Structural state diagram of concentrated suspensions of jammed soft particles in oscillatory shear flow

In a recent study [Khabaz et al., Phys. Rev. Fluids 2, 093301 (2017)], we showed that jammed soft particle glasses (SPGs) crystallize and order in steady shear flow. Here we investigate the rheology and microstructures of these suspensions in oscillatory shear flow using particle-dynamics simulations. The microstructures in both types of flows are similar, but their evolutions are very different. In both cases the monodisperse and polydisperse suspensions form crystalline and layered structures, respectively, at high shear rates. The crystals obtained in the oscillatory shear flow show fewer defects compared to those in the steady shear. SPGs remain glassy for maximum oscillatory strains less than about the yield strain of the material. For maximum strains greater than the yield strain, microstructural and rheological transitions occur for SPGs. Polydisperse SPGs rearrange into a layered structure parallel to the flow-vorticity plane for sufficiently high maximum shear rates and maximum strains about 10 times greater than the yield strain. Monodisperse suspensions form a face-centered cubic (FCC) structure when the maximum shear rate is low and hexagonal close-packed (HCP) structure when the maximum shear rate is high. In steady shear, the transition from a glassy state to a layered one for polydisperse suspensions included a significant induction strain before the transformation. In oscillatory shear, the transformation begins to occur immediately and with different microstructural changes. A state diagram for suspensions in large amplitude oscillatory shear flow is found to be in close but not exact agreement with the state diagram for steady shear flow. For more modest amplitudes of around one to five times the yield strain, there is a transition from a glassy structure to FCC and HCP crystals, at low and high frequencies, respectively, for monodisperse suspensions. At moderate frequencies, the transition is from glassy to HCP via an intermediate FCC phase.

Figure 1

(a) Configuration of a suspension with a volume fraction of 0.9 and polydispersity index of $\delta =0.1$ that is in shear flow with an applied frequency of $\omega$ and oscillation amplitude of ${\gamma }_{max}$. The flow (u), gradient $\left(\mathbf{\nabla }\right)$, and vorticity $\left(\omega \right)$ directions are shown in the figure.

Figure 2

Snapshot of simulation box of soft particle glasses under low (left) ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=1.78\times {10}^{-7}\left(\tilde{\omega }=\omega {\eta }_s/E^{*}={10}^{-8}\right)$ and high (right) ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=0.00178\left(\tilde{\omega }=\omega {\eta }_s/E^{*}={10}^{-4}\right)$ maximum shear rate after 200 cycles. The strain amplitude of the oscillations is ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y=10$. The polydispersity $\delta$ increases from top to bottom.

Figure 3

Shear stress (a), (c) and bond order parameter $Q_6$ (b), (d) for monodisperse suspensions under steady and oscillatory shear flows as a function of strain. The horizontal dashed lines show the values of these parameters for perfect FCC and HCP crystals. The applied strain amplitude ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y$ is 10, and the applied frequencies are $\tilde{\omega }=\omega {\eta }_s/\phantom{\omega {\eta }_s{E}^{*}}{E}^{*}={10}^{-8}$ (top panels) and $\tilde{\omega }=\omega {\eta }_s/\phantom{\omega {\eta }_s{E}^{*}}{E}^{*}={10}^{-4}$ (bottom panels), which correspond to maximum shear rates of ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=1.78\times {10}^{-7}$ and ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=0.00178$, respectively. The volume fraction of suspension is $\varphi =0.8$.

Figure 4

Variation of the $Q_6$ parameter during a single oscillation in (a) a low frequency of $\tilde{\omega }=\omega {\eta }_s/\phantom{\omega {\eta }_s{E}^{*}}{E}^{*}={10}^{-8}$ and (b) a high frequency of $\tilde{\omega }=\omega {\eta }_s/\phantom{\omega {\eta }_s{E}^{*}}{E}^{*}={10}^{-4}$ for monodisperse suspension with a volume fraction of 0.8. For the low-frequency oscillation, the transition from a glass to an HCP structure occurs at ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y=1$. For high-frequency oscillation, this transition requires the maximum strain amplitude ${\gamma }_{max}$ to be much greater than the yield strain ${\gamma }_y$.

Figure 5

Pair distribution functions in the flow-gradient (a) and (b) flow-vorticity (c) and (d) planes for the glassy and layered suspensions at a high [(a) and (c); ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=1.78\times {10}^{-2}$ and ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y=10]$ and a low [(b) and (d); ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=1.78\times {10}^{-7}$ and ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y=10]$ maximum shear rate.

Figure 6

(a) Shear stress and the maximum of pair distribution function in (b) flow-gradient and (c) flow-vorticity planes as a function of the strain. The maximum shear rate is ${\widetilde{\stackrel{̇}{\gamma }}}_{max}=0.0178,$ and the strain amplitude is ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y=10$. The volume fraction of the suspension is 0.8 and the polydispersity index is 0.1.

Figure 7

The state diagram of the SPGs at a volume fraction of (a) 0.7, (b) 0.8, and (c) 0.9 in the oscillatory shear flow. The lowest relative strain used to construct the state diagram is ${\gamma }_{max}/\phantom{{\gamma }_{max}{\gamma }_y}{\gamma }_y=3$.

Figure 8

The state diagram of the (a) monodisperse and (b) polydisperse $\left(\delta =0.2\right)$ SPGs at different volume fractions. The following symbols are used to distinguish the state of SPGs with different volume fraction: $\varphi =0.7$: glass $(ⵔ)$, FCC $(◊)$, HCP $(ⵔ)$, and FCC-HCP mixture $(△)$. $\varphi =0.8$: glass $(•)$, FCC $(⧫)$, HCP $(•)$, and FCC-HCP mixture $(▲)$. $\varphi =0.9$: glass $(◑)$, FCC $(⬗)$, HCP $(◑)$, and FCC-HCP mixture $(◮)$. For a layered phase in monodisperse SPGs the same symbols as the HCP phase in monodisperse SPGs are used.

Figure 9

The bond order parameter $Q_6$ (left axis) and maximum shear rate (right axis) as a function of strain for a suspension with a volume fraction of 0.9. The initial configuration of the suspension, i.e., at $\gamma =0$ is a glass. When the suspension is at rest: ${\widetilde{\stackrel{̇}{\gamma }}}_{\max }=0$.

Figure 10

Magnitude of the second peak of the $g_{u\nabla }\left(\rho \right)$ (left axis) and the maximum shear rate (right axis) as a function of the strain. The suspension has a volume fraction of 0.9 and polydispersity index of 0.2. The initial state $\gamma =0$ is a glassy structure. When the suspension is at rest, ${\widetilde{\stackrel{̇}{\gamma }}}_{\max }=0$. The shear rate evolution is shown with a dotted line.