Revisiting the Θ Point

Abstract

Using the first-order perturbation theory, we compute the osmotic second and third virial coefficients, the mean-square end-to-end distance ⟨Re2⟩, and the mean-square radius of gyration ⟨Rg2⟩ of a polymer near the Θ point. Our model is based on the discrete Gaussian chain model and includes a square-gradient term accounting for the finite-range interaction (characterized by κ), in addition to the usual monomer second and third virial coefficients (characterized by v and w, respectively). The use of the discrete model avoids the divergence problems encountered in previous studies using the continuous model. Our study identifies four special temperatures in the Θ regime: the temperature ΘN where the osmotic second virial coefficient vanishes, the critical temperature ΘNcr for phase separation, and two compensation temperatures ΘN(e) and ΘN(g) at which ⟨Re2⟩ and ⟨Rg2⟩ reach their respective ideal values. In the infinite chain-length limit N → ∞, all of these four temperatures approach Θ∞, the Θ temperature for the infinitely long chain. These temperatures differ from each other by terms of order N–1/2. In general, these temperatures follow the order ΘN > ΘNcr and ΘN > ΘN(e) > ΘN(g). Furthermore, ΘN > Θ∞, in agreement with the result obtained by Khokhlov some time ago. On the other hand, depending on the ratio w/κb, Θ∞ can be higher than ΘN(e) (for w/κb < 9.45), lower than ΘN(g) (for w/κb > 11.63), or in between ΘN(e) and ΘN(g) (for 9.45 < w/κb < 11.63). ΘNcr can be either higher or lower than Θ∞ depending on whether w/b6 is larger or smaller than 0.574. From the order of these temperatures, we conclude that the chain is mostly expanded relative to the ideal chain at its ΘN. However, at Θ∞, the chain can be either expanded or contracted, depending on the relative position of Θ∞ with respect to ΘN(e) and ΘN(g) and depending on whether the chain dimension is measured by ⟨Re2⟩ or ⟨Rg2⟩.