Skip to main content
SpringerLink
Log in

Theory of ring formation in a reversible system: general solutions

  • Original Contributions
  • Published:
Colloid and Polymer Science Aims and scope Submit manuscript

Abstract

The enumeration theory is extended in this work into a more general theory, taking back-reactions into consideration. The solutions may faithfully reproduce real processes from arbitrary starting points to a steady-state. Therefore, the presented theory includes the equilibrium theory by Jacobson-Stockmayer, the numerical solution by Gordon-Temple, and the irreversible theory by the present authors. The solutions are described first in general forms of transition probabilities {P}, and then explicitly with the aid of rate equations; simple proofs are given. The presented theory was applied to an experimental data: the distribution of cyclic species in poly(ethylene terephthalate). We shall show that agreement between theory and experiment is nearly perfect.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

N 0 :

Total number of units

V :

System volume

C 0=N 0/N A ·V :

Initial concentration (N A : Avogadro's number)

L x :

AB type chain x-mer; (AB)x

N x :

Number of AB type x-mers

R x :

Ring x-mer

N Rx :

Number of ring x-mers

E :

Small molecule eliminated by bond-formation

N E :

Number of small molecules eliminated by bond-formation

h k :

Number of reacted functional units (f.u.) in statek

ξ k :

Number of reacted functional units (f.u.) in chains in statek

Ξ k :

Total number of units in chains in statek

D=h k /N 0 :

Extent of reaction in statek

D *=ξ k /Ξ k :

Extent of reaction in chains in statek

k L :

Chain-propagation rate constant

k Rx :

Cyclization rate constant of chain x-mers

k B :

Bond breakage rate constant of chains

k B,Rx :

Bond breakage rate constant of cyclic x-mers

<k Rx > k :

Mean cyclization rate constant in statek

g(x)=k B,Rx /k B :

Ring-opening factor of cyclic x-mers

P Lx,k :

Probability that a chain x-mer will be formed in statek

{P}:

Set of transition probabilities per single jump in forward direction or reverse direction (see the text on individual transition probabilities)

M A :

Total AA monomer unit number

M B :

Total BB monomer unit number

M 0=M A +M B :

Total particle number

ζ A,i =2M A −h i :

Unreacted A functional unit (f.u.) number in statei

ζ B,i =2M B −h i :

Unreacted B f.u. number in statei

ζ Ax :

Unreacted A f.u. number on x-mers

h i :

Number of reacted A (or B) f.u. in statei

ξ i :

Number of reacted A (or B) f.u. in chains in statei

Ξ A,i =2M A −h i +ξ i :

A f.u. number in chains in statei

Ξ B,i =2M B −h i +ξ i :

B f.u. number in chains in statei

Ξ i =2(M 0−h i +ξ i ):

Total f.u. number in chains in statei

D=h i /M 0 :

Extent of reaction in statei

D * A =ξ i /Ξ A,i :

Extent of reaction of A f.u. in chains in statei

D * B =ξ i /Ξ B,i :

Extent of reaction of B f.u. in chains in statei

D *=2ξ i /Ξ i :

Extent of reaction in chains in statei

L αx :

(AA-BB)x-1-AA type chain x-mer;x=1,2,3,...

L βx :

BB-(AA-BB)x type chain x-mer;x=0,1,2,...

L γx :

(AA-BB)x type chain x-mer;x=1,2,3,...

N αx :

Number of α type x-mers

N βx :

Number of β type x-mers

N γx :

Number of γ type x-mers

References

  1. Kuhn W (1934) Kolloid Z 68:2

    Google Scholar 

  2. Jacobson H, Stockmayer WH (1950) J Chem Phys 18:1600

    Google Scholar 

  3. Jacobson H, Beckmann CO, Stockmayer WH (1950) J Chem Phys 18:1607

    Google Scholar 

  4. Flory PJ, Semlyen JA (1966) J Am Chem Soc 3209:88

    Google Scholar 

  5. Harris FE (1955) J Chem Phys 23:1518

    Google Scholar 

  6. Kilb RW (1958) J Phys Chem 969:62

    Google Scholar 

  7. Ahmad Z, Stepto RFT (1980) Colloid & Polymer Sci 258:663

    Google Scholar 

  8. Gordon M, Scantlebury GR (1966) Proc Roy Soc (London) A292:380

    Google Scholar 

  9. Spouge JL (1986) J Stat Phys 43:143

    Google Scholar 

  10. Gordon M, Roe R-J (1956) J Polym sci 21:75

    Google Scholar 

  11. Gordon M, Scantlebury GR (1968) J Polym Sci Part C No 16:3933

    Google Scholar 

  12. Gordon M, Temple WB (1972) Makromol Chem 152:277 Makromol Chem 1972 160:263

    Google Scholar 

  13. Temple WB (1972) Makromol Chem 160:277

    Google Scholar 

  14. Rolando RJ, Macosko CW (1987) Macromolecules 20:2707

    Google Scholar 

  15. Suematsu K, Okamoto T, “Theory of Ring Formation in Irreversible AB System”, see the preceding paper

  16. Yuan X, Masters AJ, Nicholas CV, Booth C (1988) Makromol Chem 189:823

    Google Scholar 

  17. Simha R (1941) J Applied Phys 12:569

    Google Scholar 

  18. Cooper DR, Semlyen JA (1973) Polymer 14:185

    Google Scholar 

  19. Suter UW, Mutter M (1979) Makromol Chem 180:1761

    Google Scholar 

  20. For example, see Ogata N (1971) “Polycondensation”, p-92, Tokyo Kagaku Doujin, etc.

  21. Hermans JJ (1965) Makromol Chem 87:21

    Google Scholar 

  22. Flory PJ (1986) “Principles of Polymer Chemistry”, Cornell University Press, Ithaca and London

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kohno Medical Institute, Tomitahama 26-14, 512, Yokkaichi City Mie Prefecture, Japan

    K. Suematsu & T. Okamoto

Authors
  1. K. Suematsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Okamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Suematsu, K., Okamoto, T. Theory of ring formation in a reversible system: general solutions. Colloid Polym Sci 270, 405–420 (1992). https://doi.org/10.1007/BF00665983

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00665983

Key words

Search

Navigation