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Polymeric Sol–Gel Transition with the Diverging Correlation Length Verified by Small-Angle X-ray Scattering

Cite this: J. Phys. Chem. Lett. 2023, 14, 46, 10396–10401
Publication Date (Web):November 13, 2023
https://doi.org/10.1021/acs.jpclett.3c02631

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Abstract

Sol–gel transitions of polymers are pivotal phenomena in material science, yet the critical phenomenon of structure during gelation has remained unclear. Here, we investigated the sol–gel transition of a fluorous polymer, poly(vinylidene fluoride-co-hexafluoropropylene), in a blend of two ionic liquids. This system features a quite high amount of cross-linker and binding sites with ion–dipole interactions between the cation and C–F dipoles, thereby facilitating easy exchange of the cross-links. Changing the mixing ratio of the two ionic liquids enabled tuning the ion–dipole interactions and inducing sol–gel transition. Notably, the correlation length and molar mass, obtained by small-angle X-ray scattering, diverged at the gelation point. Moreover, the derived critical exponents (ν = 0.85 ± 0.05) aligns remarkably well with the prediction from percolation theory (ν = 0.88). To our knowledge, this is the first report on the evident divergence during polymeric gelation by small-angle scattering and the verification of the critical exponents of the percolation theory.

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Sol–gel transitions are crucial phenomena in various fields of material science and technology, including drug delivery systems, cosmetics, paints and coatings, sensors, and optics. (1−5) The viscoelasticity significantly changes at the sol–gel transition point, making it an important point of interest. Polymeric gelation has been theoretically studied for more than 80 years, as early developed by Flory (6) and Stockmayer, (7) and is considered to be a critical phenomenon during which certain parameters diverge with the formation of a fractal structure. (8−12) One of the main structural parameters for characterizing solutions/gels and determining the mechanical properties of gels is the correlation length (ξ), denoting the length over which the polymer chains or segments may fluctuate. (1,8−12) Three-dimensional percolation theory has now been established and can predict the behavior even in the vicinity of the critical gelation point. In particular, this theory predicts that ξ may follow ξ ∼ |pcp|–ν with a critical exponent (ν) of 0.88, where p denotes the conversion or connectivity factor, and pc is p at the gelation point. (8−12) The ν value is considered to be a universal value independent of the types of polymers or interactions. Despite extensive theoretical studies, the critical phenomena of the sol–gel transitions of polymers in terms of ξ have rarely been experimentally studied. This is in contrast to the fact that experimental investigations of the critical phenomena in the rheological parameters, such as viscosity, elastic modulus, and relaxation time, have been extensively conducted. (1,12−14)

Only a limited number of experimental studies on ξ near a gelation point have been conducted. Shibayama et al. (15) and Kanaya et al. (16) did not observe any divergence in ξ determined by small-angle neutron scattering (SANS) during the gelation of poly(vinyl alcohol) in water. Similarly, no divergence of ξ was detected by small-angle X-ray scattering (SAXS) during the gelation of poly(N-isopropylacrylamide) in water. (17) Shibayama hypothesized that static scattering techniques cannot detect divergence of ξ owing to the sensitivity. (18) In addition to these studies, several instances of divergence-like upward and downward behaviors in ξ during polymeric gelation have been reported. (18−27) In most of these studies, dynamic light scattering (DLS) was employed to estimate ξ. However, estimating ξ near the gelation point by DLS is difficult due to the broad distribution of relaxation times. (28,29) Therefore, the understanding of the structural changes occurring during gelation has remained incomplete; in particular, the ν value for polymeric gelation has yet to be determined through a direct, accurate, and experimental method.

In this study, we experimentally investigate polymeric gelation by oscillatory rheometry and small-angle X-ray scattering (SAXS), with a particular focus on the critical phenomena of the structure. The system used in this study is composed of a random copolymer of vinylidene difluoride and hexafluoropropylene (PVDF-HFP) and ionic liquids (solvent). Ion–dipole interactions work between the C–F dipoles of PVDF-HFP and the cations of the ionic liquid; (30) that is, the PVDF-HFP chains are physically cross-linked via the cations. (31−33) The strength of the ion–dipole interaction;, in other words, the fraction of the cross-linking points, may be controlled by the type of ionic liquid used. In fact, the strength of ion–dipole interactions appeared to be stronger when using 1-ethyl-3-methylimidazolium trifluoromethanesulfonate ([EMIM][Otf]) than that when using 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide ([EMIM][TFSI]) (Figure 1). Thus, in a mixture of [EMIM][Otf] and [EMIM][TFSI], the strength or binding constant of the ion–dipole interactions can be regulated by changing the weight fraction of [EMIM][Otf] (w), which may induce the sol–gel transition. Compared to previously reported sol–gel transitions, (15−27) our system features a much higher amount of cross-linker ([EMIM][Otf]) and density of binding sites (C–F dipoles), and the strength of cross-linking is relatively weak. Thus, cross-linking points can be easily and rapidly exchanged.

Figure 1

Figure 1. Appearance of the solution/gel of PVDF-HFP in [EMIM][TFSI] (w = 0) and in [EMIM][Otf] (w = 1) at ϕ = 0.2, and schematic illustrations of the EMIM cation interacting with PVDF-HFP chains in the presence of the TFSI or Otf anion.

The phenomenon in which the strength of the cation–dipole interaction depends on the type of anion can be manipulated as follows. The cation–dipole interactions were sterically hindered when using larger anions (the size of the Otf anion is smaller than that of the TFSI anion; Figure 1). This effect is observed in solid polymer electrolytes (polymers containing salt without any solvents). (34) Furthermore, TFSI has two CF3 groups per anion, whereas Otf has a single CF3 group per anion. The C–F dipoles of the polymers may be screened by CF3 groups in TFSI, rather than that in the case of Otf. Therefore, an increase in w corresponds to the addition of a cross-linker, and w is proportional to the number of cross-links (i.e., p), judging from the following two points: (i) Given that the absorbance of visible light is intrinsically related to the state of the dipoles, it should reflect the ion–dipole interactions in this study. The absorbance of the polymer solutions/gels increased linearly with w (Figure S4), while the absorbance of the ionic liquid only (i.e., in the absence of PVDF-HFP) remained unchanged. This result suggests the number of ion–dipole interactions (cross-links) increases linearly with w. (ii) As [EMIM][Otf] can be regarded as the cross-linker in this system, which is written as 2D + EO ⇌ Cross-link, with the equilibrium constant K = [Cross-link]/([D]2[EO]). Here, “D”, “EO”, and “Cross-link” represent the C–F dipole, [EMIM][Otf], and cross-linking point, respectively. As [EO] is sufficiently high for the K value, (31) the concentration of [EMIM][Otf], corresponding to w, is proportional to the number of the cross-links under the constant polymer concentration.

Prior to presenting the structural data, the viscoelasticity data obtained by an oscillatory rheometry are shown to confirm the occurrence of sol–gel transition and characterize the gelation behavior in terms of viscoelasticity. Upon increasing w under a fixed polymer volume fraction (ϕ) of 0.2, a distinct gelation behavior was observed (Figure 2). At the low w, the fluid-like behavior of G′ ∼ ω2 and G″ ∼ ω was observed, where G′ and G″ denote the storage and loss moduli, respectively, and ω represents the angular frequency. The slope decreased as w increased. Furthermore, G′ and G″ became parallel, following a scaling relationship G′G″ ∼ ωn, with an n value of 0.46 ± 0.02 over the entire range of ω measured at w = 0.5–0.6. Upon further increasing w, G′ and G″ exhibited lesser dependencies on ω, and G′ was higher than G″ over the entire range of ω. This behavior is a typical characteristic of gelation observed near the critical gelation point. (1,8−12) Thus, the gelation point is between w = 0.5 and 0.6.

Figure 2

Figure 2. Storage (G′; panel a) and loss (G″; panel b) moduli and loss tangent (tan δ; panel c) as a function of ω for PVDF-HFP in mixtures of [EMIM][TFSI] and [EMIM][Otf] with different weight fractions of [EMIM][Otf] (w) at ϕ = 0.2.

The relaxation exponent n characterizes the fractal dimension and depends on the intermolecular interactions of the polymers. Previous experimental studies on polymeric gelation reported n values in the range of 0.1 to 0.9. (11,35−39) The reason for this wide range of n values is that n depends on the excluded volume effect and hydrodynamic interactions. (35,36) Thus, the rheometry does not unambiguously determine the fractal dimension, but the n value obtained in this study (0.46) indicates that the excluded volume effect is screened and the linear chain fraction in the network is relatively high.

The zero-shear viscosity obtained by η0 = limω→0(G″/ω) exhibited a critical phenomenon, which obeyed η ∼ (wcw)–k. The critical exponent k was determined by the least-squares fitting to be 0.64 ± 0.04 in the sol regime (w < wc) (Figure 3a). Here, wc denotes the w value at the critical gelation point, and wc was set at 0.56, as described later. The theoretical k values ranged from 0 to 1.35 depending on the calculation approach used, and the experimentally reported values also varied. Furthermore, the two relaxation times, τcross = 1/ω at the cross-point of G′ and G″, and τw = limω→0{G′ /(ωG″)}, are found to follow τcross ∼ (wcw)−ζ with ζ = 0.67 ± 0.02 and τw ∼ (wcw)−ζ with ζ = 1.19 ± 0.07 (Figure 3b). These ζ values are lower than the experimentally reported value (3.9) (11) and the theoretically predicted values (2.6–4.0) (11) for the longest relaxation time. These lower exponents in this study are attributed to the fact that our present system is based on the physical (transient) cross-linking with low binding constants. The comprehensive understanding of k and ζ remains arduous as these exponents depend on the interactions and gelation mechanisms. (8−14,36−38) Accordingly, within the scope of the current study, we refrain from delving further into the intricacies of these exponents.

Figure 3

Figure 3. Zero-shear viscosity (η0; panel a) and relaxation times (τcross and τw; panel b) for PVDF-HFP in mixtures of [EMIM][TFSI] and [EMIM][Otf], as a function of (wcw), at the constant volume fraction of PVDF-HFP. τcross is the 1/ω value at the intersection (G′ = G″) in the linear viscoelastic data, and τw is the terminal relaxation time expressed by τw = limω→0{G′/(ωG″)}. The solid black lines represent the least-squares fittings.

The structural changes during the gelation process were proven by SAXS measurements. The SAXS profile obtained at each w exhibited the typical profile for semidilute polymer solutions or gels (Figure 4a). (1) The scattering intensity [I(q)] had a steeper q dependence at the low-q regime, where q denotes the magnitude of the scattering vector. This low-q regime is governed by the scattering originating from the inhomogeneity or large-scale fluctuation of the concentration/cross-linking density or small quantity of aggregates. The upturn at the low-q region is observed even in physical gels (probably attributable to cooperativity of the cross-linking) (33,39) and semidilute non-cross-linked polymers. (1) Meanwhile, the high-q regime shows a less steep slope, which reflects the fluctuation of the polymer chains or network strands. This study focuses on the high-q regime, which provides information about the network structure. These small-scale fluctuations are well-known to be described by the Ornstein–Zernike function. In this study, we used the extended Ornstein–Zernike function (15) combined with the function IL(q) ∼ qa to consider the inhomogeneity (large-scale fluctuation or aggregates) as follows. (1,17,33)

I(q)IL(q)+IOZ{1+(D+1)3ξ2q2}D/2
(1)
Here, a is a constant that corresponds to the slope at the low-q upturn. IOZ denotes the intensity of the extended OZ function, which changes the relative height (forward scattering intensity) of the extended OZ function in the total scattering function. D stands for the fractal dimension of the strands, which solely influences the high-q slope. That is, in eq 1, the first term influences the low-q upturn (the larger a results in a steeper slope) while the second term (the extended OZ function) governs the middle-to-high q region. Please note that 1/ξ corresponds to the q value from the plateau to the asymptote to I(q) ∼ qD, as the arrows in Figure 4a indicate the 1/ξ values. Thus, ξ can be almost independently determined by the fitting. Therefore, each parameter in eq 1 individually influences the shape of the scattering function. The fitted curves are shown as solid black curves in Figure 4a.

Figure 4

Figure 4. SAXS profiles at various weight fractions of [EMIM][Otf] (w), at ϕ = 0.2 and at 25 °C: (a) Double-logarithmic plot, (b) Kratky plot, and (c) Ornstein–Zernike plot for the scattering intensity subtracted by IL(q). Solid black curves in panels (a) and (c) represent the fitted model curves. See also Figure S5, where the Ornstein–Zernike function is presented as the Kratky plot. Each profile is shifted vertically for clarity (A in panel (c) represents the shift factor).

Change in ξ is visually seen in the Kratky plot (Figure 4b). The very low-q regime of the negative slope in this plot reflects the inhomogeneity, which was not the focus of this study. The q-position of transitioning from the positive slope to the plateau or maximum is related to ξ. More specifically, 1/ξ corresponds to the slope (see Figure S5, showing the Ornstein–Zernike function as the Kratky plot). Therefore, it is obvious that by increasing w, ξ increases initially but then decreases with a further increase in w. In addition, in the Kratky plot where w is close to the gelation point, a maximum was observed, and the slope at the high-q region was negative. This feature may reflect critical fluctuations. In contrast, conventional semidilute solutions or gels under noncritical conditions exhibit no maximum, but the plateau at the high-q region in the Kratky plot, as observed at w = 0–0.45, 0.9, and 1.

Moreover, the Ornstein–Zernike plot (reciprocal intensity vs q2) is constructed for the scattering intensity subtracted by IL(q) (Figure 4c). I(q) – IL(q) corresponds to the Ornstein–Zernike term in eq 1 (small-scale fluctuations), and this Ornstein–Zernike plot should be a linear function as (1 + ξ2q2)/IOZ. In fact, our present data show nice linearity, indicating that the scattering intensity is well decomposed into the larger and smaller scale components (first and second terms in eq 1, respectively). This Ornstein–Zernike plot enables the precise determination of ξ values from the initial slope (q < 1/ξ).

It is remarkable that the obtained ξ and IOZ/Δρ2 exhibited distinct divergence at the gelation point (Figure 5a,c), which was determined to be wc = 0.56 ± 0.005. Here, Δρ is the contrast factor (difference in the electron density between the polymer and solvent). This behavior can be manipulated as follows: when w = 0, the main component is almost molecularly dispersed polymer chains. Thus, ξ reflects the dimensions of the polymer chains and is related to the mean-square radius of gyration (⟨S2⟩) as ξ = S2/3 (S21/2 = 3.5 nm). Even in semidilute solutions, S21/2 can be estimated although it is unperturbed state. (40) The SAXS profile at w = 0 exhibited the upturn at the low-q region, indicating the presence of larger-scale fluctuations or a small quantity of a large component (aggregate). Under these circumstances, the main component should be the molecularly dispersed polymers. (1) As w increases, cross-linking occurs; consequently, ξ and IOZ/Δρ2 increase as they now correspond to the averaged dimension and molar mass of the percolation clusters, respectively. At the critical gelation point, ξ and IOZ/Δρ2 diverge to infinity. Postgelation, ξ is regarded as the distance between the cross-linking points as the network strands fluctuate. With a further increase in w, the cross-linking density increases, leading to a decrease in ξ. The strand length at w = 1 was 2.7 nm. See the abstract graphic, which provides a schematic of the process.

Figure 5

Figure 5. Structural parameters derived from SAXS: Correlation length (ξ) as a function of w (a) and |wcw| (b). Intensity of the Ornstein–Zernike function (IOZ/Δρ2) as a function of w (c) and |wcw| (d). Blue and red circles represent the data for the pregel and postgel regimes, respectively. Solid curves and lines represent the least-squares fitting, where the two data points very close to wc were not taken into account when determining ν.

Notably, the experimentally obtained data were fitted using the equation ξ ∼ |wcw|–ν with the ν value of 0.85 ± 0.05 as well as IOZ/Δρ2 ∼ |wcw|–γ with the γ value of 1.86 ± 0.15, which nicely agrees with the 3-dimensional percolation theory (ν = 0.88 and γ = 1.74–1.80) (1,8,9) (Figure 5b,d). Further, not only the pregel regime (w < wc) but also the postgel regime (w > wc) could be well described by the same ν and γ values. Symmetrical divergence in the longest relaxation time was reported by several groups, suggesting the universality of the symmetrical divergence regardless of the type of polymers or gelation mechanisms. (14,41,42) The symmetrical divergence may be universally observed in ξ and IOZ/Δρ2, analogous to the scenario involving the longest relaxation time.

The other parameters derived from SAXS analysis (as listed in Table S2) serve to characterize the critical phenomena of gelation. The a values far from the gelation point (w = 0–0.5 and 0.8–1) fell within the range of 3 to 4. The exponents indicate that the large-scale inhomogeneity has the surface fractal structure, (43) which is commonly observed in semidilute polymer solutions or gels. (1) Whereas, near the gelation point (w = 0.525–0.7), the exponent exceeded 4 (Porod’s law). The unexpectedly high exponent may reflect the critical fluctuations occurring even at larger scales. We note that the a value governs the low-q region only, and the estimation of ξ through the fitting remains independent of a. Meanwhile, the D value was 2 at w = 0–0.45, 0.9, and 1, while D = 2.1 ± 0.05 at w = 0.5–0.8. It is generally known that D values are 2 in semidilute polymer solutions or gels. The slightly higher value (D = 2.1) near the gelation point in this study also indicates the formation of percolation clusters (branching structure) rather than linear polymer chains or a microstructure of the higher mass fractal dimension. A similar behavior was observed in another gel near the gelation point. (15)

In contrast to previous studies on the sol–gel transition proved by small-angle scattering, (15−17) our present system is formed by transient cross-linking via ion–dipole interactions in the presence of a substantial number of cross-linkers ([EMIM][Otf]). This easily exchangeable cross-linking is thought to produce an equilibrium state in the system, resulting in the distinct divergence of ξ and IOZ with the critical exponents, which agree well with that predicted by percolation theory. The transient cross-linking with a high concentration of cross-linkers is an intriguing system to further study the various aspects of the sol–gel transitions. Additionally, this study could be extended to more strongly bound cross-links, contingent upon the exchangeability, such as dynamic covalent bonds.

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  • Experimental details; visible-light spectra; Ornstein–Zernike functions; and a and D values obtained by SAXS (PDF)

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    • Kota Aoki - Department of Energy Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8603, Japan
    • Ayae Sugawara-Narutaki - Department of Energy Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8603, JapanOrcidhttps://orcid.org/0000-0003-3579-6689
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    The authors declare no competing financial interest.

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JASRI is acknowledged for providing the beamtime at SPring-8 (proposal numbers: 2022A1193, 2022B1233, 2023A1244, and 2023B1174). This work was partially supported by The Asahi Glass Foundation, Iketani Science and Technology Foundation (0341186-A), Tokai Foundation for Technology, Foundation of Kinoshita Memorial Enterprise, and The Hibi Science Foundation.

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  • Abstract

    Figure 1

    Figure 1. Appearance of the solution/gel of PVDF-HFP in [EMIM][TFSI] (w = 0) and in [EMIM][Otf] (w = 1) at ϕ = 0.2, and schematic illustrations of the EMIM cation interacting with PVDF-HFP chains in the presence of the TFSI or Otf anion.

    Figure 2

    Figure 2. Storage (G′; panel a) and loss (G″; panel b) moduli and loss tangent (tan δ; panel c) as a function of ω for PVDF-HFP in mixtures of [EMIM][TFSI] and [EMIM][Otf] with different weight fractions of [EMIM][Otf] (w) at ϕ = 0.2.

    Figure 3

    Figure 3. Zero-shear viscosity (η0; panel a) and relaxation times (τcross and τw; panel b) for PVDF-HFP in mixtures of [EMIM][TFSI] and [EMIM][Otf], as a function of (wcw), at the constant volume fraction of PVDF-HFP. τcross is the 1/ω value at the intersection (G′ = G″) in the linear viscoelastic data, and τw is the terminal relaxation time expressed by τw = limω→0{G′/(ωG″)}. The solid black lines represent the least-squares fittings.

    Figure 4

    Figure 4. SAXS profiles at various weight fractions of [EMIM][Otf] (w), at ϕ = 0.2 and at 25 °C: (a) Double-logarithmic plot, (b) Kratky plot, and (c) Ornstein–Zernike plot for the scattering intensity subtracted by IL(q). Solid black curves in panels (a) and (c) represent the fitted model curves. See also Figure S5, where the Ornstein–Zernike function is presented as the Kratky plot. Each profile is shifted vertically for clarity (A in panel (c) represents the shift factor).

    Figure 5

    Figure 5. Structural parameters derived from SAXS: Correlation length (ξ) as a function of w (a) and |wcw| (b). Intensity of the Ornstein–Zernike function (IOZ/Δρ2) as a function of w (c) and |wcw| (d). Blue and red circles represent the data for the pregel and postgel regimes, respectively. Solid curves and lines represent the least-squares fitting, where the two data points very close to wc were not taken into account when determining ν.

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    • Experimental details; visible-light spectra; Ornstein–Zernike functions; and a and D values obtained by SAXS (PDF)

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