Active Phase Separation in Mixtures of Chemically Interacting Particles

We theoretically study mixtures of chemically interacting particles, which produce or consume a chemical to which they are attracted or repelled, in the most general case of many coexisting species. We find a new class of active phase separation phenomena in which the nonequilibrium chemical interactions between particles, which break action-reaction symmetry, can lead to separation into phases with distinct density and stoichiometry. Because of the generic nature of our minimal model, our results shed light on the underlying fundamental principles behind nonequilibrium self-organization of cells and bacteria, catalytic enzymes, or phoretic colloids.

Figure 1

Mixtures of chemically interacting particles display a wealth of active phase separation phenomena. (a) Binary mixtures of producer (blue) and consumer (red)f species show, from left to right, homogeneous states with association of particles into small “molecules,” aggregation into a static dense phase that coexists with a dilute phase, and separation into two static collapsed clusters; see Movies 1–3 in the Supplemental Material [24]. (b) The static aggregate [(a), center] can undergo symmetry breaking to form a self-propelled macroscopic cluster; see Movies 8 and 9. (c) Binary mixtures of producer species (blue and red) show homogeneous states without molecule formation, separation into a static dense phase and a dilute phase that is depleted near the dense phase, and aggregation into a static collapsed cluster; see Movies 4–6. (d) Randomly generated highly polydisperse mixtures (20 different species) can remain homogeneous or undergo macroscopic phase separation; see Movies 11 and 12. Simulation parameters for each case [(a)–(c)] can be found in the description of the corresponding Movies.

Figure 2

(a) Stability diagram for mixtures of one producer and one consumer species [cf. Fig. 1], and (b) for mixtures of two producer species [cf. Fig. 1]. In (a),(b) the boxed legends attached to each quadrant symbolize the “interaction network” representing the sign of interactions between each species in the system, as described in the main text. Phase separation [aggregation in (a), separation in (b)] can be triggered by addition or removal of particles (density changes) only when interactions between the two species are intrinsically nonreciprocal. (c) Stoichiometry at the onset of the instability, obtained from 44 simulations (blue circles, see Table S1 in the Supplemental Material [24]) compared to the stability analysis prediction [Eq. (2)]. (d) Time evolution of the stoichiometry of the biggest cluster arising from aggregation of $({\alpha }_1,{\alpha }_2)=(+,-)$ mixtures, demonstrating that the long time stoichiometry is predicted by the neutrality rule [Eq. (3)] and is independent of the species’ mobility (blue: ${\tilde{\alpha }}_2=-1$, ${\tilde{\mu }}_2=8$ and 12; red: ${\tilde{\alpha }}_2=-2$, ${\tilde{\mu }}_2=4$ and 8; green: ${\tilde{\alpha }}_2=-3$, ${\tilde{\mu }}_2=3$ and 5; in all cases $N_1=800$, $N_2=200$, ${\tilde{\alpha }}_1={\tilde{\mu }}_1=1$).

Figure 3

Phase separation induced by a small amount of an active “doping agent.” (a) Simulation snapshots showing macroscopic aggregation of a previously homogeneous mixture ($N_1=N_2=500$, ${\tilde{\alpha }}_1={\tilde{\mu }}_1=1$, ${\tilde{\alpha }}_2=-1$, ${\tilde{\mu }}_2=1/2$) after addition of 5% of a third species ($N_3=50$, ${\tilde{\alpha }}_3=-5$, ${\tilde{\mu }}_3=2$), compare Movies 1 and 10 in the Supplemental Material [24]. (b) Time evolution of the size of the largest cluster (total number of particles), in the absence and presence of the third species.