatially and temporally
uncorrelated dynamical patterns, referred to as chemical tur-
bulence [13] (see also the video in the Supplemental Material
[19]). Although chaotic Turing patterns exist in deterministic
reaction-diffusion systems, such as chaotic oscillations [20],
self-replicating spots [21], and defect mediated turbulence
[17,22,23], the experimental patterns appear to be different
from the systems reported in the literature.
In this article we present an alternative model, called
“pattern-fluid” model, based on the linear superposition of
randomly oriented basic stationary patterns (fundamental
patterns), each obtained from numerically solving the LE
model (see Fig. 1 for a graphical illustration). This model
fully reproduces the turbulent patterns from the CIMA reaction
morphologically and even explains quantitative differences
between simulation and experiment for the stationary phases.
II. METHODS: SIMULATED LE PATTERNS AND
EXPERIMENTAL CIMA DATA
Our discussion of the pattern-fluid model is motivated and
validated by comparison of numeric simulations of the LE
model and experimental data for the CIMA reaction. The LE
model from Eqs. (1) and (2) is solved by a standard finite
difference method on a quadratic lattice with linear system
size L = 500 and periodic boundary conditions (which is
sufficient to exclude finite-size effects). The time and spatial
discretization was t = 0.01 and x = 1 respectively to
1539-3755/2015/91(4)/042907(6) 042907-1 ©2015 American Physical Society

SCHOLZ, SCHR
¨
ODER-TURK, AND MECKE PHYSICAL REVIEW E 91, 042907 (2015)
×φ
×
1φ
N
×
1φ
N
×
1φ
N
=
FIG. 1. (Color online) The pattern-fluid model. Heterogeneous spatial patterns are interpreted as the superposition of N + 1 randomly
arranged fundamental patterns, of weights (1 φ)/N and φ according to Eq. (3). The fundamental patterns represent solutions of the
Lengyel-Epstein equations. The random orientation and position arises either by translation and rotation or naturally from random differences
in the perturbations of the homogeneous state that lead to the patterns. The stationary ordered phases arise when the amplitude φ, of a single
pattern, here called dominant pattern, becomes significantly larger than the weight of the other patterns, i.e., φ (1 φ)/N .
ensure numerical stability. The initial conditions are given
by the uniform steady state solution perturbed by additive
uniform noise with a relative amplitude of 1%. The simulation
was iterated for T = 500 000 time steps until the system
converged, and no significant change in the concentration
profile was observed. The analysis presented in the following
was performed for parameters in the range of a [10.4,13.2],
b [0.3,0.4], and σ = 20, which is comparable to typical
experimental conditions [16]. The experimental patterns of the
CIMA reaction were provided as contrast-stretched gray-scale
images with a resolution of 512 × 480 pixels by the original
authors of [13] (a series of 590 images, six for every data
point in Fig. 10 from [13] and a time-dependent series of the
turbulent state was provided).
III. MORPHOLOGICAL DIFFERENCES BETWEEN
LE MODEL AND CIMA PATTERNS
An established method suitable for the extraction of
characteristic wavelengths of Turing patterns and instabilities
is Fourier analysis. However, it is less suitable to characterize
the finer details of the concentration profiles, owing to the
typical irregularities found in Turing patterns, such as grain
boundaries [24,25]. In particular for spatio-temporal dynamics
more advanced techniques are required [26,27].
Here we use Minkowski functionals, established robust
shape measures based on rigorous integral geometric theory
[28], to quantify the patterns morphologically. The Minkowski
functionals are defined by curvature integrals of compact
sets. In two dimensions (2D) they correspond to the area
V , perimeter S, and Euler characteristic χ, where χ is the
number difference of connected components and holes in a
set. To characterize gray-scale concentration fields such as
u(x,y), these shape measures are evaluated for the isocontours,
u(x,y) = ρ, and analyzed as a function of the isocontour
threshold value ρ.
Technically, the gray-scale experimental and simulated data
are converted to a set of binary images for a range of values of
ρ [min(u), max(u)], for which the Minkowski functionals
are calculated. For each binary image the Minkowski func-
tionals can be calculated directly by counting the number
of pixels corresponding to each phase (area), the number of
pixels at the border between the phases (perimeter) and the
difference in the number of connected regions of each phase
(Euler characteristic) [24]. However this introduces discretiza-
tion errors which persist even for any finite discretization.
Therefore we use the open source software package
PAPAYA
[29,30], which determines the contours in binary images that
separate the black and white phase using a marching squares
algorithm. From theses contours the Minkowski functionals
can be calculated by numerical integration (area enclosed by
the contour, total length of the contours, and difference of
contours enclosing the two opposite phases). As a result a set
of ρ-dependent curves is obtained for each pattern. All curves
are made dimensionless by appropriate normalization with
the total area of the image and the characteristic wavelength
λ of the pattern, extracted from the Fourier transform (V is
rescaled by the total area A
0
of the image, S is rescaled by
λ/A
0
and χ by λ
2
/A
0
). The dependence of V , S, and χ on
ρ is characteristic for different patterns and can be used to
quantitatively distinguish patterns with different morphologies
[24,25].
In Figs. 2(a)2(c) we plot two examples of stationary
patterns from the hexagonal (a = 12, b = 0.38) and stripe
phases (a = 11.6, b = 0.3), obtained by numerically solving
the LE model. As shown in Fig. 2(c) the Minkowski functionals
uniquely characterize different phases in the deterministic LE
model in terms of their ρ dependence (solid curves correspond
to hexagonal, dashed curves to stripe phase), i.e., patterns of
a certain phase are characterized by a similar set of functions
V (ρ), S(ρ), χ(ρ). Patterns with similar Minkowski functionals
can therefore be considered as statistically equivalent; for
example all stationary patterns in the considered parameter
space from the same phase have equal Minkowski functionals
(see also section 1 in the Supplemental Material [19] for an
explanation of the functional form of V (ρ), S(ρ), χ(ρ)for
typical hexagonal and stripe patterns).
Minkowski functionals do not only allow the distinction of
different phases of the LE model, but are equally useful to
compare experimental and simulation data of the same phase.
Application to experimental data of the stationary patterns in
the CIMA reaction [Fig. 3(a)], reveals substantial differences
to the same phase in the LE patterns, as shown in Fig. 3(c) for
the hexagonal phase (note the difference between the dotted
042907-2

PATTERN-FLUID INTERPRETATION OF CHEMICAL . . . PHYSICAL REVIEW E 91, 042907 (2015)
(a) LE hexagonal
(b) LE stripes
LE hexagonal
LE stripes
2
1
0
1
χ(ρ)
0
1
2
3
S(ρ)
00.20.40.60.81
0
0.2
0.4
0.6
0.8
1
ρ
V (ρ)
(c)
FIG. 2. (Color online) Use of Minkowski functionals to charac-
terize different phases of the Lengyel-Epstein model. Numerical
solutions of the LE equations in the (a) hexagonal phase, σ = 20,
a = 12, b = 0.38 and (b) stripe phase, σ = 20, a = 11.6, b = 0.3.
The gray scale corresponds to the value of the concentration profile
u(x,y), where black is the maximum and white the minimum
concentration. (c) Area V (ρ), perimeter S(ρ), and Euler characteristic
χ(ρ) as a function of the binarization threshold ρ for the hexagonal
(black solid) and stripe (blue dashed) patterns from (a) and (b). The
functional dependence on ρ is characteristic for each phase and can be
used to distinguish the phases quantitatively. See also section 1 of the
Supplemental Material [19] for details on the Minkowski functional
calculations and interpretations.
and solid curves). The reason for this difference is overall
variations of the morphology in the experimental patterns
[Fig. 3(a)]. For example, in some regions perfectly separated
hexagonal spots are observed, but in other regions these spots
become blurred or connected.
Additionally these differences persist for all experimental
images of both hexagonal and stripe patterns compared to
the LE model for various parameters (see section 1 in the
Supplemental Material [19]). This observation is inherent for
deterministic reaction-diffusion models and can also be found
compared to similar experimental systems, such as the chlorine
dioxide-iodine-malonic acid reaction in a one-sided fed reactor
[16], even when stochastic terms are added [31].
IV. PATTERN-FLUID MODEL
We propose a conceptually different model, called “pattern-
fluid, where the observed patterns are assumed to consist of
the statistical superposition of fundamental patterns,
¯
u(x,y) = φu
0
(x,y) +
1 φ
N
N
i=1
u
i
(x,y), (3)
(a)
CIMA
(b)
Pattern Fluid
LE hexagonal
CIMA
Pattern fluid
2
1
0
1
χ(ρ)
0
1
2
3
S(ρ)
00.20.40.60.81
0
0.2
0.4
0.6
0.8
1
ρ
V (ρ)
(c)
FIG. 3. (Color online) Substantial differences between the con-
centration profiles of stationary hexagonal patterns in the CIMA re-
action and the reaction-diffusion model are reconciled by the pattern-
fluid model. (a) Experimental hexagonal pattern from [13] at concen-
trations [ClO
2
]
A
= 20 mM, [H
2
SO
4
]
B
= 100 mM, [MA] = 9mM.
(b) Numerical pattern-fluid for φ = 0.425, N = 3(c)Minkowski
functionals V (ρ), S(ρ), and χ (ρ) of the hexagonal phase of the LE
model from Fig. 1(a) (dots), experimental CIMA pattern (solid) from
(a), and pattern-fluid (dashed) from (b). The Minkowski functionals
of the LE model and CIMA pattern are significantly different. A much
better agreement is found for the pattern fluid.
where u
0
, called the dominant pattern, is a solution of Eq. (1), φ
is its amplitude, and u
i
are N random rotations and translations
of u
0
v is defined analogously), as illustrated in Fig. 1.Ifu
0
is a
plane wave, this sum can be interpreted as a Fourier expansion.
However, we chose the fundamental patterns in such a way that
the spectrum of higher harmonics for the expected stationary
state u
0
is intrinsically given by the LE model.
Note that this superposition could directly correspond to a
projection of different layers onto a planar image, if the system
had finite depth. However, this practical interpretation of the
pattern-fluid model is unlikely as it was explicitly shown that
the experimental CIMA patterns are indeed two-dimensional,
i.e., the vertical extension is less than one wavelength [14].
This is further supported by the fact that simulations and
experiments of three-dimensional systems show a greater
variety of ordered phases [3236], but not turbulent behavior.
Yet, it is clear that
¯
u is, solely due to the nonlinearity of the
reaction-diffusion model, not generally a solution of Eqs. (1)
and (2), except for the trivial case φ = 1orN = 0. However,
the weight of the nonlinearity can vary throughout the (a,b)
parameter space of the LE model and for different values of
N and φ, providing different strength of the “forces” (i.e., the
degree of nonlinearity) that drives the system into a stationary
solution. Heuristically, we analyze the superposed patterns in
terms of how fast the time evolution of
¯
u is, if it followed the
042907-3

SCHOLZ, SCHR
¨
ODER-TURK, AND MECKE PHYSICAL REVIEW E 91, 042907 (2015)
dynamics of Eqs. (1) and (2). A suitable parameter for this is
the average of J
2
=(
¯
u
∂t
)
2
+ (
¯v
∂t
)
2
, which is reminiscent of
a kinetic energy term and quantifies the degree of nonlinear
interactions that drive the system towards the deterministic
solutions of the reaction-diffusion model. For parameters a,
b, φ, and N where J
2
are small,
¯
u decays more slowly, and
is hence more stable, than where J
2
is larger. (Technically
speaking, we require both J
2
and the higher moments of J
to be small.) We now provide support that the pattern-fluid
model accurately reproduces the experimental observations of
the CIMA reaction:
(1) The concentration profiles of the stationary phases are
in quantitative agreement between the pattern-fluid model and
the experimental data.
(2) The turbulent phase with spatial disorder, observed in
experiments, is well reproduced by the pattern-fluid model,
with quantitative agreement of the concentration profiles.
(3) The kinetic energy J
2
, which drives the system
towards the exact solutions of the reaction-diffusion model, is
high in regions of the parameter space where experimentally
stationary stripe patterns are prevalent, and low where turbu-
lence is found (see Fig. 6), consistent with the expectation of
the experimental phase behavior.
In order to find suitable parameter sets for φ and N
we systematically compare the fields
¯
u with concentration
profiles of experimental patterns from the turbulent and
stationary phases. A pattern fluid
¯
u is created according to
Eq. (3) from a hexagonal u
0
and parameter ranges φ [0,1]
and N = 1–32. The mean squared difference V (u,
¯
u) =
1
0
[V (u,ρ) V (
¯
u,ρ)]
2
, and analogously defined S(u,
¯
u)
and χ(u,
¯
u), provides a scalar measure for the similarity
of a pattern
¯
u with the reference image u of the CIMA
reaction, which is either hexagonal or turbulent.
2
These mean
squared differences are here used to establish quantitatively
the parameter values for the pattern-fluid model.
As shown in Fig. 4 where we plot = max(V ,S,χ)
as a function of N, φ for the stationary hexagonal patterns we
find an improved morphological agreement for nonzero values
of N and φ<1 which suggests that a pattern-fluid background
is present even for the “stationary” phase, in accordance with
the experimental observation that grain boundaries move at
a low but finite speed. For the turbulent phase we find an
excellent agreement for large N and small φ with minimized
for φ 0.1 and N 10.
Solutions with optimized parameters are shown in Fig. 3
and Fig. 5 with the corresponding shape measures V (ρ), S(ρ),
and χ (ρ) illustrating the remarkable agreement. A similar
analysis applied to the stripe phase also reveals a significantly
improved morphological agreement between experimental and
numerical patterns from the pattern-fluid model, in accordance
with our assumptions (see section 2 in the Supplemental
Material [19] for details).
As already mentioned above, due to the non-linearity of
the reaction terms
¯
u is not aprioria solution to Eq. (1). A
2
Experimental patterns have been post-processed to stretch the
contrast to the whole range of gray scales. To avoid differences
in the Minkowski functionals from this generic behavior, the same
procedure has also been applied to the numeric patterns
Hexagonal
φ =1
φ =0.4
φ =0.1
(a)
(b)
1 2 4 6 16 32
N
Turbulent
Δ
φ =1
φ =0.1
φ =0.4
(c)
2
8
2
6
2
4
2
2
2
0
φ
(d)
12461632
0.0
0.2
0.4
0.6
0.8
1.0
N
10
2.5
10
2
10
1.5
10
1
10
0.5
10
0
FIG. 4. (Color online) Optimal parameters for the pattern-fluid
model. Determination of optimal parameter values φ and N in the
pattern-fluid model by minimizing the mean-squared difference
between Minkowski functionals of simulated and experimental CIMA
patterns: = max(V ,S,χ) for the numerical pattern fluid
(averaged over 10 realizations) as a function of φ,N for a hexagonal
fundamental pattern u
0
(σ = 20, a = 12, b = 0.38) compared to
CIMA patterns from [13]. Left: Hexagonal phase [ClO
2
]
A
= 20 mM,
[H
2
SO
4
]
B
= 100 mM, [MA] = 9 mM, where (a) gives as a
function of N for fixed values of φ = 0.1, 0.4, 1 and (b) is a
gray-scale plot of in the N , φ plane. Right: Turbulent phase
[ClO
2
]
A
= 18 mM, [H
2
SO
4
]
B
= 30 mM, [MA] = 9mM,with(c)
as a function of N and (d) in the N, φ plane. Lower values mark
a better agreement.
numerical analysis within a reasonable part of the parameter
space does not reveal any region where the collective mode
¯
u
is strictly a solution of Eq. (1). However, the kinetic energy
J
2
that quantifies the degree of nonlinear interactions may
be small for parts of the (a,b) space where turbulent patterns
could be stable.
It can be seen in Fig. 6, where we show the stationary LE
patterns and the corresponding J
2
for φ = 0 and N = 12,
that for all heterogeneous patterns the kinetic energy J
2
is increasing monotonically in a above the transition from
the homogeneous to the heterogeneous state. Additionally for
the stripe phase a significant increase in the kinetic energy is
observed as a function of a.
This is in accordance with experimental observations [13]
where a transition from the stripe to the turbulent regime has
been linked to a decreasing malonic acid concentration [MA],
which increases linearly with a. Our interpretation is that the
kinetic energy of the pattern-fluid increases significantly at
larger values of [MA] so that the amplitude of the pattern fluid
decreases in favor of the stationary stripe state.
A quantitative experimental evaluation of our model
therefore requires a precise control of a and further experi-
mental investigation. In qualitative agreement with previous
042907-4

PATTERN-FLUID INTERPRETATION OF CHEMICAL . . . PHYSICAL REVIEW E 91, 042907 (2015)
(a)
CIMA
(b)
Pattern Fluid
CIMA
Pattern fluid
1
0.5
0
0.5
1
χ(ρ)
0
1
2
3
S(ρ)
00.20.40.60.81
0
0.2
0.4
0.6
0.8
1
ρ
V (ρ)
(c)
FIG. 5. (Color online) Reproduction of experimental concentra-
tion profiles of the turbulent CIMA phase by the pattern-fluid model.
(a) Experimental turbulent pattern from [13,24] with concentrations
[ClO
2
]
A
= 18 mM, [H
2
SO
4
]
B
= 30 mM, [MA] = 9mM.See
[13] for initial chemical concentrations. (b) Numerical pattern fluid
according to Eq. (3) with φ = 0.1, N = 12. (c) Area V (ρ)/A
0
,
perimeter S(ρ)λ/A
0
, and Euler characteristic χ (ρ)λ
2
/A
0
as a
function of ρ for the experimental turbulence (solid) and numerical
pattern-fluid patterns (dashed) from (a) and (b).
experiments, our model reproduces the broad range of param-
eters with a turbulent phase observed in experiments.
3
The pattern-fluid model proposed here invokes an analogy
between the treatment of nonequilibrium pattern formation
and of condensation phenomena in equilibrium thermody-
namics. A high-temperature phase, where loosely interacting
superposed patterns form fluctuating heterogeneous structures,
transforms by symmetry breaking into a condensate where the
extensive occupancy of a single pattern minimizes the free
energy of the system, leading to stationary patterns. Evidently,
thermodynamic systems are governed by a competition be-
tween entropy and energy, where a critical temperature marks
the phase transition point when thermal fluctuations tip the free
energy balance such that the high-temperature phase prevails
over the more tightly bound condensate. The counterpart of
the thermodynamic energy, quantified above by the average
squared velocity J
2
of concentration changes, relates to
the degree of nonlinearity of the fundamental patterns. The
equivalent of the ideal gas is the homogeneous solution
without any interaction between patterns. The turbulent phases
correspond to the weakly interacting collective modes (low-
3
Note that the relationship between a and the experimental
parameters is only approximate; a quantitative comparison of the
regions of stability therefore hinges on more detailed studies of this
relationship
12345
10.410.811.211.61212.412.813.2
0.30
0.32
0.34
0.36
0.38
0.40
a
b
11 12 13
a
J
2
+ c
[a.u. ]
FIG. 6. (Color online) Patterns and nonlinear interaction strength
of the (a,b) parameter space of the reaction-diffusion model. Left:
Stationary homogeneous (gray square), hexagonal (orange circle),
mixed (orange diamond) and stripe (violet triangle) patterns in the
Lengyel-Epstein reaction-diffusion model in the a,b plane with σ =
20. The solid line indicates the Turing bifurcation. Right: Nonlinear
interaction strength quantified by (
¯
u
∂t
)
2
+ (
¯v
∂t
)
2
+c in the a,b plane,
where c is a constant chosen to be equal to b for alignment with the
plot on the left. For a below the Turing bifurcation, the nonlinear
interaction strength is 0 since the fundamental patterns are spatially
homogeneous.
interaction excitations) and the stationary phases to the bound
states with maximal strength of the nonlinear interactions.
V. VARIATIONAL APPROACH
The success of the pattern fluid in this context motivates
a chemical and mathematically more rigorous treatment
which may elucidate more precisely the equivalents of the
thermodynamic concepts of energy and temperature.
However, for nonpotential systems, such as the LE model a
variational approach is not directly applicable [37,38]. For
such an approach an energy functional F [u] from which
time-evolution equations for the fundamental patterns can
be derived is required. In a variational approach with u =
(u
1
,u
2
) = (u(x,y,t),v(x,y,t)), one may attempt to write the
reaction-diffusion equations as u/∂t =−δF[u]u with
a constant and F [u] =
R
2
dx dy (
1
2
i,j
D
ij
u
i
u
j
+
(u)), with the diffusion coefficients D
11
= D
u
, D
22
= D
v
and D
12
= D
21
= 0. The scalar potential (u) needs to be
chosen such that the functional derivative (δF [u]u)
i
=
D
i
2
u
i
∂(u)/∂u
i
is identical to Eqs. (1) and (2). However,
in this simple form, this approach is not feasible since a
function (u) whose partial derivatives with respect to the
concentration fields u
1
and u
2
are the reaction rates f and g
cannot be found. The only exception where this approach is ap-
plicable is for linearized reaction rates. Consequently, for small
perturbations, where the reaction rates can be expended around
the homogeneous steady state, this energy functional can be
properly defined [31]. However, it appears more likely that a
fundamental extension to the variational approach is required.
VI. CONCLUSION
We have presented a novel interpretation of dynamic pattern
formation based on the concept of randomly overlapping
fundamental patterns. We observe significantly improved
042907-5

Citations
More filters
Posted Content
TL;DR: In this paper, a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models is presented, focusing on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowsky tensors.
Abstract: To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.

12 citations

Book ChapterDOI
01 Jan 2017
TL;DR: In this article, a theory-based simulation study of cell shape indices derived from tensor-valued intrinsic volumes, or Minkowski tensors, for a variety of common tessellation models is presented.
Abstract: To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic processes and models. In the context of applied image analysis of structured synthetic and biological materials, this question is central to the problem of inferring information about the formation process from spatial measurements of the resulting random structure. This chapter addresses this question by a theory-based simulation study of cell shape indices derived from tensor-valued intrinsic volumes, or Minkowski tensors, for a variety of common tessellation models. We focus on the relationship between two indices: (1) the dimensionless ratio 〈V 〉2∕〈A〉3 of empirical average cell volumes to areas, and (2) the degree of cell elongation quantified by the eigenvalue ratio 〈β10,2〉 of the interface Minkowski tensors W10,2. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and cell volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, Gibbs hard-core processes of spheres, and random sequential absorption processes as well as for Laguerre tessellations of configurations of polydisperse spheres, STIT-tessellations, and Poisson hyperplane tessellations. These data are complemented by experimental 3D image data of mechanically stable ellipsoid configurations, area-minimising liquid foam models, and mechanically stable crystalline sphere configurations. We find that, not surprisingly, the indices 〈V 〉2∕〈A〉3 and 〈β10,2〉 are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of these shape indices between many of the tessellation models listed above. Therefore, given a realization of a tessellation (e.g., an experimental image), these shape indices are able to narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by considering density-resolved volume-shape correlations.

6 citations


Cites background from "Pattern-fluid interpretation of che..."

  • ...) These functionals and their tensor valuations extensions, the “Minkowski tensors” are efficient numerical tools, which have been successfully applied to a variety of biological [12, 8] and physical systems [64, 65, 43] on all length scales from nuclear physics [95, 96], over condensed and soft matter [30, 39, 104, 88], to astronomy and cosmology [41, 16, 85, 26, 22, 27] as well as to pattern analysis [63, 11, 58, 87]....

    [...]

Journal ArticleDOI
TL;DR: Development of thermodynamic induction up to second order gives a dynamical bifurcation for thermodynamic variables and allows for the prediction and detailed explanation of nonequilibrium phase transitions with associated spontaneous symmetry breaking, thus solving the entropic-coupling problem.
Abstract: Development of thermodynamic induction up to second order gives a dynamical bifurcation for thermodynamic variables and allows for the prediction and detailed explanation of nonequilibrium phase transitions with associated spontaneous symmetry breaking. By taking into account nonequilibrium fluctuations, long-range order is analyzed for possible pattern formation. Consolidation of results up to second order produces thermodynamic potentials that are maximized by stationary states of the system of interest. These potentials differ from the traditional thermodynamic potentials. In particular a generalized entropy is formulated for the system of interest which becomes the traditional entropy when thermodynamic equilibrium is restored. This generalized entropy is maximized by stationary states under nonequilibrium conditions where the standard entropy for the system of interest is not maximized. These nonequilibrium concepts are incorporated into traditional thermodynamics, such as a revised thermodynamic identity and a revised canonical distribution. Detailed analysis shows that the second law of thermodynamics is never violated even during any pattern formation, thus solving the entropic-coupling problem. Examples discussed include pattern formation during phase front propagation under nonequilibrium conditions and the formation of Turing patterns. The predictions of second-order thermodynamic induction are consistent with both observational data in the literature as well as the modeling of this data.

4 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Abstract: It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

9,015 citations

Journal ArticleDOI
TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.

5,430 citations

Journal ArticleDOI
TL;DR: The experimental observation of a sustained standing nonequilibrium chemical pattern in a single-phase open reactor is interpreted as the first unambiguous experimental evidence of a Turing structure.
Abstract: We report the experimental observation of a sustained standing nonequilibrium chemical pattern in a single-phase open reactor. Considering the properties of the pattern (symmetry breaking, intrinsic wavelength), it is interpreted as the first unambiguous experimental evidence of a Turing structure.

1,042 citations

Journal ArticleDOI
09 Jul 1993-Science
TL;DR: Numerical simulations of a simple reaction-diffusion model reveal a surprising variety of irregular spatiotemporal patterns, some of which resemble the steady irregular patterns recently observed in thin gel reactor experiments.
Abstract: Numerical simulations of a simple reaction-diffusion model reveal a surprising variety of irregular spatiotemporal patterns. These patterns arise in response to finite-amplitude perturbations. Some of them resemble the steady irregular patterns recently observed in thin gel reactor experiments. Others consist of spots that grow until they reach a critical size, at which time they divide in two. If in some region the spots become overcrowded, all of the spots in that region decay into the uniform background.

862 citations

Journal ArticleDOI
01 Jan 1991-Nature
TL;DR: In this paper, the authors reported the observation of extended (quasi-two-dimensional) Turing patterns and a Turing bifurcation, a transition from a spatially uniform state to a patterned state.
Abstract: CHEMICAL travelling waves have been studied experimentally for more than two decades1–5, but the stationary patterns predicted by Turing6 in 1952 were observed only recently7–9, as patterns localized along a band in a gel reactor containing a concentration gradient in reagents. The observations are consistent with a mathematical model for their geometry of reactor10 (see also ref. 11). Here we report the observation of extended (quasi-two-dimensional) Turing patterns and of a Turing bifurcation—a transition, as a control parameter is varied, from a spatially uniform state to a patterned state. These patterns form spontaneously in a thin disc-shaped gel in contact with a reservoir of reagents of the chlorite–iodide–malonic acid reaction12. Figure 1 shows examples of the hexagonal, striped and mixed patterns that can occur. Turing patterns have similarities to hydrodynamic patterns (see, for example, ref. 13), but are of particular interest because they possess an intrinsic wavelength and have a possible relationship to biological patterns14–17.

782 citations

Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "Pattern-fluid interpretation of chemical turbulence" ?

Scholz et al. this paper proposed a multiscale simulation model for theoretical Physik at the Friedrich-Alexander-Universität Erlangen-Nürnberg.