Systematic blobbing procedure for the calculation of universal polymer properties

Abstract

A systematic renormalization group transformation along the chemical sequence of a polymer chain, systematic blobbing procedure (SBP), is introduced for the calculation of universal configurational properties. The method, which does not rely on the field theoretic analogy, gives to first order in ε (ε=4−d, d=space dimension) a firm basis to the decimation procedure suggested by de Gennes and put into practice by Gabay and Garel (GG). The renormalization transformation acts on a ‘‘Hamiltonian’’ of the system and produces Wilson-type approximate recursion relations for the coupling parameters. The SBP is used to calculate the exponent ν for a polymer chain in the good solvent region to first order in ε. The method can in principle be extended to higher orders in ε. By means of the SBP it is shown that the static universal properties of a polymer chain are not modified by the presence of a Gaussian random potential. The validity of the GG method is further strengthened by showing the equivalence, in the determination of ν, of their method and a Fixman perturbation matching procedure. The exponent ν is consequently obtained to second order in ε. All the results agree with the ‘‘corresponding’’ field theoretic calculations.