Imprinting Macroscopic Fracture during Gelation: A Mechanism for Tuning Colloidal Gels

TL;DR

The paper addresses how nonlinear oscillatory shear during the sol-gel transition controls the microstructure and mechanics of colloidal silica gels. The authors apply large-amplitude oscillatory shear during gelation and monitor long-time viscoelastic spectra, using rheology, rheo-imaging, and rheo-SAXS to connect macroscopic response to fracture patterns. They find a critical duration $\mathcal{T}_c$ and a critical strain $\gamma_c$ beyond which macroscopic cracks irreversibly weaken the gel and introduce a damage-related dissipation channel, requiring a Generalized Fractional Kelvin--Voigt description in addition to the Fractional Maxwell description. This fracture imprinting provides a tractable route to tune linear viscoelastic properties and nonlinear yielding, with potential to design soft solids with tailored ductility and energy dissipation by engineering breakage patterns during gelation.

Abstract

Colloidal gels form through the sol-gel transition of attractive particle suspensions, where local aggregation leads to a space-spanning network with solid-like properties. Their microstructure and mechanical properties are highly sensitive to external perturbations, which can substantially alter the pathway of network formation. Here, we investigate how nonlinear oscillatory shear affects the sol-gel transition of colloidal silica suspensions. Using large-amplitude oscillatory shear (LAOS), we vary both the strain amplitude and the duration of oscillatory forcing, varying between one and two times the gelation time. We find that sufficiently large strain amplitudes, or prolonged exposure to oscillations in the nonlinear regime, alter irreversibly the gel properties: the storage modulus $G'$ decreases while its frequency dependence remains unchanged. In contrast, the loss modulus $G''$, which decreases monotonically with frequency under quiescent gelation, exhibits an upturn at high frequencies when the gel is formed under strong oscillatory shear. The viscoelastic spectra of gels formed under quiescent conditions are well captured by a fractional Maxwell model, while gels formed under LAOS require an additional fractional element to account for damage-induced dissipation. Rheo-imaging experiments corroborate this interpretation by revealing the growth of cracks in gels formed under LAOS. We further show that these gels display a progressively more ductile nonlinear response for prolonged exposure to LAOS during gelation. These results demonstrate that the interplay between non-linear shear and gelation can permanently imprint a macroscopic fracture pattern into colloidal gels, offering a route to tune their viscoelastic properties.

Figures (16)

• Figure 1: Elastic modulus $G'$ as a function of normalized time $t/t_g$ during gelation, where $t_g=3610\pm150~\rm s$ is the gelation time determined by SAOS at $\omega=2\pi~\rm rad\ s^{-1}$. Each experiment was performed on a fresh sample, first under LAOS with amplitude $\gamma_{\rm OS}=40\%$ for a duration $\mathcal{T_{\rm OS}}$, and subsequently under SAOS with amplitude $\gamma_0=0.5\%$ for the rest of the experiment. Such a gelation under oscillatory shear was repeated for different $\mathcal{T_{\rm OS}}$ values: $\mathcal{T_{\rm OS}}/t_g=1.4$ (blue), $1.6$ (cyan), and $2.1$ (green). Vertical arrows mark the time at which the strain amplitude is switched from $\gamma_{\rm OS}=40\%$ to $\gamma_0=0.5\%$. The black curve corresponds to $G'_{\rm ref}(t)$, the gelation conducted entirely under SAOS with amplitude $\gamma_0=0.5\%$. Inset: normalized modulus $G'(t=4t_g)/G'_{\rm ref}(t=4t_g)$ as a function of $\mathcal{T_{\rm OS}}/t_g$. The vertical dotted line marks the critical duration $\mathcal{T}_c \simeq 1.45\,t_g$ beyond which large-amplitude oscillations irreversibly affect the gel linear viscoelastic properties. The solid black curve is a guide to the eye.
• Figure 2: Linear viscoelastic spectra measured at long time ($t=5\,t_g$) for silica gels formed (a) under SAOS ($\gamma_0=0.5\%$), and (b) under LAOS with amplitude $\gamma_{\rm OS}=40\%$ applied during $\mathcal{T_{\rm OS}}=1.6\,t_g$. (c) Loss tangent $\tan \delta =G"/G'$ for the spectra shown in (a) and (b), together with an additional gel formed under LAOS at $\gamma_{\rm OS}=40\%$ for $\mathcal{T_{\rm OS}}=1.5\,t_g$. Colors in (c) are consistent with those used in (a) and (b). The viscoelastic spectrum of the silica gel formed under SAOS is fitted by a Fractional Maxwell model [see red curves in (a) and (c), and sketch in (a)]. The viscoelastic spectra of the gels formed under LAOS are fitted by a Generalized Fractional Kelvin--Voigt model [see red curves in (b) and (c), and sketch in (b)]. Fit parameters are reported as a function of $\mathcal{T_{\rm OS}}$ in Fig. \ref{['fig:40pctFitParams']}.
• Figure 3: Model parameters extracted from fits to fractional rheological models of the long-time viscoelastic spectra ($t=5\,t_g$, see Fig. \ref{['fig:fig2']}) as a function of the oscillation duration $\mathcal{T_{\rm OS}}$ for a fixed LAOS amplitude $\gamma_{\rm OS}=40\%$ : (a) $G_0$, (b) characteristic frequency $\omega_0=(G_0/\mathbb{V})^{1/\alpha}$, (c) exponents $\alpha$ and $\beta$, and (d) characteristic frequency $\omega_1=(G_0/\mathbb{G})^{1/\beta}$. Spectra measured on gels exposed to LAOS for $\mathcal{T_{\rm OS}}<\mathcal{T}_\mathrm{c}$ were fitted with the Fractional Maxwell model [Eq. \ref{['eq:FMM']}, inset in Fig. \ref{['fig:fig2']}(a)], while those exposed to LAOS for $\mathcal{T}_{\rm OS}>\mathcal{T}_\mathrm{c}$ were fitted with the Generalized Fractional Kelvin--Voigt model for [Eq. \ref{['eq:GFKVM']}, inset in Fig. \ref{['fig:fig2']}(b)]. The vertical dotted line indicates the critical duration of oscillation $\mathcal{T}_c$ beyond which LAOS with amplitude $\gamma_{\rm OS}=40\%$ significantly alters the long-term linear viscoelastic properties of the gel. Error bars denote 95% confidence intervals from the fit. In (c), the horizontal gray line marks the fixed value $\alpha=0.43$, imposed to all fits, irrespective of $\mathcal{T}_{\rm OS}$. In (c) and (d), red curves serve guide for the eye.
• Figure 4: Linear viscoelastic spectra measured at long time ($t=5\,t_g$) for silica gels formed (a) under SAOS with amplitude $\gamma_{\rm OS}=10\%$ and (b) under LAOS with amplitude $\gamma_{\rm OS}=40\%$ over the same duration $\mathcal{T_{\rm OS}}=2\,t_g$. (c) Loss tangent $\tan \delta =G"/G'$ vs. $\omega$ measured at $t=5\,t_g$, after the sample was exposed to oscillations of strain amplitude $\gamma_{\rm OS}=1, 10, 25, 30, 40, 50,$ and $80\%$ (from dark to bright) for $\mathcal{T_{\rm OS}}=2\,t_g$. The red curves correspond to the best fits of the data using a Fractional Maxwell model for $\gamma_{\rm OS} <\gamma_{c} \simeq 20\%$ [see sketch in (a)], and using a Generalized Fractional Kelvin--Voigt model for $\gamma_{\rm OS} > \gamma_{c}$ [see sketch in (b)]. Fit parameters are reported as a function of $\gamma_{\rm OS}$ in Fig. \ref{['fig:fractionalparameters']}.
• Figure 5: Model parameters extracted from fits to fractional models of the long-time viscoelastic spectra ($t=5\,t_g$, see Fig. \ref{['fig:spectra_amplitude']}) as a function of the strain amplitude $\gamma_{\rm OS}$ for a fixed oscillation duration $\mathcal{T}_{\rm OS}=2\,t_g$: (a) $G_0$, (b) characteristic frequency $\omega_0=(G_0/\mathbb{V})^{1/\alpha}$, (c) dimensionless exponents $\alpha$ and $\beta$, and (d) characteristic frequency $\omega_1=(G_0/\mathbb{G})^{1/\beta}$. Spectra measured on gels exposed to $\gamma_{\rm OS} < \gamma_c\simeq 20\%$ were fitted by a Fractional Maxwell model [Eq. \ref{['eq:FMM']}, inset in Fig. \ref{['fig:spectra_amplitude']}(a)], while those exposed to $\gamma_{\rm OS} >\gamma_c$ were fitted with a Generalized Fractional Kelvin--Voigt model [Eq. \ref{['eq:GFKVM']}, inset in Fig. \ref{['fig:spectra_amplitude']}(b)]. The vertical dotted line indicates the critical strain amplitude $\gamma_c\simeq 20\%$ beyond which oscillations imposed throughout gelation for $\mathcal{T}_{\rm OS}=2\,t_g$ significantly impact the long-term linear viscoelastic properties of the gel. Error bars denote 95% confidence intervals from the fit. In (c), the horizontal gray line marks the fixed value $\alpha=0.43$, imposed to all fits, irrespective of $\gamma_{\rm OS}$. In (c) and (d), red curves serve guide for the eye.