Implications of the Four-Color Theorem on the Dynamics of N-Component Phase Separation

Abstract

We study phase separation dynamics of $N >2$ immiscible fluid components. In two-dimensions, as a consequence of the four-color theorem in mathematics, there is a change in topological constraints for $N = 3$ and $N \ge 4$ components. This leads to a suppression of hydrodynamic effects and we show that the diffusion-dominated coarsening dynamics for $N \ge 4$ can be collapsed to a single master curve. The behaviour extends to thin three-dimensional systems as the fluid domain size reaches the smallest system dimension. However, by varying the fluid interfacial tensions, cloaking may occur, resulting in highly complex coarsening dynamics that can differ for each component. In full three-dimensional cases, the absence of the four-color theorem (or its equivalent) means that the diffusive scaling law is only reached asymptotically for large $N$ rather than at a certain critical $N$.

Figures (3)

• Figure 1: (a) Fluid structure for $N=2$ at $t^*=1.85$ and $N=3$, $4$, $5$ and $6$ at $t^*=139$. (b) $L^*/a_N$ for varying numbers of components in a 2D periodic domain with dimensions $94.3\sqrt{N}\times94.3\sqrt{N}$. (c) Probability that an individual fluid region coalesces with a neighbour in a time interval of $\Delta t^*=1$ for a given $L^*$ with varying numbers of components. (d) Illustration demonstrating that any structure formed in the $N=6$ system is compatible with the $N=4$ system due to the four-color theorem. In contrast, regions in the $N\geq4$ phase structure with odd neighbors are incompatible with $N=3$ without coalescence.
• Figure 2: (a) $L^*/a_N$ over time in a cubic three-dimensional domain of $27.1\sqrt[3]{N}\times27.1\sqrt[3]{N}\times27.1\sqrt[3]{N}$ for $N=5$, $6$, $7$ and $8$. (b) $L^*/a_N$ over time in a domain of $67.5\sqrt{N}\times67.5\sqrt{N}\times 8.33$ for $N=3$, $4$, $5$ and $6$. (c) Three-dimensional visualisation of the thin film phase structure for $N=4$ at $t^*=3.70$ and $t^*=37.0$ (see Supplemental Movies 3 and 4 supp for full time evolution), with three fluids made transparent in the lower half of the images to highlight the change in topology over time. (d) Side view of the thin film phase structure for $N=4$ at $t^*=3.70$ and $t^*=37.0$ (see Supplemental Movie 5 supp for full time evolution).
• Figure 3: (a) Fluid structure during $N=6$ phase separation in a two-dimensional $200\times200$ domain for a cloaking and a non-cloaking surface tension configuration given in Table S2 supp (Supplemental Movie 6, 7 supp). Half the domain is shown. (b) $L^*$ over time for the cloaking and non-cloaking surface tension configurations. (c) Fluid structure during $N=6$ phase separation in a two-dimensional $267\times267$ domain where the pink fluid cloaks four of the others (Supplemental Movie 8 supp). The purple fluid only forms interfaces with the pink fluid. The specific surface tension and volume fraction configuration is given in Table S3 supp. (d) $L^*_i$ over time with the configuration in panel c, now plotted individually as the average length scale of each fluid component in the system.