Field-Embedded Particles Driven by Active Flips

Systems of independent active particles embedded into a fluctuating environment are relevant to many areas of soft-matter science. We use a minimal model of noninteracting spin-carrying Brownian particles in a Gaussian field and show that activity-driven spin dynamics leads to patterned order. We find that the competition between mediated interactions and active noise alone can yield such diverse behaviors as phase transitions and microphase separation, from lamellar up to hexagonal ordering of clusters of opposite magnetization. These rest on complex multibody interactions. We find regimes of stationary patterns, but also dynamical regimes of relentless birth and growth of lumps of magnetization opposite of the surrounding one. Our approach combines Monte Carlo simulations with analytical methods based on dynamical density functional approaches.

Figure 1

Two particles (up spins) coupled to a fluctuating field (surface plot), favoring some value ${\varphi }_0$ of the field. (Inset) Equilibrium average force as a function of the average particle separation, for the Hamiltonian described in the text. One particle is held fixed and the other one is tethered to a strong harmonic trap. (Symbols) Results of the numeric simulation (with incertitude). (Solid line) Analytical force deriving from $U(R)$ in the text. (Parameters) $r=0.01$, ${\varphi }_0=8$, $B=1$, and $\mu =0.05$.

Figure 2

(a) SFIPs in a box with periodic boundary conditions $L=150$, ${\rho }_0=0.05$, $r=0.01$, ${\varphi }_0=8$, and $B=0.07$, yielding a ferromagnetic state. Red (blue) dots indicate particles with up (down) spins. (Inset) Magnetization order parameter as a function of $B$. Light orange (dark purple) symbols correspond to $B$ increasing (decreasing). (b) SFIPs for $B=0.26$ (same other parameters) showing the coexistence of a ferromagnetic liquid and a paramagnetic gas. (c) Phase diagram of SFIPs in terms of total density and coupling strength for $r=0.01$, ${\varphi }_0=8$, and $\mu =5$. (Solid lines) Mean-field predictions for the paramagnetic-ferromagnetic transition (black) and for the binodal curve of the phase separation (green or gray). The corresponding dashed lines are the results of the Monte Carlo simulations. (d) ASFIPs for the same parameters and $\alpha =\gamma =0.1$. (Solid red line) Mean-field prediction for the transition to a patterned phase. [Yellow (light gray) zone] Beginning of segregation. [Orange (gray) zone] Ferromagnetic stripes and macroscopic clusters.

Figure 3

Snapshots of the patterns created by ASFIPs. The parameters are $r=0.01$, ${\varphi }_0=8$. Red (blue) dots indicate particles with up (down) spins. (a) Square phase obtained for symmetric flips ($B=0.26$, ${\rho }_0=0.1$, $\mu =2.5$, $\gamma =\alpha =0.005$, $L=180$). (b) Hexagonal phase of clusters ($B=0.15$, ${\rho }_0=0.4$, $\mu =5.0$, $\alpha =0.02$, $\gamma =\alpha /3$, $L=160$). (c) Striped phase obtained for symmetric flips ($B=0.15$, ${\rho }_0=0.4$, $\mu =5.0$, $\gamma =\alpha =0.02$, $L=160$). (d) The corresponding $\varphi$ field map of (c) and the time average of the fluxes of spin-up particles.