Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model

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Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model

Suropriya Saha, Jaime Agudo-Canalejo, and Ramin Golestanian

Phys. Rev. X 10, 041009 – Published 13 October 2020

Abstract

Pair interactions between active particles need not follow Newton’s third law. In this work, we propose a continuum model of pattern formation due to nonreciprocal interaction between multiple species of scalar active matter. The classical Cahn-Hilliard model is minimally modified by supplementing the equilibrium Ginzburg-Landau dynamics with particle-number-conserving currents, which cannot be derived from a free energy, reflecting the microscopic departure from action-reaction symmetry. The strength of the asymmetry in the interaction determines whether the steady state exhibits a macroscopic phase separation or a traveling density wave displaying global polar order. The latter structure, which is equivalent to an active self-propelled smectic phase, coarsens via annihilation of defects, whereas the former structure undergoes Ostwald ripening. The emergence of traveling density waves, which is a clear signature of broken time-reversal symmetry in this active system, is a generic feature of any multicomponent mixture with microscopic nonreciprocal interactions.

Received 21 May 2020
Revised 11 August 2020
Accepted 24 August 2020

DOI:https://doi.org/10.1103/PhysRevX.10.041009

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Emergence of patterns

Living matter & active matter

Theories of collective dynamics & active matter

Polymers & Soft Matter

Authors & Affiliations

Suropriya Saha ¹, Jaime Agudo-Canalejo ¹, and Ramin Golestanian ^1,2,*

¹Max Planck Institute for Dynamics and Self-Organization (MPIDS), D-37077 Göttingen, Germany
²Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom

^*ramin.golestanian@ds.mpg.de

Popular Summary

The study of active matter—composed of interacting, self-propelling units—has led to new insights across a broad range of fields, from cellular machinery (such as enzymes and molecular motors) to the flocking dynamics of birds and other animals. The behavior of active matter becomes particularly interesting when it is composed of mixtures of different species. When the component units are different, their interactions will generically violate Newton’s third law, leading to substantial departures from equilibrium behavior. Here, we use such nonreciprocal interactions to propose a new model of pattern formation in active matter composed of multiple species.

In an equilibrium multicomponent system, gradients of chemical potentials drive the system to evolve into separated components. In our model, we mathematically tweak the chemical potential of each species to include nonreciprocal interactions from other species in the system. We find that this added element leads fluid droplets of one species to attempt to collocate with other species, while the reverse is not true. This leads to the formation of complex oscillatory patterns, as one component “chases” after another.

Our model could be used as a minimal description of many multicomponent mixtures, such as cells or bacteria communicating through chemical pathways, where nonreciprocal interactions occur.

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Vol. 10, Iss. 4 — October - December 2020

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Figure 1
Phase behavior exhibited by the NRCH model. (a) Example four-component system displaying self-propelled lamellar domains in the steady state. Ripples in density travel perpendicularly to the direction of motion. (b)–(l) Binary systems displaying [(b)–(d)] bulk phase separation at low activity ( $α = 0.2$ ). At higher activity ( $α = 0.4$ ), they undergo an oscillatory instability and display [(e)–(g)] a lamellar phase with self-propelled density bands or [(h)–(l)] two-dimensional moving micropatterns, depending on the system composition. In all cases, the top (b,e,h,j) and bottom (c,f,i,k) rows of snapshots correspond to intermediate and steady states, respectively, in simulations with Gaussian white noise. Line scans of the concentration along a cross section of the corresponding steady states in a noiseless simulation are shown in panels (d,g,l). (b) Coarsening of bulk phase separated states, which proceeds through system-spanning labyrinthine patterns as in equilibrium Cahn-Hilliard, (e) coarsening of self-propelled lamellar patterns through annihilation of defects, and (h,j) moving patterns arising from ordering of self-propelled domains. For the self-propelled lamellar and 2D patterns, the concentration profiles at steady state (g,l) show a fixed wavelength and a finite separation between the peaks of the two components, sustaining the motion due to nonreciprocal interactions. The parameters corresponding to panel (a) are described in the Appendix pp1. For panels (b)–(l), we used $c_{1, 1} = 0.2$ , $c_{1, 2} = 0.5$ , $c_{2, 1} = 0.1$ , $c_{2, 2} = 0.5$ , $χ = - 0.2$ , $χ^{'} = 0$ and average densities $\bar{ϕ_{1}} = 0.35$ and $\bar{ϕ_{2}} = 0.3$ , with the exception of panels (h)–(l) for which $\bar{ϕ_{2}} = 0.42$ . The stiffness $κ = 0.0001$ is the same in all simulations. A system size of $201 \times 201$ and a time stepping of $10^{- 4}$ are used in all of the simulations presented here.
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Figure 2
Bulk phase separation in the plane of average composition $(\bar{ϕ_{1}}, \bar{ϕ_{2}})$ . (a) The spinodal region, the binodal region, the tie lines, and the two critical points are shown for $α = 0.33$ . The homogeneous state is unstable to small perturbations inside the spinodal region shaded in blue. (b) At higher activity, $α = 0.49$ , the spinodal region splits into two disconnected regions shaded in green. Both regions are surrounded by binodal regions and have developed two new critical points. (c) Change in spinodals and critical points with increasing $α$ . Splitting into two disconnected regions occurs for $α > 0.35$ . Parameters used in the free energy are $c_{1, 1} = 0.2$ , $c_{1, 2} = 0.5$ , $c_{2, 1} = 0.1$ , $c_{2, 2} = 0.5$ , $χ = - 0.2$ and $χ^{'} = 0.2$ , $κ = 0.001$ . A system size of $201 \times 201$ and a time stepping of $10^{- 4}$ are used in all of the simulations to construct the binodals.
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Figure 3
Dynamics in the NRCH model. (a) Coarsening dynamics for equilibrium bulk phase separation ( $α = 0$ ) and formation of self-propelled bands in the active case ( $α = 0.95$ ). (b) A–E show simulation snapshots over time. We note the formation of lamellar domains in C, which coarsen in D and E to the final state in Fig. 1. (c) Time evolution of the polar-order parameter as measured by the components of the fluxes $J_{x}$ and $J_{y}$ , obtained by averaging over 250 randomly generated conditions. For $α = 0.95$ , the order parameter saturates to a value of approximately 0.1, while for $α = 0.15$ , it decays to zero. Parameters for the free energy are as in Fig. 1. System sizes of $401 \times 401$ and $201 \times 201$ are used in simulations to calculate the coarsening length scale and current $J$ , respectively. Time stepping is fixed at $10^{- 4}$ .
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Figure 4
Onset of instability and comparison with the minimal oscillator model described in Sec. 4b. (a,b) Temporal variation of the density fields $ϕ_{i}$ and (c,d) corresponding phase portraits taken at a randomly chosen point in space $r$ for (a,c) $α = 0.4$ and (b,d) $α = 0.95$ . Temporal trajectories of the underlying nonconserved dynamical system are shown in panels (e,f) for $α = 0.1$ and 0.4, with initial conditions very close to (0,0). Phase portraits for $α = 0.1$ , 0.4, and 0.95 are shown in panels (g)–(i), with a trajectory originating from (0,0) highlighted in red. (g) At low $α$ , there are two stable points. (h,i) Spiral trajectories and a limit cycle arising for large enough $α$ . Note the similarity in the form of the limit cycles in panels (c) and (h), and (d) and (i). Parameters for the free energy are as in Fig. 1. The system size of $201 \times 201$ is used in all of the simulations here, with a time step of $10^{- 4}$ .
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Figure 5
Pattern formation in the plane of reciprocal and nonreciprocal interactions. A stability analysis is shown in the $(χ, α)$ plane for $c_{1, 1} = - c_{1, 2} = c_{1}$ and $c_{2, 1} = - c_{2, 2} = c_{2}$ , for (a) the general case $c_{1} \neq c_{2}$ and (b) the special case $c_{1} = c_{2}$ . The red lines indicate exceptional points at which the eigenvalues collapse and the corresponding eigenvectors become parallel. The results of the stability analysis are verified using simulations: The parameters scanned are shown using black dots, and at each dot, the steady-state limit cycle is plotted to show whether the system goes into oscillations or bulk phase separation. The stiffness parameter $κ = 0.05$ . The system size of $201 \times 201$ is used in all of the simulations here, with a time step of $10^{- 2}$ .
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Figure 6
Phase diagrams for pattern formation in the NRCH model. The spinodal region determined from the linear stability analysis is shown in panels (a) and (b) for the values of $α$ shown in the legend. (a) Spinodals for low $α$ . (b) As the activity is increased beyond $α ≳ | χ |$ , the topology of the curves consists of four arms surrounding a central circular region. (c) Phase diagram in the plane of average composition $(\bar{ϕ_{1}}, \bar{ϕ_{2}})$ for $α = 0.4$ . Bulk phase separation (four blue arms) and pattern formation (turquoise inner circle) occur in the same phase diagram. (d) Magnification of the inner circle. The gray line encloses the region where self-propelled lamellar patterns are observed [see Figs. 1], with a constant wavelength and an amplitude that decays slowly on moving away from the center of the oscillatory region. Within this region, $[ϕ_{1} (r, t), ϕ_{2} (r, t)]$ evolve towards a limit cycle independent of $r$ at all spatial points, as described in Sec. 4b. The changes in the limit cycles upon moving radially along the composition plane are depicted by plotting a scaled-down version of Fig. 4 at selected points in a few radial directions. The area of the limit cycle decreases towards the edge of this region; the color encodes the area of the limit cycle and varies as shown in the color bar. In the region between the gray and black curves, the steady state changes to a lattice of local density undulations, which moves with a constant velocity [see Figs. 1]. The trajectories change from simple limit cycles to complex repeating patterns. This complexity is reflected in the more complex phase portrait as seen in the scaled-down version plotted in the region between the gray and black boundaries. Parameters for the free energy are as in Fig. 1. The system size of $201 \times 201$ is used in all of the simulations here, with a time step of $10^{- 4}$ for panel (c) and an increased time stepping of $10^{- 3}$ for the simulations corresponding to panel (d).
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