An effective packing fraction for better resolution near the critical point of shear thickening suspensions

We present a technique for obtaining an effective packing fraction for discontinuous shear thickening suspensions near a critical point. It uses a measurable quantity that diverges at the critical point -- in this case the inverse of the shear rate $\dotγ_c^{-1}$ at the onset of discontinuous shear thickening -- as a proxy for packing fraction $φ$. We obtain an effective packing fraction for cornstarch and water by fitting $\dotγ_c^{-1}(φ)$, then invert the function to obtain $φ_{eff}(\dotγ_c)$. We further include the dependence of $\dotγ_c^{-1}$ on the rheometer gap $d$ to obtain the function $φ_{eff}(\dotγ_c,d)$. This effective packing fraction $φ_{eff}$ has better resolution near the critical point than the raw measured packing fraction $φ$ by as much as an order of magnitude. Furthermore, $φ_{eff}$ normalized by the critical packing fraction $φ_c$ can be used to compare rheology data for cornstarch and water suspensions from different lab environments with different temperature and humidity. This technique can be straightforwardly generalized to improve resolution in any system with a diverging quantity near a critical point.