Relevance of Aggregate Anisotropy in Sheared Suspensions of Carbon Black

TL;DR

This study addresses how anisotropic Carbon Black aggregates in suspensions respond to shear and how this affects rheology near the conductivity percolation threshold. It combines diffusion-limited aggregation to generate fractal, elongated aggregates with Langevin-dynamics simulations in simple shear to quantify orientation, tumbling, and viscosity. Key findings show that the long axis aligns with the flow for $Pe \gtrsim 1$, tumbling speeds up with increasing shear, and the suspension exhibits strong shear thinning, with density modulating kinetic times more than alignment. The work provides mechanistic insight into anisotropic filler behavior under flow and informs processing strategies for Carbon Black-based conductive suspensions and nanocomposites.

Abstract

Carbon Black is a filler frequently used in conductive suspensions or nanocomposites, in which it forms networks supporting electric conductivity. Although Carbon Black aggregates originate from a presumably isotropic aggregation process, the resulting particles are inherently anisotropic. Therefore, they can be expected to interact with shear flow, which significantly influences material properties. In this study, we investigate sheared suspensions of Carbon Black aggregates to elucidate the impact of aggregate anisotropy on the rheological properties. We aim at concentrations below and above the conductivity percolation threshold and comprehensively characterize particle behavior under flow conditions. Aggregates assembled by a diffusion-limited aggregation process are simulated with Langevin dynamics in simple shear flow. The simulations reveal a clear alignment of the aggregates' long axis with the flow direction, an increase in tumbling frequency with higher shear rates, and a shear-thinning response. This behavior closely parallels that of rod-like particles and underlines the significance of the anisotropic nature of Carbon Black aggregates. These findings will facilitate the optimization of nanocomposite precursor processing and the tailoring of Carbon Black-based conductive suspensions.

Figures (8)

• Figure 1: Two interacting example aggregates in a flow geometry. Each aggregate is characterized by its principal axes, long (arrows), intermediate and short (tubes) axis, respectively. The shear flow is described by the flow ($x$), velocity gradient ($y$) and vorticity ($z$) axes.
• Figure 2: Shape characterization of the aggregates via the probability distributions of their radius of gyration $R_G$ (A), axis lengths $\lambda_i$ (A), and aspect ratio $\xi$ (B). The dotted vertical lines mark the mean radius of gyration $\left< R_G\right>_N \approx 2.11\,\sigma$ and mean anisotropy $\left< \xi \right>_N \approx 2.81$. The aggregates exhibit a range of shapes and a clear deviation from the spherical limit.
• Figure 3: Time and ensemble averaged components of the gyration tensor $\left< G_{\alpha\beta} \right>_{N,t}$ plotted against the Péclet number $Pe$ for three volume fractions. As the shear rate grows, the $xx$ and $yy$-components increase, denoting an increased extent of the aggregates in flow direction and a decreased extent in gradient direction, respectively. The off-diagonal $xy$-component also indicates an emerging preferred orientation. Lines are guides to the eye.
• Figure 4: A: nematic order parameters $S$ of the three aggregate axes reveal increasing alignment of the long and short axes with increasing Péclet number $Pe$, as well as a less pronounced effect for the intermediate axis. Uncertainties of $S$ from a block bootstrapping scheme would be smaller than the symbols. B: probability distributions of $S_\mathrm{rel}$ at $\phi=6\,\%$ for $Pe = 0, 100$ show that the degree of alignment varies notably across aggregates. The kernel density estimates use a Gaussian kernel and Silverman's rule for the bandwidth determination.
• Figure 5: A: distribution of the long, intermediate, and short axes $\mathbf{e}$ of aggregates in shear flow with $Pe=100$, $\phi = 6\,\%$. The long and short axes show pronounced alignment close to the flow and vorticity directions, respectively. The fourth panel illustrates the shear flow axes within the Mollweide projection. Full symbols correspond to positive, empty symbols to negative directions. B: components of the $Q$-tensors of the three aggregate axes quantify the alignment with respect to the shear flow axes for varying Péclet number.