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XIII.—The Graph-like State of Matter. I. Statistical effects of correlations due to substitution effects, including steric hindrance, on polymer distributions.

Published online by Cambridge University Press:  14 February 2012

M. Gordon  and
T. G. Parker
M. Gordon
Affiliation:
Department of Chemistry, University of Essex
T. G. Parker
Affiliation:
Department of Chemistry, University of Essex
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Synopsis

Models which treat molecules as graphs provide the first approximation to practically all chemical and physical theories of polymers. This paper, worded so as to aim at intelligibility by both chemists and mathematicians, deals with statistical effects of correlations on distributions of molecules in their graph-like states. The correlations are formally equivalent to fertility correlations in family trees, and the theory covers the range of correlation to the point where a man's fertility expectations depend on how many brothers he has. Chemically, this describes the ‘second shell substitution effect’ which includes certain steric hindrance situations; mathematically, the problem is solved by restoring the Markovian nature of a conventional Galton-Watson process. Long-range correlations are an inevitable and troublesome aspect of the structure of gels and this work represents a step towards improving models for real gels.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 69 , Issue 3 , 1971 , pp. 181 - 198
Copyright
Copyright © Royal Society of Edinburgh 1971

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References

References to Literature

Balaban, A. T., Fӑrcaşiu, D. and Harary, F., 1971. J. Labell. Compounds, in press.Google Scholar
Butler, D. S., Malcolm, G. N. and Gordon, M., 1966. Proc. Roy. soc., 295A, 29.Google Scholar
Charlesby, A. and Pinner, S. H., 1959. Proc. Roy. soc., 249A, 367.Google Scholar
Crump, K. S. and Mode, J., 1969. J. Appl. Prob., 6, 205.Google Scholar
Dobson, G. R. and Gordon, M., 1964. J. Chem. Phys., 41, 2389.Google Scholar
Dobson, G. R. and Gordon, M., 1965. J. Chem. Phys., 43, 705.Google Scholar
Dušek, K. and Prins, W., 1969. Adv. Polym. Sci., 6, 1.Google Scholar
Essam, J. W. and Fisher, M. E., 1970. Rev. Mod. Phys., 42, 272.Google Scholar
Flory, P. J., 1941. J. Am. Chem. soc., 63, 3083, 3091, 3096.Google Scholar
Flory, P. J., 1953. The principles of polymer chemistry. Ithaca: Cornell Univ. Press.Google Scholar
Good, I. J., 1963. Proc. Roy. soc., 272A, 54.Google Scholar
Gordon, M., 1962. Proc Roy. soc., 228A, 240.Google Scholar
Gordon, M., 1970 a. J. Chem. soc., A, 729.CrossRefGoogle Scholar
Gordon, M., 1970 b. Colloquia Math. Soc. Janos Bolyai, 4, 511.Google Scholar
Gordon, M. and Scantlebury, G. R., 1964. Trans. Faraday soc., 60, 604.Google Scholar
Gordon, M. and Scantlebury, G. R., 1966. Proc Roy. soc., 292A, 380.Google Scholar
Gordon, M. and Scantlebury, G. R., 1968. J. Polym. Sci., C16, 3933.Google Scholar
Gordon, M., Parker, T. G. and Temple, W. B., 1971. J. Combin. Theory, in press.Google Scholar
Gordon, M. and Temple, W. B., 1970. J. Chem. soc., A, 729.CrossRefGoogle Scholar
Gordon, M., Ward, T. C. and Whitney, R. S., 1970. Am. Chem. Soc. Symp. (Highly Cross-linked Network Polymers, in press.) Chicago: Plenum Press.Google Scholar
Harary, F., 1967. A seminar on graph theory. New York: Holt, Rinehart and Winston.Google Scholar
Harary, F., 1969. Graph theory. London: Addison Wesley.Google Scholar
Kajiwara, K., 1971. J. Chem. Phys., 54, 296.Google Scholar
Kajiwara, K., Burchard, W. and Gordon, M., 1970. Br. Polym. J., 2, 110.Google Scholar
Kubitschek, H. E., 1967. Proc. 5th Berkeley Symp. Math. Statist. Probi., 4, 549.Google Scholar
Matula, D. W., 1969. Ann. N.Y. Acad. Sci., 159, 314.Google Scholar
Matula, D. W., Groenweghe, L. C. D., and Van Wazer, J. R., 1964. J. Chem. Phys., 41, 3105.CrossRefGoogle Scholar
Pólya, G., 1937. Acta Math., 68, 145.Google Scholar
Renyi, A., 1969. Lectures on the Theory of Search, Mimeo 600.7, Dept. Statist., UNC Chapel Hill.Google Scholar
Stockmayer, W. H., 1943. J. Chem. Phys., 11, 45.Google Scholar
Strella, S. and Bibeau, A. A., 1966. J. Macromol. Chem., 1, 417.Google Scholar
Whittle, P., 1965 a. Proc. Camb. Phil. Soc. Math. Phys. Sci., 61, 475.Google Scholar
Whittle, P., 1965 b. Proc. Roy. soc., 285A, 501.Google Scholar