# 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)

Jump to: [I. INTRODUCTION] – [II. METHODS: SIMULATED LE PATTERNS AND EXPERIMENTAL CIMA DATA] – [III. MORPHOLOGICAL DIFFERENCES BETWEEN LE MODEL AND CIMA PATTERNS] – [IV. PATTERN-FLUID MODEL] – [V. VARIATIONAL APPROACH] and [VI. CONCLUSION]

### 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] .

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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

[1–11]. 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

= cσ and nonlinear rate equations

f (u,v) = a − u − 4uv/(1 + u

2

), g(u,v) = bσ[u − uv/(1 +

u

2

)] with constants c, σ , a, and b.

The formation of stationary patterns in this model is well

understood [16–18] 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

= cσ 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

×

1−φ

N

×

1−φ

N

×

1−φ

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 [32–36], 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

dρ, 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]/δu 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]/δu)

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

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### 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]....

[...]

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