Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T17:08:56.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical processes of aggregation and polymerization

Published online by Cambridge University Press:  24 October 2008

P. Whittle
P. Whittle
Affiliation:
University of Manchester
Get access Rights & Permissions [Opens in a new window]

Abstract

We study processes in which units (particles) associate into clusters, which are then also capable of dissociation. Such processes are discussed generally in section 2, where a stochastic kinetic equation (10) is proposed which bridges the gap between the conventional kinetic equations (8) and the statistical equilibrium concept of the Gibbs distribution.

In section 4 we consider the equilibrium behaviour of a process for which the association rate of two units which are already bound to j and k other units respectively has the form (21). This is very much more general than the equi-reactive bond model usually discussed. The principal results are given in Theorem 1; from a single pair of equations (28) and (29) based on the Hj of (21) one can determine critical points, expected number of bonds, the distribution and moments of cluster size, and most other quantities of interest. This is without reference to any other consideration, such as kinetic or stoichiometric relations.

Some particular cases are worked through in section 5. The classic Flory-Stockmayer results for units with f equi-reactive sites are obtained systematically and economically, with all parameters in terms of physically given quantities. Another type of example seems to indicate the existence of a second critical point.

Corresponding results for the case of several types of unit are stated and illustrated in section 6.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 2 , April 1965 , pp. 475 - 495
Copyright
Copyright © Cambridge Philosophical Society 1965

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Band, W.Quantum statistics (Van Nostrand; Princeton, 1955).Google Scholar
(2)Bartlett, M. S.An introduction to stochastic processes (Cambridge, 1955).Google Scholar
(3)Bharucha-Reid, A. T.Elements of the theory of Markov processes and their applications (McGraw-Hill; New York, 1960).Google Scholar
(4)Born, M. and Fuchs, K.The statistical mechanics of condensing systems. Proc. Boy. Soc. Ser. A, 166 (1938), 391–414.Google Scholar
(5)Feller, W.An introduction to probability theory and its applications (Wiley, 2nd ed.; New York, 1957).Google Scholar
(6)Flory, P. J.Principles of polymer chemistry (Cornell, 1953).Google Scholar
(7)Goldberg, R. J. J.A theory of antibody-antigen reactions. I. J. Amer. Ghent. Soc. 74 (1954), 5715–5725.CrossRefGoogle Scholar
(8)Goldberg, R. J. J.A theory of antibody-antigen reactions. II. J. Amer. Chem. Soc. 75 (1953), 3127–3131.CrossRefGoogle Scholar
(9)Good, I. J.The number of individuals in a cascade process. Proc. Cambridge Philos. Soc. 45 (1949), 360–363.CrossRefGoogle Scholar
(10)Good, I. J.Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes. Proc. Cambridge Philos. Soc. 56 (1960), 366–380.CrossRefGoogle Scholar
(11)Good, I. J.Cascade theory and the molecular weight averages of the sol fraction. Proc. Roy. Soc. Ser. A, 272 (1963), 54–59.Google Scholar
(12)Gordon, M.Good's theory of cascade processes applied to the statistics of polymer distributions. Proc. Boy. Soc. Ser. A, 268 (1962), 240–259.Google Scholar
(13)Harris, T. E.The theory of branching processes (Springer; Berlin, 1963).CrossRefGoogle Scholar
(14)Mayer, J. E. and Mayer, M. G.Statistical mechanics (Wiley; New York, 1949).Google Scholar
(15)Münster, A.Prinzipien der statistischen Mechanik (Springer; Berlin, 1959).CrossRefGoogle Scholar
(16)Parzen, E.Stochastic processes (Holden-Day; San Francisco, 1962).Google Scholar
(17)Stockmayer, W. H.Theory of molecular size distribution and gel formation in branched chain polymers. J. Chem. Phys. 11 (1943), 45–55.CrossRefGoogle Scholar
(18)Stockmayer, W. H.Theory of molecular size distribution and gel formation in branched polymers. II. General cross linking. J. Chem. Phys. 12 (1944), 125–131.CrossRefGoogle Scholar
(19)Uhlenbeck, G. F. and Ford, G. W. The theory of linear graphs with applications to the theory of the virial development of the properties of gases. In Studies in statistical mechanics, vol. i (North-Holland; Amsterdam, 1962).Google Scholar
(20)Watson, G. S.On Goldberg's theory of the precipitin reaction. J. Immunology, 80 (1958), 182–185.CrossRefGoogle ScholarPubMed
(21)Whittle, P.The equilibrium statistics of a clustering process in the uncondensed phase. Proc. Roy. Soc. Ser. A (to appear).Google Scholar