Skip to main content
SpringerLink
Log in

Mean-Field Kinetic Modeling of Polymerization: The Smoluchowski Coagulation Equation

  • Chapter
  • First Online:
Grafting/Characterization Techniques/Kinetic Modeling

Part of the book series: Advances in Polymer Science ((POLYMER,volume 137))

Abstract

The Smoluchowski coagulation equation derived back in 1916 is usually linked with the diffusivity and size of aggregating particles. It can also be used as a versatile tool for mean-field kinetic analyses. In this paper it is shown to be an efficient tool for studying the relationships between the reactivity of functional groups in monomers and the size distribution of polymer species in step and chain growth polymerizations. The Smoluchowski coagulation equation and its modifications are applied as models of kinetically controlled growth reactions (with irreversible elementary reaction steps). Advantages and limitations of this method of modeling polymerization processes are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Notes

  1. Forthe process of graph-theoretical evolution of trees, which is analogous to the aggregation described by Eq. (1), the Smoluchowski equation was rederived rigorously, see ai]Balinska_K, Galina H, Quintas LV, Szymanski J (1996) Discr Appl Maths 67:13

    Google Scholar 

  2. Pauling L, Pauling P (1975) Chemistry. Freeman, San Francisco 1975

    Google Scholar 

  3. Smoluchowski M v (1916) Phys Z 17: 585

    Google Scholar 

  4. Smoluchowski M v (1917) Z Chem Phys 92: 129

    Google Scholar 

  5. Drake R ( 1972) In: Hidy GM, Brook JR (eds) Topics in current aerosol research. Pergamon, Oxford

    Google Scholar 

  6. Kuchanov SJ (1978) Methods of kinetic calculations in polymer chemistry (in Russian). Izd Chimia, Moscow

    Google Scholar 

  7. Ziff RM.(1984) In: Family F, Landau DP (eds) Kinetics of aggregation and gelation. Elsevier Sci Publ BV

    Google Scholar 

  8. Flory JP (1953) Principles of polymer chemistry. Cornell Univ Press, Ithaca

    Google Scholar 

  9. Gordon M, Scantlebury GR (1966) Proc R Soc London A 292: 380

    Article  CAS  Google Scholar 

  10. Gordon M, Scantlebury GR (1964) Trans Faraday Soc 6: 604

    Article  Google Scholar 

  11. Gordon M (1962) Proc Roy Soc (London) A268: 240

    Google Scholar 

  12. Dušek K, Prins W (1969) Adv Polym Sci 6: 1

    Article  Google Scholar 

  13. Galina H ( 1995) Makromol Theory Simul 4: 801

    Article  CAS  Google Scholar 

  14. Cheng K Ch, Chiu WY (1993) Macromolecules 26: 4658, 4665

    Google Scholar 

  15. Flory PJ (1946) Chem Rev 39: 137

    Article  CAS  Google Scholar 

  16. Stockmayer WH (1943) J Chem Phys 11: 45

    Article  CAS  Google Scholar 

  17. Spouge JL (1983) Macromolecules 16: 121

    Article  CAS  Google Scholar 

  18. McLeod JB (1962) Quart J Math (Oxford) 13: 119, 193

    Google Scholar 

  19. Drake RL (1972) In: Hidy GM, Brock GR (eds) Topics in current aerosol research, vol 3, pt 2

    Google Scholar 

  20. Ziff RM, Ernst MH, Hendriks EM (1984) J Colloid Interface Sci 100: 220

    Article  CAS  Google Scholar 

  21. Drake RL, Wright TJ (1972) J Atmos Sci 29: 548

    Article  Google Scholar 

  22. Treat RP (1990) J Phys A: Math Gen 23: 3003

    Article  Google Scholar 

  23. Spouge JL (1983) J Stat Phys 31: 363

    Article  Google Scholar 

  24. Ziff RM, Stell G (1980) J Chem Phys 73: 3492

    Article  CAS  Google Scholar 

  25. Van Dongen PGJ, Ernst MH (1987) J Colloid Interface Sci 115: 27

    Article  Google Scholar 

  26. Van Dongen PGJ, Ernst MH (1986) J Stat Phys 44: 785

    Article  Google Scholar 

  27. Leyvraz F (1984) In: Family F, Landau DP (eds) Kinetics aggregation and gelation. Elsevier Sci Publ BV

    Google Scholar 

  28. Leyvraz F, Tschudi HR (1981) J Phys A: Math Gen 14: 3389; (1982) J Phys A: Math Gen 15: 1951

    Article  CAS  Google Scholar 

  29. Hendriks EM, Ernst MH, Ziff RM (1983) J Stat Phys 31: 519

    Article  Google Scholar 

  30. Erdös P, Renýi A (1959) I Publ Math Debrecen 6: 290; (1960) MTA Mat Kut Int Kozl 5: 17

    Google Scholar 

  31. Buffet E, Pulé JV (1991) J Stat Phys 64: 87

    Article  Google Scholar 

  32. Stockmayer WH (1943) J Chem Phys 11: 45

    Article  CAS  Google Scholar 

  33. Whittle P (1965): Proc Camb Phil Soc 61: 475

    Google Scholar 

  34. Whittle P ( 1965) Proc R Soc London A 285: 501

    Google Scholar 

  35. Spouge JL (1983) J Phys A: Math Gen 16: 767

    Article  CAS  Google Scholar 

  36. Stockmayer W H (1943) J Chem Phys 11: 45

    Article  CAS  Google Scholar 

  37. Macosko CW, Miller DR (1976) Macromolecules 9: 199

    Article  CAS  Google Scholar 

  38. Miller DR, Macosko CW (1976) Macromolecules 9: 206

    Article  CAS  Google Scholar 

  39. Kuchanov SI, Povolotskaya Ye S (1982) Vysokomol Soed A24(10): 2179, 2190

    Google Scholar 

  40. Galina H, Szustalewicz A (1989) Macromolecules 22I 3124

    Google Scholar 

  41. Sarmoria C, Miller DR (1991) Macromolecules 24: 1833

    Article  CAS  Google Scholar 

  42. Faliagas AC (1993) Macromolecules 26: 3838

    Article  CAS  Google Scholar 

  43. Kuchanov SI (1979) Dokl Akad Nauk SSSR 249: 899

    CAS  Google Scholar 

  44. Galina H, Szustalewicz A (1990) Macromolecules 23: 3833

    Article  CAS  Google Scholar 

  45. Galina H, Kaczmarski K, Para B, Sanecka B (1992) Makromol Chem Theory Simul 1: 37

    Article  CAS  Google Scholar 

  46. Mikes J, Dušek K (1982) Macromolecules 15: 185

    Article  Google Scholar 

  47. Šomvársky J, Dušek K (1994) Polym Bull (Berlin) 33: 369; 377

    Article  Google Scholar 

  48. Galina H, Lechowicz J (1995) Comput Polym Sci

    Google Scholar 

  49. Dušek K (1979) Polym Bull (Berlin) 1: 523

    Article  Google Scholar 

  50. Ziff RM (1980) J Stat Phys 23: 241

    Article  Google Scholar 

  51. Good IJ (1963) Proc R Soc London A 272: 54

    CAS  Google Scholar 

  52. Van Dongen PGJ, Ernst MH, Ziff RM (1983) J Stat Phys 31: 519

    Article  Google Scholar 

  53. Fukutomi T, Koshiro Y (1996) Polym J 28: 758

    Article  CAS  Google Scholar 

  54. Irzhak TF, Peregudov NI, Irzhak VI, Rozenberg BA (1993) Vysokomol. Soed B 35:905,1545

    Google Scholar 

  55. Jackobson H, Stockmayer WH (1950) J Chem Phys 18: 1800

    Google Scholar 

  56. Flory P J (1969) Statistical mechanics of chain molecules. Interscience, New York

    Google Scholar 

  57. Friedman B, O’Shaughnessy B (1989) Phys Rev A 40: 5950

    Article  CAS  Google Scholar 

  58. Lu B, Bak TA (1988) In: Kramer O (ed) Biological and synthetic polymer networks. Elsevier, Amsterdam

    Google Scholar 

  59. Galina H (1991) Makromol Chem Macromol Symp 40: 45

    Google Scholar 

  60. Galina H, Lechowicz J, Para B, Sanecka B (1994) Polimery (Warsaw) 39: 429

    CAS  Google Scholar 

  61. Müller H (1928) Kolloidchemische Beiheft 27: 223

    Article  Google Scholar 

  62. Lushnikov AA (1974) J Colloid Interface Sci 48: 400; (1978) Izv Atm Ok Fiz 14: 738

    Article  Google Scholar 

  63. Gabellini Y, Meunier J-L ( 1992) J Phys A: Math Gen 25: 3683

    Article  Google Scholar 

  64. Kreer M, Penrose O ( 1994) J Stat Phys 75: 389

    Article  Google Scholar 

  65. Van Dongen PGJ, Ernst MH ( 1987) J Stat Phys 49: 879

    Article  Google Scholar 

  66. Stauffer D, Coniglio A, Adam M (1982) Adv Polym Sci 44: 103

    Article  CAS  Google Scholar 

  67. Villarica M, Casey MJ, Goodisman J, Chaiken J (1992) J Chem Phys 98: 4610

    Article  Google Scholar 

  68. Krivitsky DS (1996) J Phys A: Math Gen 28: 2025

    Article  Google Scholar 

  69. Ziff RM, McGrady ED (1985) J Phys A: Math Gen 18: 3027

    Article  CAS  Google Scholar 

  70. Ball JM, Carr J (1990) J Stat Phys 61: 203

    Article  Google Scholar 

  71. Costas ME, Moreau M, Vicente L ( 1995) J Phys A: Math Gen 28: 2981

    Article  CAS  Google Scholar 

  72. da Costa FP (1995) J Math Anal Appl 192:892

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Materials Chemistry, Faculty of Chemistry, Rzeszów University of Technology, 35-959, Rzeszów, Poland

    Henryk Galina

Authors
  1. Henryk Galina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaromir B. Lechowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

H. Galina Y. Ikada K. Kato R. Kitamaru J. Lechowicz Y. Uyama C. Wu

Additional information

Affectionately dedicated to Professor Manfred Gordon on his 80th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Galina, H., Lechowicz, J.B. (1998). Mean-Field Kinetic Modeling of Polymerization: The Smoluchowski Coagulation Equation. In: Galina, H., et al. Grafting/Characterization Techniques/Kinetic Modeling. Advances in Polymer Science, vol 137. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69685-7_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-69685-7_4

  • Received:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64016-5

  • Online ISBN: 978-3-540-69685-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation