American Institute of Mathematical Sciences

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January  2012, 11(1): 189-207. doi: 10.3934/cpaa.2012.11.189

An effective design method to produce stationary chemical reaction-diffusion patterns

Patrick De Kepper 1, and  István Szalai 2,

1. 

Centre De Recherche Paul Pascal, CNRS, Av. Schweitzer, 33600 Pessac, France
2. 

Institute of Chemistry, Laboratory of Nonlinear Chemical Dyanmics, Eötvös L. University, P.O. Box H-1518 Budapest 112, Hungary

Received  March 2010 Revised  July 2010 Published  September 2010

We present a semi-empirical experimental design method to produce nontrivial chemical reaction-diffusion patterns in open reactors. We specially focus on the development of stationary patterns. The method is based on autoactivated reactions that produces spatial bistability, the addition of an independent antagonist reaction to produce spatio-temporal oscillations, and the introduction of a low mobility complexing agent that rapidly and reversibly binds the main autoactivatory species. The method is presented in formal way. Actual experimental results are used for illustration. We point out the open problems of the mathematical description: they relate to the boundary conditions, to the dimensionality of the system, and to the coupled time- and space-scale changes induced by the complexing agent.
Keywords: chemical kinetics., Dynamical systems, turing patterns, pattern formation, partial differential equations, bifurcations.
Mathematics Subject Classification: Primary: 35B32, 92C15; Secondary: 80A3.
Citation: Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure and Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189
References:
[1]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Ser B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.

[2]

J. D. Murray, "Mathematical Biology I-II,'' 3rd edition, Springer Verlag, Berlin, 2002. doi: 10.1007/b98868.

[3]

H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems, Nonl. Anal. TMA, 13 (1989), 507. doi: 10.1016/0362-546X(89)90061-8.

[4]

A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities, Chaos, 4 (1994), 477. doi: 10.1063/1.166047.

[5]

G. Nicolis and I. Prigogine, "Self Organization in Nonequilibrium Systems,'' Wiley, New York, 1977.

[6]

H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, New York, 1982.

[7]

E. Ammelt, Y. A. Astrov and H. G. Purwins, Stripe Turing structures in a two-dimensional gas discharge system, Physical Review E , 55 (2001), 6731-6740. doi: 10.1103/PhysRevE.55.6731.

[8]

L. A. Lugiato, C. Oldano and L. M. Narducci, Cooperative frequency locking and stationary spatial structures in lasers, Journal of the Optical Society of America B -Optical Physics, 5 (1988), 879-888. doi: 10.1364/JOSAB.5.000879.

[9]

S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992), 1943-1967. doi: 10.2307/1941447.

[10]

F. Borgogno, P. D'Odorico and F. Laio, et al, Mathematical models of vegetation pattern formation in ecohydrology, Reviews of Geophysics, 47 (2009), RG1005. doi: 10.1029/2007RG000256.

[11]

P. De Kepper, J. Boissonade and I. Szalai, From sustained oscillations to stationnary reaction-diffusion patterns, in "Chemomechanical Instabilities in Responsive Materials'' Eds. P. Borckmans, P. De Kepper, A. R. Khokhlov, S. Métens; Springer Series A, ISBN 978-90-481-2992-8(PB), (2009) 1-33. doi: 10.1007/978-90-481-2993-5\_1.

[12]

V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953-2956. doi: 10.1103/PhysRevLett.64.2953.

[13]

K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194. doi: 10.1126/science.261.5118.192.

[14]

E. Dulos, P. W. Davies, B. Rudovics and P. De Kepper, From quasi-2D to 3D Turing patterns in ramped systems, Physica D, 98 (1996), 53-66. doi: 10.1016/0167-2789(96)00072-3.

[15]

P. De Kepper, J.-J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system, Int. J. Bifurcation and Chaos, 6 (1994), 1077-1092. doi: 10.1142/S0218127494000915.

[16]

K. J. Lee, W. D. McCormick, H. L. Swinney and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), 215-218. doi: 10.1038/369215a0.

[17]

J. Horváth, I. Szalai and P. De Kepper, An experimental design method leading to chemical turing patterns, Science, 324 (2009), 772-775. doi: 10.1126/science.1169973.

[18]

V. K. Vanag and I. R. Epstein, Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol ot microemulsion, Phys. Rev. Lett., 87 (2001), 228301. doi: 10.1103/PhysRevLett.87.228301.

[19]

P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models,'' Princeton University Press, Princeton, 1989.

[20]

J. Boissonade and P. De Kepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system, J. Phys. Chem., 84 (1980), 501-506. doi: 10.1021/j100442a009.

[21]

G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Archive for Rational Mechanics and Analysis, 97 (1987), 321-352. doi: 10.1007/BF00280410.

[22]

I. R. Epstein and J. A. Pojman, "An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos,'' Oxford University Press, New York, 1998.

[23]

P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities,'' Clarendon Press, Oxford, 1990.

[24]

P. Blanchedeau and J. Boissonade, Resolving an experimental paradox in open spatial reactors: The role of spatial bistability, Phys. Rev. Lett., 81 (1998), 5007-5010. doi: 10.1103/PhysRevLett.81.5007.

[25]

K. Benyaich, T. Erneux T, S. Metens, S, S. Villain and P. Borckmans, Spatio-temporal behaviors of a clock reaction in an open gel reactor, Chaos, 16 (2006), 037109. doi: 10.1063/1.2219703 .

[26]

J. Boissonade, E. Dulos, F. Gauffre, M. N. Kuperman and P. De Kepper, Spatial bistability and waves in a reaction with acid autocatalysis, Faraday Discuss., 120 (2001), 353-361. doi: 10.1039/b103240m.

[27]

P. Blanchedeau, J. Boissonade and P. De Kepper, Theoretical and experimental studies of spatial bistability in the chlorine-dioxide-iodide reaction, Physica D, 147 (2000), 283-299. doi: 10.1016/S0167-2789(00)00169-X.

[28]

Z. Virányi, I. Szalai, J. Boissonade and P. De Kepper, Sustained Spatiotemporal Patterns in the Bromate-Sulfite Reaction, J. Phys. Chem. A, 111 (2007), 8090-8094. doi: 10.1021/jp0723721.

[29]

I. Szalai and P. De Kepper, Spatial bistability, oscillations and excitability in the Landolt reaction, Phys. Chem. Chem. Phys., 8 (2006), 1105-1110. doi: 10.1039/b515620c.

[30]

I. Szalai and P. De Kepper, Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation, Chaos, 18 (2008), 026105. doi: 10.1063/1.2912719.

[31]

J. Horváth, I. Szalai and P. De Kepper, Pattern formation in the Thiourea-Iodate-Sulfite system: spatial bistability, waves, and stationary patterns, Physica D, 239 (2010), 776-784. doi: 10.1016/j.physd.2009.07.005.

[32]

I. Szalai and P. De Kepper, Patterns of the Ferrocyanide-Iodate-Sulfite reaction revisited: the role of immobilized carboxylic functions, J. Phys. Chem. A, 112 (2008), 783-786. doi: 10.1021/jp711849m.

[33]

S. Ponce Dawson, M. V. D'Angelo and J. E. Pearson, Towards a global classification of excitable reaction-diffusion systems, Phys. Lett. A, 265 (2000), 346-352. doi: 10.1016/S0375-9601(00)00008-6.

[34]

I. Lengyel and I. R. Epstein, A chemical approach to design Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977-3979. doi: 10.1073/pnas.89.9.3977.

[35]

J. E. Pearson and W. Bruno, Pattern formation in an N+Q component reaction-diffusion system, Chaos, 2 (1992), 513-524. doi: 10.1063/1.165893.

[36]

D. E. Strier and S. P. Dawson, Turing patterns inside cells, PLoS ONE, 2 (2007), 1053. doi: 10.1371/journal.pone.0001053.

[37]

D. Horváth and Á. Tóth, Diffusion-driven front instabilities in the chlorite-tetrathionate reaction, J. Chem. Phys., 108 (1998), 1447. doi: 10.1063/1.475355.

[38]

D. Horváth and Á. Tóth, Turing patterns in a single-step autocatalytic reaction, J. Chem. Soc. Farad. Trans., 93 (1997), 4301. doi: 10.1039/a705895k.

[39]

I. Szalai and P. De Kepper, Turing patterns, spatial bistability, and front instabilities in a reaction-diffusion system, J. Phys. Chem. A, 108 (2004), 5315-5321. doi: 10.1021/jp049168n.

[40]

I. Szalai N. Takács, J. Horváth and P. De Kepper, Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems, (2011), submitted.

[41]

B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade and P. De Kepper, Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction, J. Phys. Chem. A, 103 (1999), 1790-1800. doi: 0.1021/jp983210v.

[42]

J. A. Vastano, J. E. Pearson, W. Horsthemke and H.L. Swinney, Turing patterns in an open reactor, J. Chem. Phys., 88 (1988), 6175. doi: 10.1063/1.454456.

show all references

References:
[1]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Ser B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.

[2]

J. D. Murray, "Mathematical Biology I-II,'' 3rd edition, Springer Verlag, Berlin, 2002. doi: 10.1007/b98868.

[3]

H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems, Nonl. Anal. TMA, 13 (1989), 507. doi: 10.1016/0362-546X(89)90061-8.

[4]

A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities, Chaos, 4 (1994), 477. doi: 10.1063/1.166047.

[5]

G. Nicolis and I. Prigogine, "Self Organization in Nonequilibrium Systems,'' Wiley, New York, 1977.

[6]

H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, New York, 1982.

[7]

E. Ammelt, Y. A. Astrov and H. G. Purwins, Stripe Turing structures in a two-dimensional gas discharge system, Physical Review E , 55 (2001), 6731-6740. doi: 10.1103/PhysRevE.55.6731.

[8]

L. A. Lugiato, C. Oldano and L. M. Narducci, Cooperative frequency locking and stationary spatial structures in lasers, Journal of the Optical Society of America B -Optical Physics, 5 (1988), 879-888. doi: 10.1364/JOSAB.5.000879.

[9]

S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992), 1943-1967. doi: 10.2307/1941447.

[10]

F. Borgogno, P. D'Odorico and F. Laio, et al, Mathematical models of vegetation pattern formation in ecohydrology, Reviews of Geophysics, 47 (2009), RG1005. doi: 10.1029/2007RG000256.

[11]

P. De Kepper, J. Boissonade and I. Szalai, From sustained oscillations to stationnary reaction-diffusion patterns, in "Chemomechanical Instabilities in Responsive Materials'' Eds. P. Borckmans, P. De Kepper, A. R. Khokhlov, S. Métens; Springer Series A, ISBN 978-90-481-2992-8(PB), (2009) 1-33. doi: 10.1007/978-90-481-2993-5\_1.

[12]

V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953-2956. doi: 10.1103/PhysRevLett.64.2953.

[13]

K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194. doi: 10.1126/science.261.5118.192.

[14]

E. Dulos, P. W. Davies, B. Rudovics and P. De Kepper, From quasi-2D to 3D Turing patterns in ramped systems, Physica D, 98 (1996), 53-66. doi: 10.1016/0167-2789(96)00072-3.

[15]

P. De Kepper, J.-J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system, Int. J. Bifurcation and Chaos, 6 (1994), 1077-1092. doi: 10.1142/S0218127494000915.

[16]

K. J. Lee, W. D. McCormick, H. L. Swinney and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), 215-218. doi: 10.1038/369215a0.

[17]

J. Horváth, I. Szalai and P. De Kepper, An experimental design method leading to chemical turing patterns, Science, 324 (2009), 772-775. doi: 10.1126/science.1169973.

[18]

V. K. Vanag and I. R. Epstein, Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol ot microemulsion, Phys. Rev. Lett., 87 (2001), 228301. doi: 10.1103/PhysRevLett.87.228301.

[19]

P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models,'' Princeton University Press, Princeton, 1989.

[20]

J. Boissonade and P. De Kepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system, J. Phys. Chem., 84 (1980), 501-506. doi: 10.1021/j100442a009.

[21]

G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Archive for Rational Mechanics and Analysis, 97 (1987), 321-352. doi: 10.1007/BF00280410.

[22]

I. R. Epstein and J. A. Pojman, "An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos,'' Oxford University Press, New York, 1998.

[23]

P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities,'' Clarendon Press, Oxford, 1990.

[24]

P. Blanchedeau and J. Boissonade, Resolving an experimental paradox in open spatial reactors: The role of spatial bistability, Phys. Rev. Lett., 81 (1998), 5007-5010. doi: 10.1103/PhysRevLett.81.5007.

[25]

K. Benyaich, T. Erneux T, S. Metens, S, S. Villain and P. Borckmans, Spatio-temporal behaviors of a clock reaction in an open gel reactor, Chaos, 16 (2006), 037109. doi: 10.1063/1.2219703 .

[26]

J. Boissonade, E. Dulos, F. Gauffre, M. N. Kuperman and P. De Kepper, Spatial bistability and waves in a reaction with acid autocatalysis, Faraday Discuss., 120 (2001), 353-361. doi: 10.1039/b103240m.

[27]

P. Blanchedeau, J. Boissonade and P. De Kepper, Theoretical and experimental studies of spatial bistability in the chlorine-dioxide-iodide reaction, Physica D, 147 (2000), 283-299. doi: 10.1016/S0167-2789(00)00169-X.

[28]

Z. Virányi, I. Szalai, J. Boissonade and P. De Kepper, Sustained Spatiotemporal Patterns in the Bromate-Sulfite Reaction, J. Phys. Chem. A, 111 (2007), 8090-8094. doi: 10.1021/jp0723721.

[29]

I. Szalai and P. De Kepper, Spatial bistability, oscillations and excitability in the Landolt reaction, Phys. Chem. Chem. Phys., 8 (2006), 1105-1110. doi: 10.1039/b515620c.

[30]

I. Szalai and P. De Kepper, Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation, Chaos, 18 (2008), 026105. doi: 10.1063/1.2912719.

[31]

J. Horváth, I. Szalai and P. De Kepper, Pattern formation in the Thiourea-Iodate-Sulfite system: spatial bistability, waves, and stationary patterns, Physica D, 239 (2010), 776-784. doi: 10.1016/j.physd.2009.07.005.

[32]

I. Szalai and P. De Kepper, Patterns of the Ferrocyanide-Iodate-Sulfite reaction revisited: the role of immobilized carboxylic functions, J. Phys. Chem. A, 112 (2008), 783-786. doi: 10.1021/jp711849m.

[33]

S. Ponce Dawson, M. V. D'Angelo and J. E. Pearson, Towards a global classification of excitable reaction-diffusion systems, Phys. Lett. A, 265 (2000), 346-352. doi: 10.1016/S0375-9601(00)00008-6.

[34]

I. Lengyel and I. R. Epstein, A chemical approach to design Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977-3979. doi: 10.1073/pnas.89.9.3977.

[35]

J. E. Pearson and W. Bruno, Pattern formation in an N+Q component reaction-diffusion system, Chaos, 2 (1992), 513-524. doi: 10.1063/1.165893.

[36]

D. E. Strier and S. P. Dawson, Turing patterns inside cells, PLoS ONE, 2 (2007), 1053. doi: 10.1371/journal.pone.0001053.

[37]

D. Horváth and Á. Tóth, Diffusion-driven front instabilities in the chlorite-tetrathionate reaction, J. Chem. Phys., 108 (1998), 1447. doi: 10.1063/1.475355.

[38]

D. Horváth and Á. Tóth, Turing patterns in a single-step autocatalytic reaction, J. Chem. Soc. Farad. Trans., 93 (1997), 4301. doi: 10.1039/a705895k.

[39]

I. Szalai and P. De Kepper, Turing patterns, spatial bistability, and front instabilities in a reaction-diffusion system, J. Phys. Chem. A, 108 (2004), 5315-5321. doi: 10.1021/jp049168n.

[40]

I. Szalai N. Takács, J. Horváth and P. De Kepper, Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems, (2011), submitted.

[41]

B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade and P. De Kepper, Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction, J. Phys. Chem. A, 103 (1999), 1790-1800. doi: 0.1021/jp983210v.

[42]

J. A. Vastano, J. E. Pearson, W. Horsthemke and H.L. Swinney, Turing patterns in an open reactor, J. Chem. Phys., 88 (1988), 6175. doi: 10.1063/1.454456.

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