Binary Mixtures of Particles with Different Diffusivities Demix

The influence of size differences, shape, mass, and persistent motion on phase separation in binary mixtures has been intensively studied. Here we focus on the exclusive role of diffusivity differences in binary mixtures of equal-sized particles. We find an effective attraction between the less diffusive particles, which are essentially caged in the surrounding species with the higher diffusion constant. This effect leads to phase separation for systems above a critical size: A single close-packed cluster made up of the less diffusive species emerges. Experiments for testing our predictions are outlined.

Figure 1

Snapshots of particle configurations in a binary mixture of 500 hot particles (orange, light gray) and 500 cold particles (blue, dark gray) with diffusion constants that differ by a factor $D=D_{\text{cold}}/D_{\text{hot}}=10^{-3}$ for a packing fraction $\varphi =0.2$ at (a) early times $t=6\times {10}^3{a}^2/D_{\text{hot}}$ and (b) late times $t=6.4\times {10}^4{a}^2/D_{\text{hot}}$ of the coarsening dynamics. Initially, small shapeless clusters of cold particles form, which later coalesce and merge into a single cluster. Inset: Close-up of cluster with color-coded local hexatic order parameter ${\mathrm{\Psi }}_{6,i}=|{\mathcal{N}}_i{|}^{-1}\sum _{j\in {\mathcal{N}}_i}{e}^{\iota 6{\theta }_{ij}}$, with ${\mathcal{N}}_i$ denoting the Voronoi neighbors of particle $i$, and ${\theta }_{ij}$ is the “bond” angle between particles $i$ and $j$.

Figure 2

Fraction of cold particles in the largest cluster of the system, averaged after saturation $M_{\infty }/N_{\text{cold}}$ (symbols) as a function of the diffusion constant ratio $D=D_{\text{cold}}/D_{\text{hot}}$ of cold and hot particles for systems with $N_{\text{cold}}=N_{\text{hot}}=300$ particles. Different colors correspond to different packing fractions $\varphi$. Clustering is also observed for packing fractions larger than depicted here (see Supplemental Material [17], Fig. S6). Values $M_{\infty }/N_{\text{cold}}\gtrsim 0.8$ (shaded region) indicate that a single large and stable cluster of cold particles has formed. The saturation value $M_{\infty }$ was obtained from ten realizations. Solid lines show solutions of Eq. (1) for $t\to \infty$, where the parameter ${\alpha }_{\mathrm{int}}$ was fitted to a single set of $D$ and $\varphi$ (indicated by an asterisk); see main text and Supplemental Material [17].

Figure 3

(a) Time evolution of the fraction of cold particles in the largest cluster $M/N_{\text{cold}}$ in a system of $N_{\text{hot}}=N_{\text{cold}}=300$ hot and cold particles with diffusion constant ratio $D=1.5\times {10}^{-3}$ at total packing fraction $\varphi =0.2$. Each color (gray shade) depicts a single realization of initially randomly distributed particles. The lag time ${\tau }_{\ell }$ and the time for cluster growth ${\tau }_g$ is shown for the rightmost curve. The dashed line indicates ten collision times ${\tau }_c$. (b) Cluster size distributions of cold particles for systems with $\varphi =0.2$ and $D=0$. Different symbols and colors correspond to different particle numbers; see legend of (c). (c) Time traces of $M/N_{\text{cold}}$ [same system as (b)]. Black solid line: Numerical solution of Eq. (1) with parameters as in Supplemental Material [17], Fig. S2, started after the corresponding lag time.

Figure 4

Diffusion constant for a saturated cluster of cold particles $D_{\mathrm{cl}}$ in units of $D_{\text{hot}}$ as a function of the total number of cold particles $N_{\text{cold}}$ for various packing fractions $\varphi$ indicated in the graph; the intrinsic diffusion constant of cold particles was chosen as $D_{\text{cold}}=0$. The scaling remains unchanged for $D_{\text{cold}}>0$ (Supplemental Material [17], Fig. S7). Error bars indicate 1 standard deviation of the distribution of $D_{\mathrm{cl}}$ values obtained from ten independent simulations. The three straight lines indicate power law scalings $D_{\mathrm{cl}}\sim N_{\text{cold}}^{-\kappa }$: $\kappa =1.5$ (dashed), $\kappa =1.0$ (solid), $\kappa =0.5$ (dotted). The best fit was obtained for $\kappa =1$.

Figure 5

(a) Illustration of caging. Two cold particles (blue, dark gray) are initially ($t=0$) in contact and homogeneously surrounded by hot particles (orange, light gray). The figure shows two simulations at time $t=2a^2/D_{\text{cold}}$ with the same diffusion constant of cold particles $D_{\text{cold}}$. For each time increment of $0.01a^2/D_{\text{cold}}$ a transparent disk with radius $a$ is drawn, such that the density of disks depicts the positions of particles during the simulation. Top: Simulation with equal diffusion constants $D_{\text{hot}}=D_{\text{cold}}$. Bottom: Same simulation but with increased diffusion constant of hot particles $D_{\text{hot}}=10D_{\text{cold}}$. (b) Radial pair distribution function $g(r)$ for two cold particles surrounded by 11 hot particles at $\varphi =0.13$ for three different diffusion constant ratios $D=D_{\text{cold}}/D_{\text{hot}}$. Solid black line: $g(r)$ of hard disks in the dilute limit ($\varphi \to 0$).