## Abstract

A real-space renormalization group method is used to calculate the swelling factor of a three-dimensional, randomly-jointed chain with hard-sphere excluded-volume interactions. This method differs from more conventional procedures patterned on the field theoretic approach pioneered by Gell-Mann and Low and Wilson. It is specifically designed to produce estimates of the swelling factor for finite values of the chain length and for a broad range of [excluded-volume] bare coupling coefficients. In addition to predictions specific to chains of finite length, the theory produces two distinct power-law scaling formulas for the asymptotic, long-chain limit of the swelling factor. One of these is descriptive of an ideal chain and is associated exclusively with very small values of the bare excluded-volume interaction parameter. The other is appropriate to chains with larger values of the interaction parameter and which exhibit significant deviations from ideality. The `critical exponent' associated with this second class of chains is equal to v = 0.5916, a value which agrees quite well with the results of previous investigations. Our renormalization group calculations are based on a pair of functional equations, one for an effective coupling function and another for the swelling factor.

Original language | English (US) |
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Pages (from-to) | 22-47 |

Number of pages | 26 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 272 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 1999 |

### Bibliographical note

Funding Information:This research has been supported by grants from the Theoretical and Computational Chemistry Program of the National Science Foundation. JIS gratefully acknowledges financial support through a Dreyfus New Faculty Award and a McKnight/Land-Grant Assistant Professorship. We also gratefully acknowledge grants of computer time from the Minnesota Supercomputer Institute. Finally, we are indebted to Professor Leo Kadanoff for encouragement and helpful communications.