Shear-driven memory effects in carbon black gels

TL;DR

This study uncovers how shear history imprints lasting memory in low-volume fraction carbon black gels by coupling rheology with ultra-small-angle X-ray scattering. By quantifying a Mason-number–driven transition, it reveals two memory-encoding pathways: at high Mn memory is stored in the size of shear-induced clusters $\xi_1$, producing a homogeneous post-flow network, while at low Mn memory arises from densification and large-scale heterogeneity of a double-fractal network with mesh size $\xi_2$ extending to tens of micrometers. A three-level cluster-of-clusters framework links microstructure to rheology, with $G^{\prime}$ modeled as $G^{\prime} = \frac{U}{a\delta^2}\phi\left(\frac{\xi_1}{\xi_2}\right)^{d_{f_2}-2}\left(\frac{r_0}{\xi_1}\right)^{d_{f_1}-2}$, and structural parameters extracted from USAXS explain the non-monotonic elasticity and memory effects. The findings highlight the broad, tunable structure–memory landscape of disordered colloidal gels, enabling design principles for smart materials whose mechanical response encodes flow history, with potential applications in additive manufacturing and beyond.

Abstract

In recent years, significant effort has been devoted to developing smart materials whose mechanical properties can adapt under physical stimuli. Particulate colloidal gels, which behave as solids but can also flow under stress, have emerged as promising candidates. Resulting from the attractive interaction between their constituents, their network architecture exhibit solid-like properties even at very low volume fractions. This structural flexibility allows them to adopt various configurations and store structural information making them highly susceptible to memory effects. Shear flow, applied through rheometry, offers a simple and effective way to tune their properties and imprint a ``rheological memory'' of the flow history. However, the precise relationship between flow history and viscoelastic response remains elusive, largely due to the limited structural characterization of these systems during flow and after flow cessation. Here, we use ultra-small angle X-ray scattering (USAXS) to reveal a strong structural memory in the solid state, where the microstructure formed under shear is retained after flow cessation. We identify two distinct mechanisms of structural memory, as governed by the ratio of viscous to attractive forces, namely, the Mason number. Using recently developed fractal scaling laws, we show that the rheology is fully determined by the gel microstructure. Notably, these gels exhibit a double-fractal architecture, highlighting the remarkably broad range of length scales over which these disordered materials are structured. By clarifying how memory is encoded, our results offer strategies to tune shear sensitivity of colloidal gels and design smart materials.

Figures (12)

• Figure 1: (a) Flow curve of the $1.6~\%$ carbon black dispersion measured during a continuous ramping down of the shear rate. Colored markers indicate final stresses from flow step-down experiments at various shear rates (inset). (b) Example of viscoelastic spectra for the $1.6~\%$ dispersion after pre-shearing at $\dot{\gamma} = 45~\rm s^{-1}$ (orange) and $\dot{\gamma} = 1~\rm s^{-1}$ (green). Solid curves represent the best fits using a Kelvin-Voigt fractional model. The crossover point $(G_c, \omega_c)$, where $G^{\prime}_c = G^{\prime\prime}_c$, is highlighted with black markers. The inset shows $G_c$ as a function of $\omega_c$. (c) Rescaled crossover modulus $\tilde{G_c} = G_c(\dot{\gamma}) / G_{c}^p$, where $G_{c}^p$ is the modulus after rejuvenation at $\dot{\gamma} = 10^{3}~\rm s^{-1}$, plotted as a function of the pre-shear rate $\dot{\gamma}$. Color codes for the volume fraction of primary particles $\phi_{r_0}$. (d) Power-law exponent $\Delta$ from $G_c \propto \phi_{r_0}^{\Delta}$, as a function of pre-shear rate. The vertical line correspond to the critical shear rate $\dot{\gamma^*}=42$ s$^{-1}$ at which $\Delta$ is maximum. The shaded area indicates the antithixotropic regime, where the exponent value (determined for a pre-shear time $t = 200~\rm s$) depends on the shearing time for $t < 10^4~\rm s$. Inset shows examples of power-law fits.
• Figure 2: (a) Averaged scattering intensity $I(q)$ vs wave vector $q$, measured for the $1.6~\%$ dispersion at various shear rates. Dotted and solid curves represent the structure under flow and after flow cessation, respectively. A horizontal shift and Kratky plots ($q^2 I(q)$ vs $q$) are employed to enhance data visualization. (b) Schematic representation of the hierarchical fractal model used to describe the three structural levels in carbon black dispersions: primary particles ($r_0$), clusters ($\xi_1$), and the network ($\xi_2$). (c)-(d) Dependence of the correlation lengths and fractal dimensions on the shear rate. For the cluster level (blue markers), empty and filled symbols correspond to measurements under flow and after flow cessation, respectively.
• Figure 3: Interplay between the gel structure and its mechanical properties. (a) Schematic illustrating that shear history shape the hierarchical organization of the clusters. (b) The schematic depicts the microstructure of the CB gel, composed of clusters $\xi_1$ (blue dotted lines) assembled into agglomerates $\xi_2$ (black dotted lines). From right to left: ① $Mn \gg 1$, homogeneous network of mesh size $\xi_2$ composed of small clusters $\xi_1$; ② $Mn \sim 1$, homogeneous network of mesh size $\xi_2$ composed of large clusters $\xi_1$; ③ $Mn \ll 1$, heterogeneous network of dense mesh size $\xi_2$ composed of small clusters $\xi_1$. (c) Crossover modulus $G_c$ of the $1.6~\%$ dispersion as a function of $\dot{\gamma}/\dot{\gamma^*}$, where $\dot{\gamma^*} = 40~\mathrm{s}^{-1}$ is the critical shear rate determined from Fig. \ref{['fig:rheol']}. In the text, we discuss that this normalized shear rate can be identified with the Mason number $Mn$. The solid line represents $G_c^{model}$ (Eq \ref{['eq:G']}), with structural parameters obtained from USAXS and ($U = 38~k_B T$,$\delta = 4.4~\mathrm{nm}$). Dash line represent $G_c^{model}$ by varying ($U$, $\delta$) by $\pm$ 25%.
• Figure 4: (a) Effect of shearing time at $\dot{\gamma} = 1~\mathrm{s}^{-1}$ on the shear stress (crosses) and gel elasticity (dots) after flow cessation for the $3.2~\%$ dispersion. The inset shows the crossover modulus $G_c$ as a function of the shear stress measured just before flow cessation. The black line represents the best linear fit. (b) Flow curve of the $3.2~\%$ CB dispersion. Solid lines correspond to the steady-state flow curve, while the dotted line represents the transient flow curve obtained during a fast flow sweep. The discontinuity at $\dot{\gamma} \approx 10~\mathrm{s}^{-1}$ marks the upper limit of the antithixotropic regime. Markers indicate the time evolution of the stress shown in (a), using the same color code. Inset displays example of viscoelastic spectra.
• Figure 5: Example of results for the pre-shear protocol with a pre-shear time of $200~\mathrm{s}$, corresponding to volume fractions of CB particles of $1.6$ [(a)-(c)], 2.5 [(d)-(f)] and $3.2~\%$ [(g)-(i)]. Each set of panel displays the shear stress vs time during the pre-shear step, the elastic modulus vs time during rest following flow cessation and the viscoelastic spectra of the aged gels. Black markers correspond to the crossover point $(G_c, \omega_c)$, where $G^{\prime}_c(\omega) = G^{\prime\prime}_c(\omega)$.