Convective Cage Release in Model Colloidal Glasses

The mechanism of flow in glassy materials is interrogated using mechanical spectroscopy applied to model nearly hard sphere colloidal glasses during flow. Superimposing a small amplitude oscillatory motion orthogonal onto steady shear flow makes it possible to directly evaluate the effect of a steady state flow on the out-of-cage ($\alpha$) relaxation as well as the in-cage motions. To this end, the crossover frequency deduced from the viscoelastic spectra is used as a direct measure of the inverse microstructural relaxation time, during flow. The latter is found to scale linearly with the rate of deformation. The microscopic mechanism of flow can then be identified as a convective cage release. Further insights are provided when the viscoelastic spectra at different shear rates are shifted to scale the alpha relaxation and produce a strain rate-orthogonal frequency superposition, the colloidal analogue of time temperature superposition in polymers with the flow strength playing the role of temperature. Whereas the scaling works well for the $\alpha$ relaxation, deviations are observed both at low and high frequencies. Brownian dynamics simulations point to the origins of these deviations; at high frequencies these are due to the deformation of the cages which slows down the short-time diffusion, while at low frequency, deviations are most probably caused by some mild hydroclustering.

Figure 1

(a) The orthogonal frequency sweeps at $\varphi =0.61$ for $\mathrm{Pe}=2\times 10^{-1}$ (filled star), $2\times 10^{-2}$ (filled diagonal), $2\times 10^{-3}$ (filled circle), and $2\times 10^{-4}$ (filled square). SROFS data at $\varphi =0.60$ (b), $\varphi =0.61$ (c), and $\varphi =0.64$ (d). The color map in (a),(b),(c), and (d) from blue to red indicates steady shear from low to high Pe. Solid black lines indicate $G^{\prime }$ (closed/half closed symbols) and $G^{\prime \prime }$ (open symbols) fits for $G^{\prime \prime }>G^{\prime }$.

Figure 2

(a) ${\sigma }_o^{\text{orth}}$ vs Pe at $\omega =0.1$ (filled diagonal), 1 (filled circle), and $10\mathrm{rad}/\mathrm{s}$ (filled square). (b) ${\sigma }_o^{\text{orth}}$ vs $\omega$ at $\mathrm{Pe}=1$ (filled diagonal), 0.02 (filled star), $2\times 10^{-4}$ (filled circle), and 0 (filled square) for $\varphi =0.61$. Closed symbols indicate solidlike response ($G^{\prime }>G^{\prime \prime })$ and open symbols a liquidlike response ($G^{\prime }<G^{\prime \prime }$).

Figure 3

(a) Flow curves superimposed with the orthogonal steady state stress at different Pe [$\varphi =0.64$ ($\mathbf{-}$), $\varphi =0.61$ ($\mathbf{-}$), $\varphi =0.60$ ($\mathbf{-}$)]. ($\mathbf{-}$) indicates the flow curve at $\varphi =0.635$ with the sudden upturn indicative of shear thickening. (b) Crossover frequency ${\omega }_c$ vs Pe. The horizontal shifting factor $a$ and (d) the vertical shifting factor $b$ used in obtaining the scaling for SROFS (Fig. 1). The vertical dashed lines indicate the frequency regime where ${\omega }_c$ is accessible. The symbols represent $\varphi =0.64$ (filled square), $\varphi =0.61$ (filled diagonal), and $\varphi =0.60$ (filled circle).

Figure 4

(a) Mean square displacement from BD simulations for $\varphi =0.62$ when $\mathrm{Pe}=0$ ($\mathbf{-}$), 0.01 ($\mathbf{-}$), and 0.1 ($\mathbf{-}$) during steady flow. (b) Shear rate orthogonal frequency superposition data deduced from BD. $G^{\prime }$ ($\mathbf{-}$) and $G^{\prime \prime }$ ($·-·$) obtained by converting MSD data of (a) for $\mathrm{Pe}=0$ ($\mathbf{-}$), 0.01 ($\mathbf{-}$), and 0.1 ($\mathbf{-}$) [40]. (c) $g(r)$ in the velocity-gradient plane for $\mathrm{Pe}=0.1$ in steady state flow. (d) $t_c/t_B$ calculated from ${\omega }_c$ for $\varphi =0.64$ (filled square), $\varphi =0.61$ (filled diagonal), and $\varphi =0.60$ (filled circle). (black filled triangle) is the $t_c/t_B$ extracted from BD simulation for $\varphi =0.62$. (square), (star) represents $t_{\alpha }/t_B$ and $t_2/t_B$, respectively, for $\varphi =0.62$ calculated from Ref. [23].