Gelation dynamics of charged colloidal rods: critical behaviour and time-connectivity superposition principle

Abstract

Charged colloidal particles can self-assemble into gel networks upon screening of electrostatic repulsion by added salt. While gelation of spherical colloids has been extensively studied, much less is known about the gelation dynamics of anisotropic colloids. Here, we focus on cellulose nanocrystals (CNCs) as prototypical rigid, highly charged rod-like colloids. In aqueous solution with salt, CNCs display a rich phase diagram ranging from gel at low solid content to glassy phases at higher concentrations. Building on our previous work [Morlet-Decarnin et al., ACS Macro Lett., 2023, 12, 1733], we present an extensive study of the mechanical recovery dynamics of CNC suspensions following a strong shear. Time-resolved mechanical spectroscopy reveals a liquid-to-solid transition characterized by a well-defined critical gel point. The evolving viscoelastic spectra can be rescaled onto master curves, demonstrating a time-connectivity superposition principle and critical dynamics on both sides of the gel point. By varying the CNC weight fraction and salt concentration, we identify a boundary between gel and attractive glass states marked by clear changes in rheological observables, including the elastic and viscous moduli at the gel point and their high-frequency power-law exponents. Analysis of dynamic critical exponents and hyperscaling reveals pronounced asymmetry between pre-gel and post-gel dynamics and non-universal values of the dynamic exponent. These findings highlight gelation mechanisms specific to highly charged rod-like colloids and call for complementary microstructural characterization and theoretical modeling.

Figures (12)

• Figure 1: Transmission electron microscopy images of a dilution of the commercial CNC suspension from Celluforce (courtesy of Bruno Jean and Jean-Luc Puteaux). The 6.4 wt% CNC suspension was exposed to mechanical stirring during 5 min at 2070 rpm (IKA RW 20 Digital mixer equipped with an R1402 blade dissolver) prior to dilution.
• Figure 2: Time-resolved mechanical spectroscopy during the recovery of a $3.2$ wt% CNC suspension containing $12$ mM NaCl. (a) Elastic modulus $G'$ ($\bullet$) and viscous modulus $G"$ ($\circ$) as a function of angular frequency $\omega$ at different times across the sol-gel transition ($t=28$, 1330, and 10000 s, from yellow to dark blue). Inset: corresponding loss factor $\tan \delta=G"/G'$ as a function of $\omega$. The data shown in the inset correspond to the time points highlighted with larger colored symbols in (b) and (c). (b) Temporal evolution of the frequency-scaling exponents $\beta'$ ($\bullet$) and $\beta"$ ($\circ$), extracted from power-law fits of $G'(\omega,t)$ and $G"(\omega,t)$ respectively. (c) Temporal evolution of the loss factor $\tan \delta (t)$ measured at the five frequencies of the multiwave signal: $\omega/(2\pi)=0.3$ Hz ($\bullet$), 0.6 Hz ($\blacktriangle$), 1.2 Hz ($\blacksquare$), 3 Hz ($\blacktriangleleft$), and 6 Hz ($\blacktriangleright$). Inset: same data displayed in linear scales close to the gel point. Darker colors correspond to higher frequencies. The vertical green dotted lines in (b) and (c) highlight the gelation time $t_g$, identified by the frequency-independent loss factor.
• Figure 3: Temporal evolution of the elastic modulus $G'$ ($\bullet$) and viscous modulus $G"$ ($\circ$) measured at fixed frequency $\omega/(2\pi)=1.2$ Hz following a 60 s preshear at $\dot{\gamma}=1000$ s$^{-1}$. The vertical dotted line at $t_c=165$ s marks the crossover time at which $G'=G"$ at this frequency. The vertical dotted line at $t_g=1330$ s indicates the gelation time identified from time-resolved mechanical spectroscopy by the frequency-independent loss factor. The vertical dashed line at $t^*=1314$ s denotes the inflection time introduced in Ref. Morlet-Decarnin:2022. Inset: Viscoelastic spectra $G'(\omega)$ ($\bullet$) and $G"(\omega)$ ($\circ$) measured after $10000$ s of recovery under small-amplitude oscillatory shear (strain of amplitude 0.2 %) by sweeping down the angular frequency logarithmically from 20 to 0.1 rad.s$^{-1}$ within 1500 s. Experiment performed on the same 3.2 wt% CNC suspension containing 12 mM of NaCl as in Fig. \ref{['fig:reprise_multiwave_tg']}.
• Figure 4: Time-resolved mechanical spectroscopy of the sol-gel transition in a $3.2$ wt% CNC suspension containing $12$ mM of NaCl. (a) Loss factor $\tan \delta$ as a function of angular frequency $\omega$ at different times after shear cessation, from $t=8$ up to $10000$ s (from yellow to dark blue). (b) Temporal evolution of the horizontal shift factor $a(t)$, with the initial value arbitrarily set to $a(0)=1$ s.rad$^{-1}$. Red curves show the best power-law fits of the data in the vicinity of the gel point, $a\sim\varepsilon^{-y}$, where $\varepsilon=|t-t_g|/t_g$, yielding exponents $y_l=14.4$ for $t<t_g$ and $y_g=11.6$ for $t>t_g$. Inset: $a$ plotted as a function of $\varepsilon$ on logarithmic scales together with the corresponding fits, restricted to the fitting intervals. (c) Master curve for $\tan \delta$ as a function of the rescaled angular frequency $\tilde{\omega} = a(t)\omega$. Red curves show the best fits obtained using a fractional Maxwell model for $t<t_g$ (upper curve and sketch) and a fractional Kelvin-Voigt model for $t>t_g$ (lower curve and sketch). (d) Elastic modulus $G'$ ($\bullet$) and viscous modulus $G"$ ($\circ$) as a function of $\omega$ at the same times as in (a). (e) Temporal evolution of the vertical shift factor $b(t)$, with $b(0)=1$ Pa$^{-1}$. Red curves show the best power-law fits of the data in the vicinity of the gel point, $b\sim\varepsilon^{-z}$, yielding exponents $z_l=3.9$ for $t<t_g$ and $z_g=2.7$ for $t>t_g$. Inset: $b(t)$ as a function of $a(t)$, with the red line indicating a power-law relation of exponent $\beta=0.25$. (f) Master curve for the rescaled elastic modulus $\Tilde{G'}$ and viscous modulus $\Tilde{G"}$ as a function of $\tilde{\omega} = a(t)\omega$. Red curves show the best fits obtained using a fractional Maxwell model for $t<t_g$ (lower curves) and a fractional Kelvin-Voigt model for $t>t_g$ (upper curves). The dotted lines at $\tan \delta(t_g)=0.44$ in (a) and (c), and at $t_g=1330$ s in (b) and (e) indicate the gel point.
• Figure 5: Time-resolved mechanical spectroscopy of the sol-gel transition in CNC suspensions with varying CNC and salt concentrations. From left to right: 1 wt% CNC and 17 mM NaCl, 2.6 wt% CNC and 14 mM NaCl, 4 wt% CNC and 10 mM NaCl, and 5.5 wt% CNC and 6.5 mM NaCl. (a)--(d) Master curves for the loss factor $\tan \delta$ as a function of the rescaled angular frequency $\tilde{\omega} = a(t)\omega$. Insets: temporal evolution of the horizontal shift factor $a(t)$, with the initial value arbitrarily set to $a(0)=1$ s.rad$^{-1}$. (e)--(h) Master curves for the rescaled elastic modulus $\tilde{G'}$ ($\bullet$) and viscous modulus $\tilde{G"}$ ($\circ$) as a function of $\tilde{\omega} = a(t)\omega$. Insets: temporal evolution of the vertical shift factor $b(t)$, with $b(0)=1$ Pa$^{-1}$. In all graphs, dotted lines indicate the gel point $t_g$. Red curves in the main graphs show the best fits of the data using a fractional Maxwell model for $t<t_g$ (upper curve) and a fractional Kelvin-Voigt model for $t>t_g$ (lower curve). Red curves in the insets show the best power-law fits in the vicinity of the gel point, $a\sim\varepsilon^{-y}$, with exponents $y_l$ for $t<t_g$ and $y_g$ for $t>t_g$, and $b\sim\varepsilon^{-z}$, with exponents $z_l$ for $t<t_g$ and $z_g$ for $t>t_g$, where $\varepsilon=|t-t_g|/t_g$.