Linear Rheology and Structural Characterization

We investigated the viscoelastic behavior of the PBDT gels resulting from the experiments shown in Figure 4.2 with linear rheology. We performed isothermal frequency sweeps from 100 to 0.1 rad s−1 _{between 25 and 75 °C and applied time-temperature superposition (TTS) to}

horizontally shift each isotherm along the frequency axis to construct master curves at 𝑇-(T = 25 °C spanning nearly 10 decades in frequency for each concentration (see Figure D.2a). Furthermore, we collapsed the concentration series using time-concentration superposition (TCS) to generate a single master curve spanning over 20 decades in frequency and over 5 decades in dynamic moduli, shown in Figure 4.3a.

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Figure 4.3. Linear rheology of PBDT gels. (a) Storage (𝐺’, filled symbols) and loss (𝐺”, open symbols) moduli master curve as a function of shifted frequency from TTS at 𝑇-.+ = 25 °C and TCS at 𝑐-.+ = 30 wt.%. The network lifetime (1/𝜏) is found at the crossover of 𝐺’ and 𝐺". The inset shows the shift factors derived from TCS as function of concentration and fit using the William-Landel-Ferry (WLF) equation given by Eqn. 4.2. (b) Digital photograph of a 25 wt.% shear-activated gel after loading into the rheometer, evidencing solid-like behavior. (c)𝐺’ at 10 rad s−1 _{and 25 °C (filled circles) for the gels as a}

function of concentration. The calculated 𝐺’ based on polyelectrolyte scaling theory (𝐺’ ≈ 𝑘/𝑇/𝜉,) is shown (filled squares). The extrapolated 𝐺’ for an isotropic methyl cellulose (MC) gel is given for comparison (dashed line) from ref236_{. (d) Distance between physical crosslinks (}𝑙

0) calculated from Eqn. 4.3 as a function of concentration. The extrapolated 𝑙0 for MC gels is given for comparison.

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The collapse of the data using TTS and TCS indicates that the nature of the gel structure is concentration independent. The TCS shift factors 𝑎8 are plotted in the inset of Figure 4.3a and are well-described by a modified William-Landel-Ferry (WLF) equation

log 1/𝑎₈ = d(Q 6 Q")*)

'X(Q 6 Q")*) (Eqn. 4.2)

where 𝐴 and 𝐵 are fitting constants and 𝑐-(T = 30 wt.%. The gels, shown in Figure 4.3a, exhibit a solid-like viscoelastic response (𝐺\ > 𝐺") with a nearly frequency independent modulus (𝐺\ ~

𝜔0.1_{) over a wide range of frequencies down to the inverse network lifetime} 1/𝜏 ≈ 106$[_{rad s}−1_.

At frequencies below 1/𝜏, 𝐺\ and 𝐺" are parallel and scale as ~ 𝜔$/4 over the 5 lowest decades in frequency – no terminal flow behavior is observed (𝐺\ ~ 𝜔2_, 𝐺" _~ 𝜔1_{). A Cole-Cole plot lacks}

the signature semi-circle shape of an ideal Maxwell fluid, as is found for WLMs,234 _confirming

that the gels have a broad spectrum of relaxation times (see Figure D.2b).232 _{This behavior is}

similar to that observed in stiff biopolymer networks whose dynamics are governed by transient physical crosslinks.237

The low values of 𝐺\, given in Figure 4.3c, highlights the unusual structure of the gels. For comparison, we calculate 𝐺\ using polyelectrolyte scaling theory43, 191𝐺′ ≈ 𝑘

'𝑇/𝜉%, where 𝜉 is the lateral correlation length determined from small-angle X-ray scattering (SAXS, see Figure D.3 and Figure D.4), and extrapolate the trend in 𝐺\ for an isotropic, fibrillar methyl cellulose (MC) gel to equivalent concentrations.236 _{The PBDT gels exhibit a remarkably low} 𝐺\ _{(1–30 kPa)}

when compared to the scaling calculation and extrapolated modulus of the MC gel. We postulate that the lower 𝐺\ arises from the local nematic alignment that reduces the number density of physical crosslinks. Above 18 wt.%, the concentration scaling of 𝐺\ is stronger (𝐺\~ 𝑐5) than both the scaling calculations (𝐺\~ 𝑐$.%M) and typical fibrillar gels (𝐺\~ 𝑐464.M),236, 238-240 _likely

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To understand the nature of the gel structure, we calculate the distance between physical crosslinks 𝑙Q using a model for the linear elastic modulus 𝐺. of fibrillar gels241

𝐺_. = 6𝜌𝑘_'𝑇;+%

;_,$ (Eqn. 4.3)

where 𝜌 = 𝜙 𝜋𝑟⁄ 4 is the filament-length density, 𝑟 is the fibril radius, and 𝑙J is the persistence length. The value of 𝑙J for PBDT rods is at least 200 nm based on the I-N transition concentration reported here, but could be up to 1 µm.67_{Figure 4.3d shows the calculated} 𝑙

Q using various values of 𝑙J (200, 400, and 800 nm). The 𝑙Q ≈ 100–1000 nm values for PBDT shear- activated gels are much higher than the model MC gel, where the extrapolated 𝑙Q ≈ 1–10 nm,236,

239 _{consistent with physical crosslinking occurring at the ends, rather than along the body, of}

PBDT rods.

Further, we characterized the gel structure pre- and post-shear through SAXS and WAXS, see Figure D.3, Figure D.4, Figure D.5, and related discussion in Appendix D. We observe an increase in low-𝑞 scattering post-shear, suggesting the onset of network formation on mesoscopic length scales. The position and shape of the structure factor, corresponding to the lateral rod-rod correlation length, does not change after shear, indicating a constant rod diameter. This is in contrast to thermoreversible gels of rodlike polymers and virus, where the suppression of the structure factor is correlated with the onset of gelation.200, 242 _{The WAXS spectra, which}

reports on the local polymer-polymer packing within the rods, is unchanged after gelation, ruling out shear-induced crystallization as a potential gelation mechanism. Observations of a freeze- fractured 16 wt.% post-shear gel via scanning electron microscopy (SEM) were consistent with network formation, see Figure D.6. Our SEM observations are similar to what is observed after irreversible gelation of WLMs due to branching and junction formation of micellar bundles.212

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We investigated the role of electrostatic interactions on gel formation by measuring the SAXS spectra and flow curve of a 3 wt.% PBDT solution with increasing concentration of monovalent salt (NaCl), see Figure D.7a—c and related discussion in Appendix D. The addition of salt gradually suppresses the structure factor peak, arising from rod-rod electrostatic repulsion, indicating a suppression of positional correlations between rods. Moreover, the nonlinear rheological behavior of the PBDT solution with added salt evidences a weak gel with no measurable hysteresis. Both of these observations are in contrast to the behavior observed when PBDT gels are formed under shear, suggesting a different mechanism for gel formation.