Revisiting the Θ Point

Using the first-order perturbation theory, we compute the osmotic second and third virial coefficients, the mean-square end-to-end distance ⟨R_e²⟩, and the mean-square radius of gyration ⟨R_g²⟩ 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 Θ_N where the osmotic second virial coefficient vanishes, the critical temperature Θ_N^(cr) for phase separation, and two compensation temperatures Θ_N^((e)) and Θ_N^((g)) at which ⟨R_e²⟩ and ⟨R_g²⟩ 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 Θ_N > Θ_N^(cr) and Θ_N > Θ_N^((e)) > Θ_N^((g)). Furthermore, Θ_N > 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 Θ_N^((e)) (for w/κb < 9.45), lower than Θ_N^((g)) (for w/κb > 11.63), or in between Θ_N^((e)) and Θ_N^((g)) (for 9.45 < w/κb < 11.63). Θ_N^(cr) 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 Θ_N. However, at Θ∞, the chain can be either expanded or contracted, depending on the relative position of Θ∞ with respect to Θ_N^((e)) and Θ_N^((g)) and depending on whether the chain dimension is measured by ⟨R_e²⟩ or ⟨R_g²⟩.