Nonequilibrium structure of colloidal dumbbells under oscillatory shear

We investigate the nonequilibrium behavior of dense, plastic-crystalline suspensions of mildly anisotropic colloidal hard dumbbells under the action of an oscillatory shear field by employing Brownian dynamics computer simulations. In particular, we extend previous investigations, where we uncovered nonequilibrium phase transitions, to other aspect ratios and to a larger nonequilibrium parameter space, that is, a wider range of strains and shear frequencies. We compare and discuss selected results in the context of scattering and rheological experiments. Both simulations and experiments demonstrate that the previously found transitions from the plastic crystal phase with increasing shear strain also occur at other aspect ratios. We explore the transition behavior in the strain-frequency phase and summarize it in a nonequilibrium phase diagram. Additionally, the experimental rheology results hint at a slowing down of the colloidal dynamics with higher aspect ratio.

Figure 1

Hard dumbbell phase diagram in the volume fraction $\varphi$ to aspect ratio $L^{*}$ plane [27]. The state points for dumbbells A ($\varphi =0.55,L^{*}=0.24$), B ($\varphi =0.55,L^{*}=0.3$), and C ($\varphi =0.55,L^{*}=0.1$) considered in this work are marked by symbols $\left(\blacksquare \right)$. The state point S $(•)$ denotes an almost hard-sphere-like reference system $(\varphi =0.55,L^{*}=0.02)$. For better orientation, the freezing $\left({\varphi }_{\mathrm{freeze}}^{\mathrm{HS}}\right)$, melting $\left({\varphi }_{\mathrm{melt}}^{\mathrm{HS}}\right)$, and close-packed $\left({\varphi }_{\mathrm{CP}}^{\mathrm{HS}}\right)$ packing densities of the HS system are indicated at the vertical axis.

Figure 2

Sketch of the imposed shear flow and the used coordinate system where the velocity is in $x$, the velocity gradient in $y$, and the vorticity is along the $z$ direction. The dumbbells' bead diameter is $\sigma$ and the aspect ratio (elongation) is defined as $L^{*}=L/\sigma$, where $L$ is the center-to-center distance of both beads. In the simulations, we apply Lees-Edwards boundary conditions [43].

Figure 3

Out-of-equilibrium states [I: twinned fcc $\left(▴\right)$, II: disordered $(\circ )$, III: sliding layers $\left(\blacksquare \right)$] observed in the frequency-strain $\left(f\tau \text{−}{\gamma }_{\max }\right)$ plane of the parameter space in our BD simulations at state point A, cf. Fig. 1. The dashed lines are tentative boundaries of the respective nonequilibrium states.

Figure 4

Scattering intensities $I\left(\mathbf{q}\right)$ (250 cycles averaged) in the $q_y=0$ plane of sheared dumbbell suspensions ($L^{*}=0.30,\varphi =0.55$, state point B) at frequency $f=5{\tau }^{-1}$. From top left to bottom right: increasing strain amplitude ${\gamma }_{\text{max}}$: $0.05$ (a), $0.1$ (b), $0.2$ (c), $0.3$ (d), $0.5$ (e), and $1.0$ (f).

Figure 5

Order parameters for state points A and B at $f=5{\tau }^{-1}$. (a) Averaged orientational order parameter in flow direction $〈P_2^x〉$, and (b) translational order characterized by $〈Q_4〉$ (filled symbols) and $〈W_4〉$ (open squares).

Figure 6

Simulation snapshots at prominent points in the shear cycle for ${\gamma }_{\text{max}}=0.3,f=1{\tau }^{-1}$ for state point A in the twinned fcc state I at (a) maximum, (b) zero, and (c) minimum instantaneous strain in the shear cycle. The particle radii are scaled by $1/2$ for a better view of the structure.

Figure 7

Time-resolved orientation within one cycle (averaged over 250 cycles) at frequency $f=5{\tau }^{-1}$, elongation $L^{*}=0.30$ (state point B), volume fraction $\varphi =0.55$, and for different strain amplitudes: (a) ${\gamma }_{\max }=0.05$, (b) ${\gamma }_{\max }=0.10$, and (c) ${\gamma }_{\max }=0.15$.

Figure 8

System (B) in the fully disordered state II at ${\gamma }_{\max }=0.2$, snapshot (a) and cycle-averaged orientational order parameters ${〈P_2^{\alpha }〉}_{\mathrm{cycle}}\left(t_c\right)$ (b).

Figure 9

Snapshot in the flow-gradient $\left(x-y\right)$ plane (a) and oriental order parameters (b) in velocity $\left(x\right)$, gradient $\left(y\right)$, and vorticity $\left(z\right)$ directions versus strain cycle at ${\gamma }_{\text{max}}=1.00$ for state point B in the high-shear state III, where the orientations exhibit a finite order modulated by the imposed shear.

Figure 10

(a) Orientational relaxation constants from the decays of the $C_1$ correlation functions for state points A, B, and C, and (b) contact values of the radial distribution function $g\left(r\right)$ on increasing strain amplitude for state point A.

Figure 11

Phase equilibrium (a) of hard dumbbells at $L^{*}\approx 0.3$ from crystallization experiments. The experimental phase diagram (b) [denoted by red squares ()] is compared with the prediction of MC simulations (solid black line $\left(•\right)$ [27]) for $L^{*}=0.24$ and $L^{*}=0.30$.

Figure 12

(a) Dependence of $G^{\prime }$ and $G^{\prime \prime }$ on increasing ${\gamma }_{\max }$ for hard dumbbells with $L^{*}\approx 0.30$ in the plastic crystal phase under oscillatory shear at $f=1\mathrm{Hz}$ (circles) and $f=5\mathrm{Hz}$ (squares). This measurement is performed with the default setting $\left(100\mathrm{s}/\mathrm{point}\right)$. (b) Shear moduli versus strain amplitude ${\gamma }_{\max }$ for the same system phase that is measured under oscillatory shear of $f=1\mathrm{Hz}$ with $500\mathrm{s}/\mathrm{point}$. The filled symbols denote $G^{\prime }$, while open symbols represent $G^{\prime \prime }$.

Figure 13

Yielding behavior of the hard dumbbells with $L^{*}\approx 0.30$ (the DPM_b microgels) in the fully crystalline phase $\left({\varphi }_{\mathrm{eff}}=0.60\right)$ in the oscillatory shear of $5\mathrm{Hz}$ and the corresponding two-dimensional scattering patterns are measured by SANS. Along the dependence of $G^{\prime }$ and $G^{\prime \prime }$ on various strains: (a) at rest (b) $16\%$ (${\mathrm{Pe}}_r=0.29$, the end of the linear regime), (c) $42.3\%$ (${\mathrm{Pe}}_r=0.75$ the plateau of $G^{\prime }$, the minimum for $G^{\prime \prime }$), (d) $51.5\%$ (${\mathrm{Pe}}_r=0.90$, the maximum for $G^{\prime \prime }$), (e) $92.6\% \left({\mathrm{Pe}}_r=1.62\right)$, (f) $300\% \left({\mathrm{Pe}}_r=5.37\right)$, and (g) $1000\% \left({\mathrm{Pe}}_r=17.84\right)$.