PHYSICAL REVIEW E 91 , 042907 (20)5)
Pattern-fluid interpretation of chemical turbulence Christian S cholz,1,2 G erd E. Schroder-Turk,3-2 and K laus M ecke2 1Institute fo r Multiscale Simulation, Friedrich-Alexander-Universitat Erlangen-Niimberg, Nagelsbachstrufie 49b, 91052 Erlangen, Germany 2Theoretische Physikl, Friedrich-Alexander-Universitdt Erlangen-Niimberg, Staudtstrafie 7b, 91058 Erlangen, Germany 3Murdoch University, School o f 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)]. PACS number(s); 05.45.—a, 82.40.Bj, 82.40.Ck
DOI; 10.1103/PhysRevE.91.042907
I. INTRODUCTION Spatially heterogeneous concentration profiles that form spontaneously in chem ical reaction-diffusion system s repre sent the epitom ical exam ple o f non-linear pattern form ation, w ith im m ediate analogies to many other nonlinear system s [1-11]. W hile Turing predicted their spontaneous form ation by a diffusion-driven instability in driven tw o-com ponent system s in the 1950s [12], it took four decades to realize these experim entally in a chem ical system [2,13,14], One o f these system s is the chlorile-iodide-m alonic acid (CIM A ) reaction [2], In this nonequilibrium reaction, chem i cals are continuously fed through a gel reactor and reactiondiffusion patterns are observed on the surface. M athem atically this system can be treated as a twocom ponent reaction-diffusion model for the spatial concen tration fields u ( x , y ) o f chlorite and v ( x , y ) o f iodide. This so-called Lengyel-Epstein (LE) model [15] is given by ~
= D „A u + f ( u , v ) ,
(1)
^
= D t At) + g(w,w),
(2)
at
at
w ith the L aplace operator A = + -jfe, effective1 diffusion constants D u = 1 and D v — c a and nonlinear rate equations f ( u , v ) = a — u - 4 mi>/(1 + u 2), g ( u , v ) — b o [ u - u v / ( 1 + w2)] w ith constants c, a , a, and b. The form ation o f stationary patterns in this model is well understood [16-18] w ith three types o f observed patterns: hexagonal honeycom b, hexagonal dot, and stripe patterns. The param eter space for these phases is consistent with
1D,, = co 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 a a reaction constant that enhances the diffusivity difference. 1539-3755/2015/91 (4)/042907(6)
experim ents in term s o f qualitative pattern m orphology and quantitative characteristic wavelengths [16]. There are, however, tw o essential inconsistencies betw een the experim entally observed patterns and the num erical solu tions o f the nonlinear models: (1) A detailed com parison betw een the solutions to the LE model and the CIM A patterns obtained in [13] reveals significant m orphological differences in the concentration profiles betw een experim ent and sim ulation, w hich cannot be attributed to slight variations of the involved param eters. (2) E xperim ents have observed spatially and tem porally uncorrelated dynam ical patterns, referred to as chem ical tur bulence [13] (see also the video in the Supplem ental M aterial [19]). A lthough chaotic Turing patterns exist in determ inistic reaction-diffusion system s, such as chaotic oscillations [20], self-replicating spots [21], and defect m ediated turbulence [17,22,23], the experim ental patterns appear to be different from the system s reported in the literature. In this article we present an alternative model, called “pattern-fluid” model, based on the linear superposition of random ly oriented basic stationary patterns (fundam ental patterns), each obtained from num erically solving the LE model (see Fig. 1 for a graphical illustration). This model fully reproduces the turbulent patterns from the C IM A reaction m orphologically and even explains quantitative differences betw een sim ulation and experim ent for the stationary phases. II. METHODS: SIMULATED LE PATTERNS AND EXPERIMENTAL CIMA DATA O ur discussion o f the pattern-fluid m odel is m otivated and validated by com parison o f num eric sim ulations o f the LE model and experim ental data for the C IM A reaction. The LE model from Eqs. (1) and (2) is solved by a standard finite difference m ethod on a quadratic lattice w ith linear system size L = 500 and periodic boundary conditions (which is sufficient to exclude finite-size effects). T he tim e and spatial discretization was A t = 0.01 and A.r = 1 respectively to
042907-1
©2015 American Physical Society
SCHOLZ, SCHRODER-TURK. AND MECKE
PHYSICAL REVIEW E 91, 042907 (2015)
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 -)/N and 0 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 , of a single pattern, here called dominant pattern, becomes significantly larger than the weight of the other patterns, i.e„ 4> » (1 -
Pattern-fluid interpretation of chemical turbulence Christian S cholz,1,2 G erd E. Schroder-Turk,3-2 and K laus M ecke2 1Institute fo r Multiscale Simulation, Friedrich-Alexander-Universitat Erlangen-Niimberg, Nagelsbachstrufie 49b, 91052 Erlangen, Germany 2Theoretische Physikl, Friedrich-Alexander-Universitdt Erlangen-Niimberg, Staudtstrafie 7b, 91058 Erlangen, Germany 3Murdoch University, School o f 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)]. PACS number(s); 05.45.—a, 82.40.Bj, 82.40.Ck
DOI; 10.1103/PhysRevE.91.042907
I. INTRODUCTION Spatially heterogeneous concentration profiles that form spontaneously in chem ical reaction-diffusion system s repre sent the epitom ical exam ple o f non-linear pattern form ation, w ith im m ediate analogies to many other nonlinear system s [1-11]. W hile Turing predicted their spontaneous form ation by a diffusion-driven instability in driven tw o-com ponent system s in the 1950s [12], it took four decades to realize these experim entally in a chem ical system [2,13,14], One o f these system s is the chlorile-iodide-m alonic acid (CIM A ) reaction [2], In this nonequilibrium reaction, chem i cals are continuously fed through a gel reactor and reactiondiffusion patterns are observed on the surface. M athem atically this system can be treated as a twocom ponent reaction-diffusion model for the spatial concen tration fields u ( x , y ) o f chlorite and v ( x , y ) o f iodide. This so-called Lengyel-Epstein (LE) model [15] is given by ~
= D „A u + f ( u , v ) ,
(1)
^
= D t At) + g(w,w),
(2)
at
at
w ith the L aplace operator A = + -jfe, effective1 diffusion constants D u = 1 and D v — c a and nonlinear rate equations f ( u , v ) = a — u - 4 mi>/(1 + u 2), g ( u , v ) — b o [ u - u v / ( 1 + w2)] w ith constants c, a , a, and b. The form ation o f stationary patterns in this model is well understood [16-18] w ith three types o f observed patterns: hexagonal honeycom b, hexagonal dot, and stripe patterns. The param eter space for these phases is consistent with
1D,, = co 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 a a reaction constant that enhances the diffusivity difference. 1539-3755/2015/91 (4)/042907(6)
experim ents in term s o f qualitative pattern m orphology and quantitative characteristic wavelengths [16]. There are, however, tw o essential inconsistencies betw een the experim entally observed patterns and the num erical solu tions o f the nonlinear models: (1) A detailed com parison betw een the solutions to the LE model and the CIM A patterns obtained in [13] reveals significant m orphological differences in the concentration profiles betw een experim ent and sim ulation, w hich cannot be attributed to slight variations of the involved param eters. (2) E xperim ents have observed spatially and tem porally uncorrelated dynam ical patterns, referred to as chem ical tur bulence [13] (see also the video in the Supplem ental M aterial [19]). A lthough chaotic Turing patterns exist in determ inistic reaction-diffusion system s, such as chaotic oscillations [20], self-replicating spots [21], and defect m ediated turbulence [17,22,23], the experim ental patterns appear to be different from the system s reported in the literature. In this article we present an alternative model, called “pattern-fluid” model, based on the linear superposition of random ly oriented basic stationary patterns (fundam ental patterns), each obtained from num erically solving the LE model (see Fig. 1 for a graphical illustration). This model fully reproduces the turbulent patterns from the C IM A reaction m orphologically and even explains quantitative differences betw een sim ulation and experim ent for the stationary phases. II. METHODS: SIMULATED LE PATTERNS AND EXPERIMENTAL CIMA DATA O ur discussion o f the pattern-fluid m odel is m otivated and validated by com parison o f num eric sim ulations o f the LE model and experim ental data for the C IM A reaction. The LE model from Eqs. (1) and (2) is solved by a standard finite difference m ethod on a quadratic lattice w ith linear system size L = 500 and periodic boundary conditions (which is sufficient to exclude finite-size effects). T he tim e and spatial discretization was A t = 0.01 and A.r = 1 respectively to
042907-1
©2015 American Physical Society
SCHOLZ, SCHRODER-TURK. AND MECKE
PHYSICAL REVIEW E 91, 042907 (2015)
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 -)/N and 0 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 , of a single pattern, here called dominant pattern, becomes significantly larger than the weight of the other patterns, i.e„ 4> » (1 -
