Self-consistent variational theory for globules

– A self-consistent variational theory for globules based on the uniform expansion method is presented. This method, first introduced by Edwards and Singh to estimate the size of a self-avoinding chain, is restricted to a good solvent regime, where two-body repulsion leads to chain swelling. We extend the variational method to a poor solvent regime where the balance between the two-body attractive and the three-body repulsive interactions leads to contraction of the chain to form a globule. By employing the Ginzburg criterion, we recover the correct scaling for the θ-temperature. The introduction of the three-body interaction term in the variational scheme recovers the correct scaling for the two important length scales in the globule—its overall size R, and the thermal blob size ξT . Since these two length scales follow very different statistics—Gaussian on length scales ξT , and space filling on length scale R—our approach extends the validity of the uniform expansion method to non-uniform contraction rendering it applicable to polymeric systems with attractive interactions. We present one such application by studying the Rayleigh instability of polyelectrolyte globules in poor solvents. At a critical fraction of charged monomers, fc, along the chain backbone, we observe a clear indication of a first-order transition from a globular state at small f to a stretched state at large f ; in the intermediate regime the bistable equilibrium between these two states shows the existence of a pearl-necklace structure. Introduction. – The uniform expansion method of Edwards and Singh is a well known self-consistent variational approach to study the size and the probability distribution of the end-to-end distance of a chain in a good solvent [1, 2]. The method is based on the uniform expansion of a chain in terms of the expansion in an unknown step length b1 such that the size of the self-avoiding chain consisting of N segments is governed by Gaussian statistics, R = Nb1. The variational parameter b1 is then determined self-consistently by a perturbative calculation, usually truncated at first order. In particular, the method recovers the correct scaling, 〈 R 2 〉 ≈ b(v/b)N, for the size of an excluded volume chain, a result which is consistent with the classical Flory theory [3]. In contrast to other analytical approaches—for instance, the self-consistent field theory by Edwards [4], de Gennes [5], and the more rigorous renormalization group treatment by Freed [6, 7]—the advantage of the uniform expansion method is its mathematical simplicity in treating the complex excluded volume problem, which renders it applicable to more complicated systems with additional interactions. It has