Theta point (‘‘tricritical’’) region behavior for a polymer chain: Transition to collapse

Abstract

The conformational space renormalization group method is generalized further to describe excluded volume effects in finite molecular weight polymers in the theta point region where contributions from effective three-body interactions become appreciable. The theory builds upon our previous description of the good solvent region and uses t’Hooft-Veltman style dimensional regularization along with the renormalization group (RG) to determine the general analytic structure of measurable quantitites of interest. Our formalism is compared in detail with that employed in field theory to describe tricritical behavior. Although many results are in qualitative agreement between the two approaches, there are some numerical differences. Our method considers renormalization for chains with fixed length, and this introduces differences into the renormalization scheme from that used in field theory, differences dictated by the physical differences in the types of systems considered. Our treatment of finite length chains contrasts with the field theoretical expansions about the unphysical limit of infinitely long polymers; thus it entirely avoids the use of the method of insertions. The RG theory is used to calculate the mean square end-to-end distance 〈R2〉 and the second and third virial coefficients A2 and A3 in the theta region as a function of both the two- and three-body interactions and of the chain length. We compute the temperature difference between the point at which A2 vanishes and at which 〈R2〉 is the pure Gaussian limiting value. The virial coefficients are used to evaluate the dilute solution portion of the coexistence curve, providing an estimate of the collapse transition temperature. The presence of essential logarithmic dependences of the coexistence curve on chain length implies that mean field theory is not valid for describing measurable quantities in this region.