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Published December 8, 2020 | Supplemental Material
Journal Article Open

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⟩.

Additional Information

© 2020 American Chemical Society. Received: June 4, 2020; Revised: September 27, 2020; Published: November 16, 2020. This work was conducted jointly by King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia, and California Institute of Technology (Caltech) under a collaborative research program in catalysis. The authors gratefully acknowledge the support provided by KFUPM and Caltech. Pengfei Zhang also acknowledges the financial support provided by the National Natural Science Foundation of China (21803011 and 22073016). The authors declare no competing financial interest.

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