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Encyclopedia of Complexity and Systems Science pp 7565–7583Cite as

Reaction Kinetics in Fractals

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Definition of theSubject

Processes involving reaction among atoms, molecules, andparticles in general, are ubiquitous both in nature and inlaboratories. After the introduction of the concept of fractalsby B. Mandelbrot [89] in the 1980s, it hasbeen realized that a wide variety of reactions take placein fractal media, leading to the observation of anomalousbehavior, i.?e. the so called “fractal reactionkinetics” [33,67,81]. The study of thissubject has received considerable theoreticalattention [38,43,76,77,103] and a hugenumerical effort has been made to improve itsunderstanding [13,17,18,20,34,80,92]. Furthermore, among thepractical examples of fractal reaction kinetics one can quoteexciton annihilation in composite materials, chemical reactionsin membranes pores, charge recombination in colloids and clouds,coagulation, polymerization and growth ofdendrites, [15,79,83,94] etc. Another sceneryfor the study of reaction kinetics infractals...

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Abbreviations

Fractals :

Fractal geometry is a mathematical tool well suited to treating complex systems that exhibit scale invariance or, equivalently, the absence of any characteristic length scale. Scale invariance implies that objects are self-similar: if we take a part of the object and magnify it by the same magnification factor in all directions, the obtained object cannot be distinguished from the original one. Self-similar objects are often characterized by non-integer dimensions, a fact that led B. Mandelbrot, early in the 1980s, to coin the name “fractal dimension”. Also, all objects described by fractal dimensions are generically called fractals. Within the context of the present work, fractals are the underlying media where the reaction kinetics among atoms, molecules, or particles in general, is studied.

Reaction kinetics :

The description of the time evolution of the concentration of reacting particles (? i , where \( { i = 1,2,\dots,N } \) identifies the type of particle) is achieved, far from a stationary regime, by formulating a kinetic rate equation \( { \dot{\rho_{i}}(t) = F[\rho_{i}(t)] } \), where F is a function. This description, known in physical chemistry as the law of mass action, states that the rate of a chemical reaction is proportional to the concentration of reacting species and was formulated by Waage and Guldberg in 1864. Often, especially when dealing with reactions occurring in homogeneous media, F involves integer powers (also known as the reaction orders) of the concentrations, leading to classical reaction kinetics. However, as in most cases treated in this article, if a reaction takes place in a fractal, one may also have kinetic rate equations involving non-integer powers of the concentration that lead to fractal reaction kinetics . Furthermore, it is usual to find that the slowest step involved in a kinetic reaction determines its rate, leading to a process-limited reaction, where e.?g. the process could be diffusion or mass transport, adsorption, reaction, etc.

Heterogeneously catalyzed reactions :

A reaction limited by at least one rate-limiting step could be prohibitively slow for practical purposes when, e.?g., it occurs in a homogeneous media. The role of a good solid-state catalyst in contact with the reactants – in the gas or fluid phase – is to obtain an acceptable output rate of the products. Reactions occurring in this way are known as heterogeneously catalyzed. This type of reaction involves at least the following steps: (i) the first one comprises trapping, sticking and adsorption of the reactants on the catalytic surface. Particularly important, from the catalytic point of view, is that molecules that are stable in the homogeneous phase, e.?g. H 2, N2, O2, etc., frequently undergo dissociation on the catalyst surface. This process is essential in order to speed up the reaction rate. (ii) After adsorption, species may diffuse or remain immobile (chemisorbed) on the surface. The actual reaction step occurs between neighboring adsorbed species of different kinds. The result of the reaction is the formation of products that can either be intermediates of the reaction or its final output. (iii) The final step is the desorption of the products, which is essential not only for the practical purpose of collecting and storing the desired output, but also in order to regenerate the catalytically active surface sites.

Bibliography

Primary Literature

  1. Albano EV (1990) Finite-size effects in kinetic phase transitions of a model reaction on a fractal surface: Scaling approach and Monte Carlo investigation. Phys Rev B 42:R10818–R10821

    Google Scholar 

  2. Albano EV (1990) Monte Carlo simulation of the oxidation of carbon monoxide on fractal surfaces. Surf Sci 235:351–359

    ADS  Google Scholar 

  3. Albano EV (1991) On the self-poisoning of small particles upon island formation of the reactants in a model for a heterogenoeusly catalyzed reaction. J Chem Phys 94:1499–1504

    ADS  Google Scholar 

  4. Albano EV (1992) Critical exponents for the irreversible surface reaction \( { A + B \rightarrow AB } \) with B-desorption on homogeneous and fractal media. Phys Rev Lett 69:656–659

    ADS  Google Scholar 

  5. Albano EV (1992) Irreversible phase transitions in the dimer-monomer surface reaction process on fractal media. Phys Lett A 168:55–58

    Google Scholar 

  6. Albano EV (1992) Study of the critical behaviour of an irreversible phase transition in the \( { A + B \rightarrow AB } \) reaction with B-desorption on a fractal surface- Phys Rev A 46:5020–5025

    ADS  Google Scholar 

  7. Albano EV (1994) Critical behaviour of the irreversible phase transitions of a dimer-monomer process on fractal media. J Phys A (Math Gen) 27:431–436

    MathSciNet  ADS  Google Scholar 

  8. Albano EV (1995) Spreading analysis and finite-size scaling study of the critical behavior of a forest fire model with immune trees. Physica A 216:213–226

    ADS  Google Scholar 

  9. Albano EV (1996) The Monte Carlo simulation method: A powerful tool for the study of reaction processes. Heter Chem Rev 3:389–418

    Google Scholar 

  10. Albano EV (2000) Borówko M (ed) Computational methods in surface and colloid Science. Marcel Dekker Inc, New York, pp 387–435

    Google Scholar 

  11. Albano EV (2001) Monte Carlo simulations of teh short-time dynamics of a first-oreder irreversible phase transition. Phys Lett A 288:73–78

    MathSciNet  ADS  MATH  Google Scholar 

  12. Albano EV, Mártin HO (1988) Study of recombination reactions of particles adsorbed on fractal and multifractal substrata. Appl Phys A 47:399–407

    Google Scholar 

  13. Albano EV, Mártin HO (1988) Temperature-programmed reactions with anomalous diffusion. J Phys Chem 92:3594–3597

    Google Scholar 

  14. Alexander S, Orbach R (1982) Density of states on fractals: Fractons. J Physique Lett 43:L625–L631

    Google Scholar 

  15. Anacker LW, Kopelman R (1984) Fractal chemical kinetic: Simulations and experiments. J Chem Phys 81:6402–6403

    ADS  Google Scholar 

  16. Anacker LW, Kopelman R (1987) Steady-state chemical kinetics on fractals: Segregation of reactants. Phys Rev Lett 58:289–291

    Google Scholar 

  17. Anacker LW, Kopelman R, Newhouse JS (1984) Fractal chemical kinetics: Reacting random walkers. J Stat Phys 36:591–602

    ADS  MATH  Google Scholar 

  18. Anacker LW, Parson RP, Kopelman R (1985) Diffusion-controlled reaction kinetics on fractal and Euclidian lattices: Transient and steady-state annihilation. J Phys Chem 89:4758–4761

    Google Scholar 

  19. Argyrakis P, Kopelman R (1986) Fractal behaviour of correlated random walk on percolation clusters. J Chem Phys 84:1047–1048

    ADS  Google Scholar 

  20. Argyrakis P, Kopelman R (1992) Diffusion-controlled binary reactions in low dimensions: Refined simulations. Phys Rev A 45:5814–5819

    ADS  Google Scholar 

  21. Avnir D (1991) Fractal geometry: a new approach to heterogeneous catalysis. Chem Ind 12:912–916

    Google Scholar 

  22. Avnir D (1997) Applications of fractal geometry methods in heterogeneous catalysis. In: Ertl G, Knozinger H, Weitkamp J (eds) Handbook of heterogeneous catalysis, vol 2. Wiley-VCH, Weinheim, pp 598–614

    Google Scholar 

  23. Avnir D, Farin D, Pfeifer P (1983) Chemistry in noninteger dimensions between two and three. II Fractal surfaces of adsorbents. J Chem Phys 79:3566–3571

    MathSciNet  ADS  Google Scholar 

  24. Bab MA, Fabricius G, Albano EV (2005) Critical behavior of an Ising system on the Sierpinski carpet: A short-time dynamics study. Phys Rev E 71:036139

    ADS  Google Scholar 

  25. Bab MA, Fabricius G, Albano EV (2006) Discrete scale invariance effects in the nonequilibrium critical behavior of the Ising magnet on a fractal substrate. Phys Rev E 74:041123

    ADS  Google Scholar 

  26. Bab MA, Fabricius G, Albano EV (2008) On the occurrence of oscillatory modulations in the power-law behavior of dynamic and kinetic processes in fractal media. EPL 81(1)

    Google Scholar 

  27. Barabási AL, Stanley HE (1995) Fractal concepts in surface growth. Cambridge University Press, Cambridge

    Google Scholar 

  28. Barber MN (1983) Domb C, Lewobitz JL (eds) Phase transitions and Critical Phenomena, vol II. Academic Press, London, p 146

    Google Scholar 

  29. Barzykin AV, Tachiya M (2007) Stochastic models of charge carrier dynamics in semiconducting nanosystems. J Phys C Condens Matter 19:065113

    ADS  Google Scholar 

  30. Bell AT (2003) The impact of nanoscience on heterogeneous catalysis. Science 14:1688–1691

    ADS  Google Scholar 

  31. Bena I, Droz M, Martens K, Rácz Z (2007) Reaction-diffusion fronts with inhomogeneous initial conditions. J Phys C Condens Matter 19:065103

    Google Scholar 

  32. ben-Avraham D, Considine DB, Meakin P, Redner S, Takayasu H (1990) Saturation transition in a monomer-monomer model of heterogeneous catalysis. J Phys A (Math Gen) 23:4297–4312

    Google Scholar 

  33. ben-Avraham D, Havlin S (2000) Diffussion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge

    Google Scholar 

  34. Berry H (2002) Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation. Biophys J 83:1891–1901

    Google Scholar 

  35. Binder K, Heermann D (2002) Monte Carlo simulations in statistical physics. An introduction, 4th edn. Springer, Berlin

    Google Scholar 

  36. Block JH, Ehsasi M, Gorodetskii V (1993) Dynamic Studies of surface reactions with microscopic techniques. Prog Surf Sci 42:143–168

    ADS  Google Scholar 

  37. Bordogna CM, Albano EV (2007) Statistical methods applied to the study of opinion formation models: a brief overview and results of a numerical study of a model based on the social impact theory. J Phys C Condens Matter 19:065144

    ADS  Google Scholar 

  38. Bramson M, Lebowitz JL (1988) Asymptotic behavior of densities in difussion-dominated annihilation reactions. Phys Rev Lett 61:2397–2400

    ADS  Google Scholar 

  39. Brosilow BJ, Ziff RM (1992) Comment on: NO-CO reaction on square and hexagonal surfaces: A Monte Carlo simulation. J Catal 136:275–278

    Google Scholar 

  40. Bunde A, Havlin S (1995) A brief introduction to fractal geometry. In: Bunde A, Havlin S (eds) Fractals in science. Springer, Berlin, pp 1–25

    Google Scholar 

  41. Casties A, Mai J, von Niessen W (1993) A Monte Carlo study of the CO oxidation on probabilistic fractal. J Chem Phys 99:3082–3091

    ADS  Google Scholar 

  42. Clar S, Drossel B, Schwabl F (1996) Forest fires and other examples of self-organized criticality. J Phys C Condens Matter 8:6803–6824

    ADS  Google Scholar 

  43. Clément E, Kopelman R, Sanders LM (1991) The diffusion-limited reaction \( { A + B \rightarrow 0 } \) on a fractal lattice. J Stat Phys 65:919–924

    Google Scholar 

  44. Clément E, Leroux-Hugon P, Argyrakis P (1994) Catalysis on a fractal lattice: A model for poisoning. Phys Rev E 49:4857–4864

    Google Scholar 

  45. Clément E, Sanders LM, Kopelman R (1989) Steady-state diffusion-controlled \( { A + A \rightarrow 0 } \) reaction in Euclidean and fractal dimensions: Rate laws and particle self-ordering. Phys Rev A 39:6472–77

    Google Scholar 

  46. Considine DB, Redner S, Takayasu H (1989) Comment on: Noise-induced bistability in a Monte Carlo surface-reaction model. Phys Rev Lett 63:2857

    ADS  Google Scholar 

  47. Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys 65:851–1112

    ADS  Google Scholar 

  48. Dornic I, Chaté H, Chave J, Hinrichsen H (2001) Critical coarsening without surface tension: The universality class of the Voter model. Phys Rev Lett 87:045701

    Google Scholar 

  49. Ehsasi M, Matloch M, Franck O, Block JH, Christmann K, Rys FS, Hirschwald W (1989) Steady and nonsteady rates of reaction in a heterogeneously catalyzed reaction: Oxidation of CO on platinum, experiments and simulations. J Chem Phys 91:4949–4960

    ADS  Google Scholar 

  50. Engel T (1978) A molecular beam investigation of He, CO, and O2 scattering from Pd(111). J Chem Phys 69:373–385

    ADS  Google Scholar 

  51. Engel T, Ertl G (1978) A molecular beam investigation of the catalytic oxidation of CO on Pd(111). J Chem Phys 69:1267–1281

    ADS  Google Scholar 

  52. Ertl G (1990) Oscillatory catalytic reactions at single-crystal surfaces. Adv Catal 37:213–277

    Google Scholar 

  53. Even U, Rademann K, Jortner J, Manor N, Reisfeld R (1984) Electronic energy transfer on fractals. Phys Rev Lett 42:2164–2167

    Google Scholar 

  54. Fichthorn K, Gulari E, Ziff RM (1989) Noise-induced bistability in a Monte Carlo surface-reaction model. Phys Rev Lett 63:1527–1530

    ADS  Google Scholar 

  55. Frachebourg L, Krapivsky PL (1996) Exact results for kinetics of catalytic reactions. Phys Rev E 53:R3009–R3012

    ADS  Google Scholar 

  56. Gallos LK, Argyrakis P (2007) Influence of complex network substrate on reaction-diffusion processes. J Phys C Condens Matter 19:065123

    ADS  Google Scholar 

  57. Gao Z, Yang ZR (1999) Dynamic behavior of the Ziff–Gulari–Barshad model on regular fractal lattices: The influence of lacunarity. Phys Rev E 59:2795–2800

    ADS  Google Scholar 

  58. Gefen Y, Aharony A, Mandelbrot BB (1983) Phase transitions on fractals. I Quasi-linear lattices. J Phys A (Math Gen) 16:1267–1278

    MathSciNet  ADS  Google Scholar 

  59. Gefen Y, Aharony A, Mandelbrot BB (1984) Phase transitions on fractals. III Infinitely ramified lattices. J Phys A (Math Gen) 17:1277–1289

    MathSciNet  ADS  Google Scholar 

  60. Gefen Y, Mandelbrot BB, Aharony AY (1980) Critical phenomena on fractal lattices. Phys Rev Lett 45:855–858

    MathSciNet  ADS  Google Scholar 

  61. Gouyet JF (1996) Physics and fractal structures. Springer, Paris

    Google Scholar 

  62. Grassberger P (1989) Directed percolation in 2+1 dimensions. J Phys A (Math Gen) 22:3673–3679

    ADS  Google Scholar 

  63. Grassberger P, de la Torre A (1979) Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour. Ann Phys (New York) 122:373–396

    ADS  Google Scholar 

  64. Grinstein G, Lai Z-W, Browne DA (1989) Critical phenomena in a nonequilibrium model of heterogeneous catalysis. Phys Rev A 40:4820–4823

    ADS  Google Scholar 

  65. Grinstein G, Muñoz MA (1997) Garrido PL, Marro J (eds) Fourth Granada lectures in computational physics. Springer, Berlin, p 223

    Google Scholar 

  66. Gálfi L, Rácz Z (1988) Properties of the reaction front in an A+B=C type reaction-diffusion process. Phys Rev A 38:3151–3154

    Google Scholar 

  67. Havlin S, ben-Avraham D (1987) Diffusion in disordered media. Adv Phys 36:695–798

    Google Scholar 

  68. Hinrichsen H (2000) Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv Phys 49:815–958

    ADS  Google Scholar 

  69. Huang Y, Ouillon G, Saleur H, Sornette D (1997) Spontaneous generation of discrete scale invariance in growth models. Phys Rev E 55:6433–6447

    ADS  Google Scholar 

  70. Imbhil R, Ertl G (1995) Oscillatory kinetics in heterogeneous catalysis. Chem Rev 95:697–733

    Google Scholar 

  71. Imbihl R (1993) Oscillatory reactions on single crystal surfaces. Prog Surf Sci 44:185–343

    ADS  Google Scholar 

  72. Jensen I (1991) Non-equilibrium critical behaviour on fractal lattices. J Phys A (Math Gen) 24:L1111–L1117

    ADS  Google Scholar 

  73. Jensen I, Fogedby H, Dickman R (1990) Critical exponents for an irreversible surface reaction model. Phys Rev A 41:R3411–R3414

    Google Scholar 

  74. Johansen A, Sornette D (1998) Evidence of discrete scale invariance in DLA and time-to-failure by canonical averaging. Int J Mod Phys C 9:433–447

    ADS  Google Scholar 

  75. Johnson BFG (2003) Nanoparticles in catalysis. Topics catal 24:147–159

    ADS  Google Scholar 

  76. Kang K, Redner S (1984) Scaling approach for the kinetics of recombination processes. Phys Rev Lett 52:955–958

    MathSciNet  ADS  Google Scholar 

  77. Kang K, Redner S (1985) Fluctuation-dominated kinetics in diffusion-controlled reactions. Phys Rev A 32:435–447

    ADS  Google Scholar 

  78. Kittel C (1986) Introduction to solid state physics, 8th edn. Wiley, New York

    Google Scholar 

  79. Kopelman R (1976) In: Fong FK (ed) Radiationless processes in molecules and condensed phases, vol 15. Springer, Berlin, p 297

    Google Scholar 

  80. Kopelman R (1986) Rate processes on fractals: Theory, simulations, and experiments. J Stat Phys 42:185–200

    ADS  Google Scholar 

  81. Kopelman R (1988) Fractal reaction kinetics. Science 241:1620–1626

    ADS  Google Scholar 

  82. Krapivsky PL (1992) Kinetics of monomer-monomer surface catalytic reactions. Phys Rev A 45:1067–1072

    ADS  Google Scholar 

  83. Lee Koo Y-E, Kopelman R (1991) Space- and time-resolved diffusion-limited binary reaction kinetics in capillaries: experimental observation of segregation, anomalous exponents, and depletion zone. J Stat Phys 65:893–918

    ADS  Google Scholar 

  84. Liggett TM (1985) Interacting particle systems. Springer, New York

    MATH  Google Scholar 

  85. Loscar E, Albano EV (2003) Critical behaviour of irreversible reaction systems. Rep Prog Phys 66:1343–1382

    ADS  Google Scholar 

  86. Mai J, Casties A, von Niessen W (1992) A model for the catalytic oxidation of CO on fractal lattices. Chem Phys Lett 196:358–362

    ADS  Google Scholar 

  87. Mai J, Casties A, von Niessen W (1993) A Monte Carlo simulation of the catalytic oxidation of CO on DLA clusters. Chem Phys Lett 211:197–202

    ADS  Google Scholar 

  88. Maltz A, Albano EV (1992) Kinetic phase transitions in dimer-dimer surface reaction models studied by means of mean-field and Monte Carlo methods. Surf Sci 277:414–428

    ADS  Google Scholar 

  89. Mandelbrot BB (1983) The fractal geometry of nature. Freeman, San Francisco

    Google Scholar 

  90. Marro J, Dickman R (1999) Nonequilibrium phase transitions and critical phenomena. Cambridge Univ Press, Cambridge

    Google Scholar 

  91. Meakin P, Scalapino DJ (1987) Simple models for heterogeneous catalysis: Phase transition-like behavior in nonequilibrium systems. J Chem Phys 87:731–741

    ADS  Google Scholar 

  92. Newhouse JS, Kopelman R (1985) Fractal chemical kinetics: Binary steady-state reaction on a percolating cluster. Phys Rev B 31:1677–1678

    ADS  Google Scholar 

  93. Newhouse JS, Kopelman R (1988) Steady-state chemical kinetics on surface clusters and islands: Segregation of reactants. J Phys Chem 92:1538–1541

    Google Scholar 

  94. Parus SJ, Kopelman R (1989) Self-ordering in diffusion-controlled reactions: Exciton fusion experiments and simulations on naphthalene powder, percolation clusters, and impregnated porous silica. Phys Rev B 39:889–892

    ADS  Google Scholar 

  95. Pfeifer P, Avnir D (1983) Chemistry in noninteger dimensions between two and three. I Fractal theory of heterogeneous surfaces. J Chem. Phys 79:3558–3565

    MathSciNet  Google Scholar 

  96. Rammal R, Toulouse G (1983) Random walks on fractal structures and percolation clusters. J Physique Lett 44:L13–L22

    Google Scholar 

  97. Rozenfeld AF, Albano EV (2001) Critical and oscillatory behavior of a system of smart preys and predators. Phys Rev E 63:061907

    ADS  Google Scholar 

  98. Shipilevsky BM (2007) Formation of a finite-time singularity in diffusion-controlled annihilation dynamics. J Phys C Condens Matter 19:065106

    ADS  Google Scholar 

  99. Sornette D (1998) Discrete scale invariance and complex dimensions. Phys Rep 297:239–270

    MathSciNet  ADS  Google Scholar 

  100. Stanley HE (1971) Introduction to phase transitions and critical phenomena. Oxford University Press, New York

    Google Scholar 

  101. Stauffer D, Aharoni A (1992) Introduction to the percolation theory, 2nd edn. Taylor and Francis, London

    Google Scholar 

  102. Suchecki K, Holyst JA (2006) Voter model on Sierpinski fractals. Physica A 362:338–344

    Google Scholar 

  103. Toussaint D, Wilczek F (1983) Particle-antiparticle annihilation in diffusive motion. J Chem Phys 78:2642–2647

    ADS  Google Scholar 

  104. Voigt CA, Ziff RM (1997) Epidemic analysis of the second-order transition in the Ziff–Gulari–Barshad surface-reaction model. Phys Rev E 56:R6241–R6244

    ADS  Google Scholar 

  105. Weiss GH (1995) A primer of random walkology. In: Bunde A, Havlin S (eds) Fractals in science. Springer, Berlin, p 119

    Google Scholar 

  106. Wieckowski A, Savinova ER, Vayenas CG (2003) Catalysis and electrocatalysis at nanoparticle surfaces. Marcel Dekker, New York, pp 1–959

    Google Scholar 

  107. Witten TA, Sander LM (1983) Diffusion-limited aggregation. Phys Rev B 27:5686–5697

    MathSciNet  ADS  Google Scholar 

  108. Yaldram K, Khan MA (1991) NO-CO reaction on square and hexagonal surfaces: A Monte Carlo simulation. J Catal 131:369–377

    Google Scholar 

  109. Zhdanov VP, Kasemo B (1994) Kinetic phase transitions in simple reactions on solid surfaces. Surf Sci Rep 20:111–189

    ADS  Google Scholar 

  110. Ziff RM, Brosilow BJ (1992) Investigation of the first-order phase transition in the \( { A-B_2 } \) reaction model using a constant-coverage kinetic ensemble. Phys Rev A 46:4630–4633

    ADS  Google Scholar 

  111. Ziff RM, Gulari E, Barshad Y (1986) Kinetic phase transitions in an irreversible surface-reaction model. Phys Rev Lett 56:2553–2556

    ADS  Google Scholar 

Books and Reviews

  1. Binder K (1997) Applications of Monte Carlo methods to statistical physiscs. Rep Prog Phys 60:487–560

    ADS  Google Scholar 

  2. Bunde A, Havlin S (1991) Fractals and disordered media. Springer, New York

    Google Scholar 

  3. Cardy J (1997) Goddard P, Yeomas JM (eds) Scaling and renormalization in statistical physics. Cambridge University Press, Cambridge

    Google Scholar 

  4. Christmann K (1991) Introduction to surface physical chemistry. Steinkopff, Darmstadt, p 1

    Google Scholar 

  5. Egelhoff WF Jr (1982) King DA, Woodruff DP (eds) Fundamental studies of heterogeneous catalysis, vol 4. Elsevier, Amsterdam

    Google Scholar 

  6. Lindenberg K, Oshanin G Tachiya M (2007) Chemical kinetics beyond the texbook: Fluctuations, Many-particle effects and anomalous dynamics. Special issue of J Phys C Condens Matter 19

    Google Scholar 

  7. Walgraef D (1997) Spatio-temporal pattern formation. Springer, New York

    Google Scholar 

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  1. Facultad de Ciencias Exactas, UNLP, CONICET, Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA) CCT La Plata, La Plata, Argentina

    Ezequiel V. Albano

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Albano, E.V. (2009). Reaction Kinetics in Fractals. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_447

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