Abstract
Branched Polymers in solution are known to undergo a collapse transition driven by the quality of the solvent at the Θ-point. The collapse of the polymer is in a characteristic length, usually taken to be the root mean square radius of gyration, R, of the polymer. In the “good solvent” regime, one expects that R ~ M ν, where M is the molecular mass of the polymer. ν is a critical exponent, commonly called the metric exponent (and it describes the scaling of R with M). In three dimensions, it is believed that \(\nu = \tfrac{1}{2}\), and the branched polymer is said to be “expanded”. Beyond the collapse transition, it is believed that \(v = \frac{1}{2} \) , so that the polymer scales like a solid object (d is the spatial dimension). Branched polymers can be modeled as trees in the cubic lattice, with a short-ranged interaction between vertices which are nearest neighbour in the lattice. Trees can be efficiently sampled by a Metropolis Monte Carlo algorithm. We collect data on the Θ-transition by finding the peak in the specific heat of trees using a Robbins-Monro scheme. In addition, Monte Carlo simulations on trees over a wide range of the short-ranged force using umbrella sampling is described. The data strongly support the notion that the collapse transition is a continuous (second order) transition with a divergent specific heat. We also report values of computed critical exponents.
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References
G. Parisi and S. Sourlas, Phys. Rev. Lett., 46 (1981) 871.
N. Madras, C.E. Soteros, S.G. Whittington, J.L. Martin, M.F. Sykes, S. Flesia and D.S. Gaunt, J. Phys. A: Math. Gen.,23 (1990) 5327.
D.S. Gaunt and S. Flesia, J. Phys. A: Math. Gen.,91 (1991) 2127.
D.C. Rapaport J. Phys. A: Math. Gen.,10 (1977) 637.
H.A. Lim, A. Guha and Y. Shapir, Phys. Rev. A, 38 (1988) 3710.
B. Derrida and H.J. Herrmann, J. Physique, 44 (1983) 1365.
D.S. Gaunt and S. Flesia Physica A, 168 (1990) 602.
E.J. Janse Van Rensburg and N. Madras, J. Phys. A: Math. Gen., 25 (1992) 303.
H. Robbins and S. Monro, Ann. Math. Stat., 22 (1951) 400.
G.M. Torrie and J.P. Valleau, J. Comput. Phys., 23 (1977) 187.
I.D. Lawrie and S. Sarlbach, Theory of Tricritical Points, in Phase Transitions and Critical Phenomena, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1984), 9 (1984) 1.
A.L. Owzarek, T. Prellberg and R. Brak, J. Phys. A: Math. Gen., 26 (1993) 4565.
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys., 21 (1953) 1087.
C. Vanderzande, Phys. Rev. Lett., 70 (1993) 3595.
S. Flesia, D.S. Gaunt, C.E. Soteros and S.G. Whittington, J. Phys. A: Math. Gen., 25 (1992) 3515.
S. Flesia, D.S. Gaunt, C.E. Soteros and S.G. Whittington, J. Phys. A: Math. Gen., 26 (1993) L993.
S. Flesia, D.S. Gaunt, C.E. Soteros and S.G. Whittington, J. Phys. A: Math. Gen., 27 (1994) 5831.
F. Seno and C. Vanderzande, Preprint (1995).
P.M. Lam, Phys. Rev. B, 38 (1988) 2813.
I.S. Chang and Y. Shapir, Phys. Rev. B, 38 (1988) 6736.
A.L. Stella, E. Orlandini, I. Beichl, F. Sullivan, M.C. Tesi and T.L. Einstein, Phys. Rev. Lett., 69 (1992) 3650.
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Van Rensburg, E.J.J., Madras, N. (1998). Monte Carlo Simulation of the Θ-Point in Lattice Trees. In: Whittington, S.G. (eds) Numerical Methods for Polymeric Systems. The IMA Volumes in Mathematics and its Applications, vol 102. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1704-6_9
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DOI: https://doi.org/10.1007/978-1-4612-1704-6_9
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