Abstract
The non-Gaussian distribution function of the square distance of the center of mass from one fixed end of a polymer chain is discussed in detail. The characteristic functions of the distribution function with and without interactions are represented by the Feynman path integral.The path integral is evaluated in closed form exactly for the non-interacting case and approximately for the interacting case by the use of a trial path integral. The distribution function and some statistical quantities are calculated analytically and numerically in two dimensions. It is shown that the most probable peak of the distribution function is pushed out and pulled in by repulsive forces and attractive forces, respectively.
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H. Yamakawa, “Modern Theory of Polymer Solutions,” Harper and Row, N. Y., 1971.
S. F. Edwards, Proc. Phys. Soc., 85, 613 (1965).
S. F. Edwards, Proc. Phys. Soc., 88, 265 (1966).
H. P. Gills and K. F. Freed, Chem. Phys. Lett., 8, 234 (1971).
J. Naghizadeh, J. Chem. Phys., 48, 265 (1968).
J. des Cloizeaux, Phys. Rev., A, 10, 1665 (1974).
D. J. Bruch and M. A. Moore, J. Phys., A, 9, 435 (1976).
P. G. de Gennes, Phys. Lett., A, 38, 339 (1973).
Z. Alexandrowicz and Y. Accad, J. Chem. Phys., 54, 5338 (1971).
N. C. Smith and R. J. Fleming, J. Phys., A, 8, 929 (1975).
N. C. Smith and R. J. Fleming, J. Phys., A, 8, 538 (1975).
F. L. MacCrackin, J. Mazur, and C. M. Guttman, Macromolecules, 6, 859 (1973).
C. Domb, Adv. Chem. Phys., 15, 229 (1969).
C. Domb, J. Chem. Phys., 38, 2957 (1963).
M. E. Fisher and B. J. Hiley, J. Chem. Phys., 34, 1253 (1961).
T. Minato and A. Hatano, Polym. Prepr., 25, 322 (1976).
M. Fixman, J. Chem. Phys., 36, 306 (1962).
W. C. Forsman and R. E. Hughes, J. Chem. Phys., 38, 2118 (1963).
W. C. Forsman, J. Chem. Phys., 42, 2829 (1965).
S. R. Coriell and J. L. Jackson, J. Math. Phys., 8, 1276 (1967).
R. F. Hoffman and W. C. Forsman, J. Chem. Phys., 52, 2222 (1970).
S. K. Gupta and W.C. Forsman, J. Chem. Phys., 55, 2594 (1971).
H. Fujita and T. Norisue, J. Chem. Phys., 52, 1115 (1970).
R. Koyama, J. Phys. Soc., 24, 580 (1968).
S. K. Gupta and W. C. Forsman, Macromolecules, 5, 779 (1972).
N. Wiener, Acta. Math., 55, 117 (1930).
R. P. Feynman, Rev. Mod. Phys., 20, 367 (1948).
K. F. Freed, Adv. Chem. Phys., 22, 1 (1972).
F. W. Weigel, Physics Report, 16C (1975).
S. F. Edwards and Y. B. Gulyaev, Proc. Phys. Soc., 83, 495 (1964).
M. Saitoh, private communication.
T. Minato, Polym. J. (to be published).
R. P. Feynman, Phys. Rev., 97, 660 (1955).
I. C. Sanchez and C. von Frankenberg, J. Chem. Phys., 52, 1617 (1970). These authors confused Z2 with the value of the three-dimensional case in their notation. The correct one is Z2=iNq in the two-dimensional case. Refer to 38.
P. J. Flory, “Principles of Polymer Chemistry,” Cornell University Press, Ithaca, N.Y., 1953.
J. de Cloizeaux, J. Phys., (Paris), 31, 715 (1970).
I. M. Gelfand and A.M. Yaglom, J. Math. Phys., 1, 48 (1960).
S. G. Brush, Rev. Mod. Phys., 33, 79 (1963).
R. P. Feynman and A. R. Hibbs, “Quantum Mechanics and Path Integrals,” McGraw-Hill, N.Y., 1965.
R. P. Feynman, “Statistical Mechanics,” Benjamin, N.Y., 1972.
T. Minato and A. Hatano, J. Phys. Soc. Jpn. 42, 1992 (1977).
K. Sole, Macromolecules, 5, 705 (1972).
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Minato, T., Hatano, A. Distribution Functions of Polymers with and without Interactions. I. The Distribution Function of the Square Distance of the Center of Mass from One Fixed End of a Polymer Chain. Polym J 9, 239–251 (1977). https://doi.org/10.1295/polymj.9.239
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DOI: https://doi.org/10.1295/polymj.9.239