Tagged-particle dynamics in confined colloidal liquids

We present numerical results for the tagged-particle dynamics by solving the mode-coupling theory in confined geometry for colloidal liquids (cMCT). We show that neither the microscopic dynamics nor the type of intermediate scattering function qualitatively changes the asymptotic dynamics in vicinity of the glass transition. In particular, we find similar characteristics of confinement in the low-frequency susceptibility spectrum which we interpret as footprints of parallel relaxation. We derive predictions for the localization length and the scaling of the diffusion coefficient in the supercooled regime and discover a pronounced nonmonotonic dependence on the confinement length. For dilute liquids in the hydrodynamic limit we calculate an analytical expression for the intermediate scattering functions, which is in perfect agreement with event-driven Brownian dynamics simulations. From this, we derive an expression for persistent anticorrelations in the velocity autocorrelation function (VACF) for confined motion. Using numerical results of the cMCT equations for the VACF we also identify a crossover between different scalings corresponding to a transition from unconfined to confined behavior.

Figure 1

Normalized incoherent intermediate scattering function $S_0^{\left(s\right)}(q_m,t)/S_0^{\left(s\right)}(q_m,0)$ for accessible width $L=2.0\sigma$ for wave number $q_m\sigma =6.52$ corresponding to the first sharp diffraction peak in the structure factor. The control parameter $\epsilon =(\phi -{\phi }_c)/{\phi }_c=\pm {10}^{-n/3}, n\in \mathbb{N},$ increases from left to right for $\epsilon <0$ and from bottom to top for $\epsilon >0$. The critical correlator for $\phi ={\phi }_c$ (or $\epsilon =0$) is displayed as thick line and labeled as “c.”

Figure 2

Comparison of Brownian and Newtonian dynamics in the normalized incoherent intermediate scattering function $S_0^{\left(s\right)}(q_m,t)/S_0^{\left(s\right)}(q_m,0)$ for accessible width $L=2.0\sigma$, control parameter $\epsilon =-{10}^{-5}$, and wave number $q_m\sigma =6.52$. The result for Newtonian dynamics corresponds to $\epsilon =-{10}^{-5}$ shown in Fig. 1 in Ref. [56].

Figure 3

Frequency-dependent susceptibility of the self-dynamics ${\chi }_0^{\prime \prime \left(s\right)}(q_m,\omega )$ for the same parameters as in Fig. 1. The dashed black line shows a Debye peak, ${\chi }_D^{\prime \prime }\left(\omega \right)=2{\chi }_{\text{max}}\omega {\tau }_D/\left[1+{\left(\omega {\tau }_D\right)}^2\right]$ (${\chi }_{\text{max}}=0.37, {\tau }_D=1.56\times {10}^8{\sigma }^2{D}_0^{-1}$) for comparison.

Figure 4

Frequency-dependent susceptibility of the self-dynamics ${\chi }_0^{\prime \prime \left(s\right)}(q_m,\omega )$ for channel width $L=1.0\sigma$. The parameters for the Debye peak (dashed black line) are ${\chi }_{\text{max}}=0.41, {\tau }_D=1.45\times {10}^{11}{\sigma }^2{D}_0^{-1}$.

Figure 5

Mean-square displacement $\delta r_{\parallel }^2\left(t\right)$ for accessible width $L=2.0\sigma$ asymptotically close to the glass transition. The color code is the same as used in Fig. 1.

Figure 6

Localization length $l_{\parallel }$ close to the glass transition with $\epsilon =(\phi -{\phi }_c)/{\phi }_c$. The data were extracted using Eq. (32). The lines correspond to fits $l_{\parallel }\left(\epsilon \right)=l_{\parallel }^c-h_l\sqrt{\epsilon }$.

Figure 7

Inverse diffusion constant close to the glass transition. The fitted (and shifted) power laws plotted in black are $\gamma (L=1.0\sigma )=2.81$ (full), $\gamma (L=2.0\sigma )=2.19$ (long-dashed), and $\gamma \left(\text{bulk}\right)=2.52$ (short-dashed). The data were extracted using Eq. (31).

Figure 8

Velocity autocorrelation function close to the glass transition for $\epsilon <0$ (a) and $\epsilon >0$ (b) and accessible width $L=2.0\sigma$. The critical correlator is fitted by a power law with exponent $a+2=2.34$, shown as a dashed black line. The dotted black line displays as reference a power law with exponent $2-b=1.31$. The color code is the same as used in Fig. 1.

Figure 9

Dependence of the critical localization length $l_{\parallel }$ and scaling exponent $\gamma$ on the accessible width $L$. The exponents were determined from the asymptotic analysis as described in Ref. [56]. The arrows indicate the value of the exponents in the bulk limit for hard spheres (arrows pointing to the right [69]) and hard disks (arrows pointing to the left [70]).

Figure 10

Mean-square displacement $\delta r_{\parallel }^2\left(t\right)$ at time $t^{*}=100{\sigma }^2{D}_0^{-1}$ for different accessible widths $L$ and packing fractions $\phi$.

Figure 11

Coherent intermediate scattering function for small $q_s\sigma =0.272$ at packing fraction $\phi =0.1$ and $L=1.0\sigma$. Results are shown for event-driven Brownian simulations and for the theoretical prediction Eqs. (35) and (37).

Figure 12

Incoherent intermediate scattering function for small $q_{s^{\prime }}\sigma =0.388$ at packing fraction $\phi =0.1$ and $L=1.0$. Results are shown for event-driven Brownian simulations and for the theoretical prediction Eqs. (36) and (37). The value for $D_{\parallel }$ was determined from the mean-square displacement of the colloids in the EDBD simulations.

Figure 13

Velocity autocorrelation function determined with cMCT for $\phi =0.005$. The figure shows results for confined and bulk colloidal liquids. Black lines correspond to power laws with exponents 2.0 (dashed) and 2.5 (dotted).