Wetting and Pattern Formation in Non-Reciprocal Ternary Phase Separation

TL;DR

This work addresses wetting and pattern formation in a non-equilibrium ternary mixture by formulating a minimal symmetric ternary Cahn–Hilliard framework with a non-reciprocal coupling between two active order parameters, plus a spectator phase. The authors combine a Flory–Huggins-like equilibrium free energy with a non-reciprocal dynamics given by $\partial_t \phi_1 = \nabla^2\left[\frac{\delta \mathcal{F}}{\delta \phi_1} + \alpha \phi_2\right]$, $\partial_t \phi_2 = \nabla^2\left[\frac{\delta \mathcal{F}}{\delta \phi_2} - \alpha \phi_1\right]$, and analyze linear stability, weakly non-equilibrium behaviour, and strong non-equilibrium pattern formation. The study reveals quasi-static states with near-120° contact angles, a limit-cycle regime where phases rotate around triple lines, and traveling-wave states that restabilise at high non-reciprocity; a perturbative calculation yields a lamellar travelling speed $v = \alpha \frac{\int_{-L}^{L} (\phi_1^{(0)}\phi_2^{(0)\prime} - \phi_2^{(0)}\phi_1^{(0)\prime})\,d\xi}{\int_{-L}^{L} (\Psi_1^{(0)}\phi_1^{(0)\prime} + \Psi_2^{(0)}\phi_2^{(0)\prime})\,d\xi}$ with $\Psi_i^{(0)}(\xi) = \int^{\xi} \phi_i^{(0)}(\tilde{\xi}) d\tilde{\xi}$. These findings extend classical equilibrium wetting to active interfaces and provide a minimal framework for non-reciprocal three-phase dynamics, including an exceptional-point transition and a conserved Hopf-like instability, with potential experimental realizations in non-reciprocal droplet mixtures and active emulsions.

Abstract

Non-reciprocal interactions are among the simplest mechanisms that drive a physical system out of thermal equilibrium, leading to novel phenomena such as oscillatory pattern formation. In this paper, we introduce a ternary phase separation model, with non-reciprocal interactions between two of the three phases and a spectator phase that mimics a boundary. Through numerical simulations, we uncover three distinct phase behaviours: a quasi-static regime, characterized by well-defined non-equilibrium contact angles at the three phase contact line; a limit cycle regime, with the three bulk phases rotating around the three phase contact line; and a travelling wave regime, featuring persistent directional motion. We complement our numerical findings with analytical examination of linear stability and the wave propagation speed near equilibrium. Our model provides a minimal framework for extending classical equilibrium wetting theory to active and non-equilibrium systems.

Figures (7)

• Figure 1: Left panel: Static phase diagram of the symmetric ternary phase separation model. Inside the binodal region (bounded by the green curves), coexistence is favoured over mixing. The intersection points of these curves mark the vertices of the triple phase separation region, inside which the coexistence of the three phases is globally energetically preferred. The spinodal region (shaded area) indicates compositions where spontaneous phase separation occurs. Notably, the spinodal region does not fully cover the triple-phase separation region, since compositions near those of the three coexisting phases, when homogeneous, are trapped in local free energy minima. Middle panel: An example of an equilibrium state of coexistence (under periodic boundary conditions on the square domain), featuring a three-phase coexistence whose contact line is bounded by 120-degree angles. Right panel: An example of an equilibrium lamellar state of coexistence without three phase contact lines. The parameters used throughout the simulations in this paper are $a=-0.1$, $b=9$, $c=-1$ and $\kappa=0.4$.
• Figure 2: Upper panels: Linear stability analysis of homogeneous compositions at different levels of non-reciprocity, measured by the dimensionless quantity $|\frac{\alpha}{a}|$. Eigenvalue behaviour is categorised into unstable oscillatory (real parts of complex eigenvalues are positive for certain wavenumbers $q$, labeled as orange crosses), unstable stationary (real parts of all complex eigenvalues (if any) are negative, labeled as gray triangles), and purely stable (labeled as light gray dots). Lower panels: Eigenvalue spectra as a function of wavenumber $q$, with solid and dashed lines representing the real and imaginary parts of the eigenvalues respectively. Stability types are indicated in the lower left of each panel using the same markers as in the upper panels. Panels (a)-(e) illustrate an exceptional point transition as non-reciprocity is increased for a fixed composition. Panels (e)-(f) exhibit the conserved Hopf bifurcation induced by varying composition for fixed non-reciprocity.
• Figure 3: Illustration of quasi-static and non-quasi-static behaviour of Neumann angles. First three rows: snapshots, time evolution, and histograms of Neumann angles at varying levels of non-reciprocity. Snapshots include the linear/circular fits (plotted as black lines/arcs) used to obtain the angles. Time evolutions are initialised in the equilibrium configuration and recorded from $t=3\times 10^4$ after initial transients from the equilibrium configuration. Histograms include the mean and standard deviation of the angle fluctuations. Final row: snapshots of destabilised three phase contact lines joined by three hexagonal compartments. Movies exhibiting quasi-static and chaotic behaviours are provided in the Supplemental Material as ‘Mov 3a’ and ‘Mov 3b’, corresponding to the second row and the final row respectively.
• Figure 4: Upper left panel: Spatiotemporal plot showing stable propagation of lamellar patterns for relatively small non-reciprocity $|\alpha/a|=0.1$. Upper right panel: Propagation speed analytically predicted \ref{['eq:SpeedPredict']} and numerically measured from simulations performed in two dimensions. Lower panel: Concentration profiles perpendicular to the chaser (green) - chased (red) interface at varying levels of non-reciprocity (near-equilibrium, stable propagation, and near onset of instability respectively), obtained from a 1D cross section of the 2D system.
• Figure 5: Larger values of non-reciprocity ($|\alpha/a|=0.5$ in first row of the figure) disrupt the quasi-static regime and enable the system to transition into a limit cycle regime. In this figure, time progresses from left to right, with the chaser (green) phase chasing the chased (red) phase around the three phase contact line. Further increase of non-reciprocity ($|\alpha/a|=1$ in second row of the figure) causes three phase contact lines to break, and after undergoing a chaotic transient, the system eventually evolves into a travelling pattern (which itself is unstable at $|\alpha/a|=1.0$, as further illustrated in Fig. \ref{['fig:WaveStab']} and Mov 6a). See also corresponding Movies titled ‘Mov 5a’ and ‘Mov 5b’ in the Supplemental Material.