Phase transitions and phase coexistence: equilibrium systems versus externally driven or active systems - Some perspectives

Figures & data

Figure 1. Snapshot pictures of the time development of a system with $N=4000$ particles in $L\times L$ box with $L=40$ and $v_0=0.01$ at 50 (a) and 3000 (b) steps. Adapted from Czirok et al.^[4] © IOP publishing. Reproduced with permission. All rights reserved

Figure 2. Simulation results for the phase diagram of active Brownian particles in $d=2$ dimensions, plotting the state of the system in the plane of variables packing fraction $\varphi$ and Péclet number $Pe$. Within the region confined by the binodal (dashed and solid line), two-phase coexistence occurs between a gas-like phase and a liquid like-phase, as indicated. Open circles show the coexisting packing fractions of the gas $({\varphi }_g)$ and liquid $({\varphi }_{\mathrm{\ell }})$, respectively, and also the packing fractions at the rectilinear diameter ${\varphi }_d=({\varphi }_g+{\varphi }_{\mathrm{\ell }})/2$ are indicated (red circles). The location of the critical point (diamond) was estimated by a subsystem finite size scaling analysis. Reprinted figure with permission from J.T. Siebert et al., physical review E 98, 030601 (2018) ^[16] (https://doi.org/10.1103/PhysRevE.98.030601). Copyright 2018 by the American Physical Society

Figure 3. (a) Snapshot configuration of the two-dimensional active Brownian particle model, for a large volume fraction $\eta =0.64$, in a square $L\times L$ box with periodic boundary conditions and $D_r=0.2$, $v_0=5.0$. (b) Enlarged portion of the high density cluster. Colors encode the direction of the self-propulsion. (c,d) Windows from the bulk where the velocities of the particles are shown by blue arrows, showing aligned and vortex domains, respectively. So one can see that there is little correlation between these directions for neighboring particles, even if strong correlations between their velocities (shown by arrows) occur. Reprinted figure with permission from L. Caprini et al., physical review letters 124, 078001 (2020) ^[18] (https://doi.org/10.1103/PhysRevLett.124.078001). Copyright 2020 by the American Physical Society

Figure 4. The swim pressure of a three-dimensional system of active Brownian particles plotted vs packing fraction $\varphi$, for various Péclet numbers: Pe = 9.8 (purple), 29.5 (olive), 44.3 (green), 59.0 (blue), and 295.0 (red). The values are normalized by the ideal (bulk) pressure of individual active Brownian particles as at the same Péclet number. From Winkler et al. ^[86] – Published by the royal society of chemistry

Figure 5. Phase diagram of the AO model for ${\sigma }_p/{\sigma }_c=0.8$ in the plane of variables ${\eta }_c$ and ${\eta }_p$, as well as using ${\eta }_c$ and ${\eta }_p^r$ (insert). Continuous curves show the so-called free volume approximation (which implies a mean-field-type critical behavior) while symbols show Monte Carlo data (which imply a critical behavior compatible with the Ising model universality class). The square indicates the location of the critical point. Reprinted from R.L.C. Vink and J. Horbach, journal of chemical physics, 121, 3253, (2004), ^[96] with the permission of AIP Publishing

Figure 6. (a) Snapshot picture of the active polymer-colloid mixture at the state point ${\eta }_c=0.15$, ${\eta }_p=0.20$ with $f_A=10$, for a box with linear dimensions $L_x\times L_y=12$, $L_z=48$. Colloids are shown in yellow, polymers in black. (b) Profiles of ${\eta }_p$ and ${\eta }_c$ in z-direction. (c) Corresponding profiles for the various components of T(z) as obtained from the equipartition theorem. Reprinted from B. Trefz et al., journal of chemical physics, 144, 144902, (2016), ^[10] with the permission of AIP publishing

Figure 7. (a) Schematic representation of the simulation box used for the cumulant analysis. Simulations are done at medium packing fraction of particles with an edge length ratio of 3:1 and periodic boundary conditions. This results in a two-phase coexistence with a slab geometry of the liquid domain. The slab is always aligned with the short axis. Two $\mathrm{\ell }\times \mathrm{\ell }$ subboxes are then placed in the liquid domain, such that the x-coordinates of their center of mass coincides with the x-coordinate of the liquid domain. The other two subboxes are shifted by $3\mathrm{\ell }$ along the x-axis, so they are centered in the vapor region. (b) Cumulant ratio $Q_{\mathrm{\ell }}=⟨m^2{⟩}_{\mathrm{\ell }}^2/⟨m^4{⟩}_{\mathrm{\ell }}$ for several choices of $\mathrm{\ell }$, namely $\mathrm{\ell }=8,10,12$ and 15 lattice spacings, for the Ising/lattice gas model in $d=2$ dimensions, plotted vs. temperature $T$ (in units of the exchange constant $J/k_B$). (c) Cumulants for the two-dimensional active Brownian particle model plotted vs. the Péclet numbers for four choices of $\mathrm{\ell }$ (in units of $\sigma$) as indicated. A crossing of $Q_{\mathrm{\ell }}(Pe)$ can be seen for roughly $Pe=40\pm 2$. The dashed lines are guides to the eye only. Reprinted figure with permission from J.T. Siebert et al., physical review E 98, 030601 (2018) ^[16] (https://doi.org/10.1103/PhysRevE.98.030601). Copyright 2018 by the American Physical Society

Siebert, J. T.; Dittrich, F.; Schmid, F.; Binder, K.; Speck, T.; Virnau, P. Critical Behavior of Active Brownian Particles. Phys. Rev E. 2018, 98(3), 030601. DOI: https://doi.org/10.1103/PhysRevE.98.030601.

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Caprini, L.; Marconi, U. M. B.; Puglisi, A. Spontaneous Velocity Alignment in Motility-Induced Phase Separation. Phys. Rev. Lett. 2020, 124(7), 078001. DOI: https://doi.org/10.1103/PhysRevLett.124.078001.

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Vink, R. L. C.; Horbach, J. Grand Canonical Monte Carlo Simulation of a Model Colloid–polymer Mixture: Coexistence Line, Critical Behavior, and Interfacial Tension. J. Chem. Phys. 2004, 121(7), 3253. DOI: https://doi.org/10.1063/1.1773771.

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Trefz, B.; Das, S. K.; Egorov, S. A.; Virnau, P.; Binder, K. Activity Mediated Phase Separation: Can We Understand Phase Behavior of the Nonequilibrium Problem from an Equilibrium Approach? J. Chem. Phys. 2016, 144(14), 144902. DOI: https://doi.org/10.1063/1.4945365.

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