Non-reciprocal interactions between condensates in chemically active mixtures

Abstract

We study the behaviour of catalytically active droplets in multi-component conserved mixtures affected by noise. Working in the thin interface limit, we analytically determine the state diagram of the system, characterized by multiple dynamical regimes, and verify our findings using numerical simulations. In particular, we show the emergence of a non-reciprocal, chemically-mediated interaction between the droplets, which leads to the formation of (meta-)stable clusters of droplets of different species. We find that the clusters can display self-propulsion in a large part of the parameter space, including regions where the non-reciprocal interactions between the droplets are purely attractive. This surprising feature arises from the non-local nature of the chemical interactions, and points to locality violations as a general mechanism for energy dissipation and emergence of out-of-equilibrium steady states in active matter.

Figures (6)

• Figure 1: (a) The species gather into droplets due to their interaction. Spatial variations in the densities of the species ($\phi_1$ and $\phi_2$) create a spatially varying chemical field ($c$) as they produce and consume the chemical. (b) The chemical affects the intra-species interaction and mediates effective inter-species interactions. (c) Configurations of droplets: (I) disjoint, (II) partially overlapping and (III) fully overlapping. (d) Attractive interactions give rise to chasing-behaviour. The centre of the droplet is an emitter (E) of a chemical signal and the interfaces are sensors (S). An example where purely attractive interactions lead to chasing (left) and its implementation with two droplets (right).
• Figure 2: (a) State diagram obtained by simulating Eq. \ref{['eq:eom']}, presented a radial plot with radius $\mathcal{R} = R_1 / R_2$ from 0 to 1 and angle $\vartheta = \arg(\mathcal{C}_{1\mathrel{ \mkern-4mu\hbox{)}}2}+\mathrm{i}\,\mathcal{C}_{2\mathrel{ \mkern-4mu\hbox{)}}1})$. The black [Eq. \ref{['eq:stabingl']}] and orange [Eq. \ref{['eq:stabrep']}] lines delimit the boundaries between the different configurations predicted by theory. (b) Effective potentials for $W_{-}(\bm x)$ in the various state. (c) Steady-state velocity and relative position from Eq. \ref{['eq:releqdist']} and \ref{['eq:velocity']} (dashed black), and comparison with the result from simulations, where colour indicaters time, with the steady-state shown in orange. The simulations are obtained with $T = 10^{-5}$.
• Figure 3: (a) Histogram of the jumping times for the relative position in the overlapping regime at $T=2\times10^{-2}$, following a Poisson distribution with measured jumping time $\langle\tau\rangle = 1.29 \times 10^{3}$, compared with $\tau_K = 1.44\times 10^3$. Inset: Plot of centre positions show a clear distinction between running and tumbling. (b) MSD of the cluster from simulations of Eq. \ref{['eq:eom']} in the overlapping (red) and engulfed (blue) regimes. The latter follows an AOU process with coefficients obtained from Eq. \ref{['eq:dynLangevin']} (dashed orange). (c) Steady-state probability distribution for $x$. Points show simulations of overlapping (red) and attractive (blue) regimes, and compared with the predictions from Eq. \ref{['eq:dynLangevin']} (dashed lines).
• Figure 4: (a) State diagram at finite (large disk) and vanishing (small disk) decay rate $\gamma$ in $d=3$. (b) Results from simulations of \ref{['eq:eom']} in $d=2$ at $T = 10^{-4}$. (b) Droplet trajectories (droplet 1 in blue, droplet 2 in black). Inset: Snapshot of the densities $\phi_a$, red and blue, and the chemical field $c$. (c) MSD of the midpoint for droplets in the attractive (blue) and overlapping (red) configurations. (d) The effective potential for $\bm{x}$, inferred from the probability distribution of the relative position, which is consistent with the theoretical analysis. Dashed lines are running averages to guide the eye.
• Figure 5: Simulations with non-zero $\chi$. To the left: state diagram categorizing the behaviour of the system as in Fig. \ref{['fig:phase']}. The green triangle indicates states where all interactions are attractive, but the system still self-propels. To the right: the corresponding steady-state velocity. Parameters: $R_1 = 1$, $R_2 = 1.5$ ($\mathcal{R} = \frac{2}{3}$), and $\partial_c \sigma_2(0)= 0.8$.