Dynamic heterogeneities and non-Gaussian behavior in two-dimensional randomly confined colloidal fluids

A binary mixture of superparamagnetic colloidal particles is confined between glass plates such that the large particles become fixed and provide a two-dimensional disordered matrix for the still mobile small particles, which form a fluid. By varying fluid and matrix area fractions and tuning the interactions between the superparamagnetic particles via an external magnetic field, different regions of the state diagram are explored. The mobile particles exhibit delocalized dynamics at small matrix area fractions and localized motion at high matrix area fractions, and the localization transition is rounded by the soft interactions [T. O. E. Skinner et al., Phys. Rev. Lett. 111, 128301 (2013)]. Expanding on previous work, we find the dynamics of the tracers to be strongly heterogeneous and show that molecular dynamics simulations of an ideal gas confined in a fixed matrix exhibit similar behavior. The simulations show how these soft interactions make the dynamics more heterogeneous compared to the disordered Lorentz gas and lead to strong non-Gaussian fluctuations.

Figure 1

(a) Schematic of the experiment, a binary system of small and large particles confined between two glass slides (particle diameters to scale). The large particles support the top slide. The magnetic field $B$ tunes the effective interaction between the particles. (b) Mean-squared displacement for the fluid particles in a very dilute 2D cell. A dashed line indicating diffusive behavior, $\delta r^2\left(t\right)\sim t$, is shown as a guide to the eye. (c) State diagram for the effective area fractions of the fluid (${\mathrm{\Phi }}_F$) vs the matrix particles (${\mathrm{\Phi }}_M$). (d) Snapshot of the system at state point L1P6 in a quadrant of size $214\times 171\mu \mathrm{m}$.

Figure 2

Experiment: single-particle probability distributions from 2D histograms of all particle positions in a quadrant measured for state points (a) L1P1, (b) L1P6, (c) L2P1, and (d) L2P6. The distributions are normalized such that the total probability of the whole quadrant is unity. The size of the particles is indicated by the red circles under the scale bars in each plot, and the size of the hard-core excluded area for centers of mobile particles is indicated in (a) by the blue circle under the scale bar.

Figure 3

Simulation: single-particle probability distributions from 2D histograms in the confined ideal gas case. Shown are square sections of length $20{\sigma }_M$ and the histograms are normalized such that the total probability integrated over the shown section is unity.

Figure 4

Experiment: partial radial distribution functions, for the (a) matrix-matrix interaction $g_{MM}\left(r\right)$, (b) fluid-fluid interaction $g_{FF}\left(r\right)$, and (c) fluid-matrix interaction $g_{MF}\left(r\right)$ for each state point along line 1 and line 2. Line 2 is shifted by 2 in each plot.

Figure 5

Simulation: partial radial distribution functions, for the (a) matrix-matrix interaction $g_{MM}\left(r\right)$, (b) fluid-fluid interaction $g_{FF}\left(r\right)$, and the fluid-matrix interaction $g_{MF}\left(r\right)$ for the single-energy and confined-ideal-gas cases [the latter data is shifted upwards by 5 in (b) and by 1 in (c)].

Figure 6

Experiment: self-part of the intermediate scattering function $F_{\mathrm{s}}(q,t)$ for the fluid particles for a range of wave numbers $q$ relating to state points (a) L1P1, (b) L1P6, (c) L2P1, and (d) L2P6 (in colors), as well as the corresponding Gaussian approximations (in gray). A measure of the nonergodicity parameter is obtained with $f_{\mathrm{s}}\left(q\right)\approx F_{\mathrm{s}}(q,t\approx 3300s)$, indicated by the dotted line, and shown in Fig. 8.

Figure 7

Simulation: self-part of the intermediate scattering function $F_{\mathrm{s}}(q,t)$ in the simulation for the single-energy case (colored dashed lines) and the confined ideal gas case (colored solid lines) for particle diameters (a) ${\sigma }_F=0.2$, (b) 0.45, and (c) 0.7, as well as the corresponding Gaussian approximations (in gray).

Figure 8

(a) Long-time limit of the SISF $f_{\mathrm{s}}\left(q\right)$ of the experiment along lines 1 and 2 in semilogarithmic presentation. (b) Long-time limit of the SISF $f_{\mathrm{s}}\left(q\right)$ in the simulation for the single-energy (dashed) and ideal-gas case (solid).

Figure 9

Mean-quartic displacement $\delta r^4\left(t\right)$ for experiment (a) and simulation (b) as a function of time. In the simulation, single-energy case (dashed lines) and confined-ideal-gas case (solid lines) are shown. The straight gray lines $\sim t^2$ and $\sim t^{4/\widehat{z}}$ with $\widehat{z}\approx 2.955$ serve as a guide to the eye.

Figure 10

Non-Gaussian parameter ${\alpha }_2\left(t\right)$ for the experiment (a), and for the simulation in the (b) single-energy and (c) confined-ideal-gas cases.

Figure 11

Experiment: mean-squared displacement $\delta r^2\left(t\right)$ for the matrix particles at all state points (color), the center of mass of the matrix particles (red), and the fluid particles at the lowest and highest state points (gray) along line 0 (a), line 1 (b), and line 2 (c).

Figure 12

Experiment: histograms of the maximum distance $r^{*}$ that each particle travels away from its initial position at $t=0$ over the whole duration of the experiment for both the matrix and fluid particles along line 0 (a) and (b), line 1 (c) and (d), and line 2 (e) and (f). The reclassification cutoff distance $r_{rc}^{*}$ is marked by vertical line in the histograms of the fluid particles.

Figure 13

Experiment: single-particle probability distributions from all small particle positions in a quadrant measured for state points (a) L0P1 and (b) L0P5. Normalized so that the total probability is unity. The size of the colloidal particles is indicated as red circles and the size of the hard-core excluded area for centers of mobile particles is indicated in (a) as blue circle.

Figure 14

Experiment: partial radial distribution functions: (a) for the matrix-matrix interaction $g_{MM}\left(r\right)$, (b) fluid-fluid interaction $g_{FF}\left(r\right)$, and (c) fluid-matrix interaction $g_{MF}\left(r\right)$ for each state point along line 0.

Figure 15

Experiment: self-part of the intermediate scattering function $F_{\mathrm{s}}(q,t)$ for the fluid particles for a range of wave numbers $q$ relating to state points (a) L0P1 and (b) L0P5 (in colors), as well as the corresponding Gaussian approximations (in gray). A measure of the nonergodicity parameter is obtained with $f_{\mathrm{s}}\left(q\right)\approx F_{\mathrm{s}}(q,t\approx 3300s)$, indicated by the dotted line, and shown in (c).

Figure 16

Experiment: (a) mean-squared displacement $\delta r^2\left(t\right)$ for the state point along line 0. The straight gray lines indicate $\sim t$ and $\sim t^{2/z}$ with $z\approx 3.036$ and serve as a guide to the eye. (b) Mean-quartic displacement $\delta r^4\left(t\right)$ for the state points along line 0. The straight gray lines indicate $\sim t^2$ and $\sim t^{4/\widehat{z}}$ with $\widehat{z}\approx 2.955$ and serve as guide to the eye. (c) The corresponding non-Gaussian parameter ${\alpha }_2\left(t\right)$ for the state point along line 0.