Designing the Morphology of Separated Phases in Multicomponent Liquid Mixtures

Phase separation of multicomponent liquid mixtures plays an integral part in many processes ranging from industry to cellular biology. In many cases the morphology of coexisting phases is crucially linked to the function of the separated mixture, yet it is unclear what determines the morphology when multiple phases are present. We developed a graph theory approach to predict the topology of coexisting phases from a given set of surface energies, enumerate all topologically distinct morphologies, and reverse engineer conditions for surface energies that produce the target morphology.

Figure 1

Morphologies of three coexisting phases $R$ (red), $G$ (green), and $B$ (blue) are determined by the magnitudes of surface tensions (${\gamma }_{RB}\ge {\gamma }_{RG}\ge {\gamma }_{BG}>0$) and volume fractions. First row: schematics of local arrangements of phases and corresponding graph representations for (a) partial wetting (${\gamma }_{RB}<{\gamma }_{RG}+{\gamma }_{GB}$), and (b) complete wetting (${\gamma }_{RB}>{\gamma }_{RG}+{\gamma }_{GB}$). Second row: representative simulation snapshots at ${10}^6$ time steps (see Video S1 for the time evolution [28]). The blue phase is semitransparent for the snapshots with unequal volume fractions. The simulation parameters are given in Table S1 [28].

Figure 2

Prediction of the topology of separated phases from the set of surface energies $\{{\gamma }_{IJ}\}$. Surface energies are used to produce the set of graphs of triplets of phases, from which we construct the connectivity graph describing the topology of separated phases (see text). (a) When all four triplet connectivity graphs are fully connected, then the connectivity graph with four vertices is also fully connected. (b),(c) Different sets of graphs (yellow boxes indicate the difference) can produce the same topology of separated phases. The last column shows representative simulation snapshots at ${10}^6$ time steps. The simulation parameters are given in Table S1 [28].

Figure 3

Graph representations and simulation snapshots at ${10}^6$ time steps for all (a)–(f) six distinct topologies of four coexisting phases with equal volume fractions (top) and nonequal volume fractions with transparent white phase (bottom). See Video S2 for time evolution [28]. The simulation parameters are given in Table S1 [28].

Figure 4

Reverse engineering of (a), (b) two target structures. To reverse engineer the model parameters for target structures, we first construct a connectivity graph, which is then divided into subgraphs of triplets of phases that are associated with inequalities of surface energies. The subgraphs highlighted with yellow boxes do not provide any constraints on surface energies. The linear programming is used to find a set of surface energies that satisfy these inequalities (see text), which are then converted to interaction parameters ${\chi }_{ij}$. The average volume fractions $\{{\overline{\varphi }}_i\}$ of components are chosen such that the volume fractions of separated phases are consistent with the target structure. The resulting simulation snapshots at $4\times {10}^5$ time steps are shown on the right (see Video S3 for time evolution [28]). The simulation parameters are given in Table S1 [28].