Fig. 1

Rheology of suspensions (packing fraction \(\phi\)). a Flow curves and their fits to Eq. (1). Fitting parameters are as follows ; \(\phi = 0 : \tau _{\mathrm{y}} = 0\) (Pa), \(n = 0.998\), \(K = 0.10\) \((\text{ Pa } \text{ s}^{n})\), \(\phi = 0.1 : \tau _{\mathrm{y}} = 0.07\) (Pa), \(n\ = 0.96, K = 0.14\) \((\text{ Pa } \text{ s}^{n})\), \(\phi = 0.3 : \tau _{\mathrm{y}} = 1.2\) (Pa), \(n = 0.91, K = 0.50\) \((\text{ Pa } \text{ s}^{n})\), \(\phi = 0.4 : \tau _{\mathrm{y}} = 4.7\) (Pa), \(n = 0.87, K = 2.0\) \((\text{ Pa } \text{ s}^{n})\), and \(\phi = 0.5 : \tau _{\mathrm{y}} = 27\) (Pa), \(n = 0.75, K = 19.2\) \((\text{ Pa } \text{ s}^{n})\). b The \(\phi\) dependence of relative viscosity \(\eta _{\mathrm{r}} = \eta / \eta _{\mathrm{l}}\). A red curve indicates \(\eta _{\mathrm{r}}\) calculated from Eq. (2) with \(\phi _{\mathrm{c}} = 0.60\) (indicated by a blue broken line) . c Stress sweeps of storage (\(G'\) : large markers) and loss (\(G''\) : small markers) moduli measured under small to large amplitude oscillatory shear at \(f = 1\) Hz. Arrows indicate the linear and non-linear regions of the \(\phi = 0.4\) suspension. \(G' \simeq 0\) of \(\phi = 0, 0.1\) suspensions are not plotted. \(\times\) indicates the \(G'\), \(G''\) crossover which defines the yield stress \(\tau _{\mathrm{y}}\) (corresponding to a strain \(\gamma \sim 4 \times 10^{-3}\)). d Frequency sweeps of \(G'\) (large markers) and \(G''\) (small markers), measured under a small amplitude (\(\gamma = 10^{-4}\)) oscillatory shear. The thick (thin) broken line indicates the instrumental inertia effect (\(G_{\mathrm{inertia}}\)) of the spindle used to measure the \(\phi = 0\) (\(\phi \ge 0.1\)) suspension (Additional file 7: Eq. (3)). e Creep curves of \(\phi = 0.4\) suspension sheared under the \(\tau\) indicated in the legend. \(\tau\) was applied during \(0 \le t \le 5\) s and released at \(t = 5\) s (response time \(< 0.01\) s). The \(\gamma\) is normalized by the respective maximum values \(\gamma _{\mathrm{max}}\) at \(t \simeq 5\) s, which increases from \(\gamma _{\mathrm{max}} = 1.4 \times 10^{-3}\) (\(\tau = 0.2\) Pa) to \(\gamma _{\mathrm{max}} = 6.17\) (\(\tau = 5\) Pa). f \(\phi\) dependence of the yield stress \(\tau _{\mathrm{y}}\) (with errors) obtained from the flow curves (a) and stress sweeps (c). The bubble buoyancy pressures p (Eq. (5)) are plotted for comparison