Temperature Dependence of Polymer Network Diffusion

Takeshi Fujiyabu, Takamasa Sakai, Ryota Kudo, Yuki Yoshikawa, Takuya Katashima, Ung-il Chung, and Naoyuki Sakumichi

Phys. Rev. Lett. 127, 237801 – Published 30 November 2021

Abstract

The swelling dynamics of polymer gels are characterized by the (collective) diffusion coefficient $D$ of the polymer network. Here, we measure the temperature dependence of $D$ of polymer gels with controlled homogeneous network structures using dynamic light scattering. An evaluation of the diffusion coefficient at the gelation point $D_{gel}$ and the increase therein as the gelation proceeds $Δ D \equiv D - D_{gel}$ indicates that $Δ D$ is a linear function of the absolute temperature with a significantly large negative constant term. This feature is formally identical to the recently discovered “negative energy elasticity” [Y. Yoshikawa et al. Phys. Rev. X 11, 011045 (2021)], demonstrating a nontrivial similarity between the statics and dynamics of polymer networks.

Received 30 June 2021
Revised 20 September 2021
Accepted 22 October 2021
Corrected 3 December 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.237801

Physics Subject Headings (PhySH)

Diffusion Elastic modulus Elasticity Network diffusion

Gels Polymer gels

Light scattering Rheology techniques

Condensed Matter, Materials & Applied Physics

Corrections

3 December 2021

Correction: The previously published Fig. 2 contained errors in the T₀ values given in the upper panels of part (b) and has been replaced.

Authors & Affiliations

Takeshi Fujiyabu ^‡, Takamasa Sakai ^*,‡, Ryota Kudo, Yuki Yoshikawa , Takuya Katashima, Ung-il Chung, and Naoyuki Sakumichi ^†

Department of Bioengineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

^*Corresponding author. sakai@tetrapod.t.u-tokyo.ac.jp
^†Corresponding author. sakumichi@tetrapod.t.u-tokyo.ac.jp
^‡These authors have contributed equally to this work.

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Vol. 127, Iss. 23 — 3 December 2021

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Images

Figure 1
Temperature ( $T$ ) dependence of (collective) diffusion coefficient $D$ and its components ( $D_{gel}$ and $Δ D$ ) in four gel samples with different network connectivities ( $p = 0.7$ , 0.8, 0.9, and 1.0) for a polymer concentration of $c = 60 g / L$ . (see Supplemental Material, Fig. S1 for $c = 30$ , 90, and $120 g / L$ [9].) The black and gray circles represent the experimental results of $D (T, p)$ for each sample. By extrapolating $D (T, p)$ to the gelation point ( $p \to p_{gel}$ ), we obtained $D_{gel} (T)$ (red circles) [see Fig. 2]. Then, we obtained $Δ D$ as $Δ D (T, p) = D (T, p) - D_{gel} (T)$ (colored squares) for each sample. The lines represent the least-squares fits of $D$ , $D_{gel}$ , and $Δ D$ . We determine the temperature $T_{D}$ at which $Δ D$ does not contribute to $D$ , i.e., $D (T_{D}, p) = D_{gel} (T_{D})$ and $Δ D (T_{D}, p) = 0$ . (Schematic illustration) The gel samples were synthesized by $A B$ -type cross-end coupling of two precursors (tetra-arm polymers) to tune $p$ ( $0 \leq p \leq 1$ ) after the completion of the reaction, while maintaining $c$ .
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Figure 2
Formally identical relationships governing the shear modulus $G$ and $Δ D$ . We synthesized the gel samples with 16 different network structures ( $c = 30$ , 60, 90, $120 g / L$ and $p = 0.7$ , 0.8, 0.9, 1.0). We measured $G$ and $D$ at $T = 288$ , 293, 298, 303, and 308 K. The data of $G$ are partly taken from Ref. [5]. (a) Shear modulus dependence of $D$ at constant $T$ . The open circles represent the experimental results, and the lines represent the least-squares fits of $D = D (G) = D_{gel} + α G$ . Here, $D (G = 0)$ corresponds to $D_{gel}$ (filled circles) because of the definition in Eq. (3). (b) Temperature dependence of $G$ (upper panels) and $D_{gel}$ and $Δ D \equiv D - D_{gel}$ (lower panels). Insets in the upper panels depict the same results in different ranges of $T$ . The open circles represent the experimental results, and the lines represent the least-squares fits of the $T$ dependences of $G$ , $D_{gel}$ , and $Δ D$ . All extrapolations of $G = G (T)$ and $Δ D = Δ D (T)$ with the same $c$ pass through $T_{0}$ and $T_{D}$ on the $T$ axis, leading to Eqs. (1) and (5), respectively. The value of $T_{0}$ or $T_{D}$ in each graph represents the average of the four samples with different $p$ , and the values in parentheses represent the standard deviation.
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Figure 3
(a) Log-log plots of the $c$ dependence of $f / η$ , indicating $f / η \sim c^{2 ν / (3 ν - 1)} ≃ c^{1.54}$ for $ν ≃ 0.5876$ . Here, each friction coefficient between the polymer and the solvent $f$ is obtained from the linear fits in Fig. 2 by assuming Eq. (6) and $β ≃ 0.563$ . (b) Log plots of the $T$ dependence of $α η$ . The viscosity of the solvent (water) $η$ is taken from Ref. [35]. The solid lines serve as a guide to the eye.
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