Derivation and analysis of a phase field crystal model for a mixture of active and passive particles

Abstract

We discuss an active phase field crystal (PFC) model that describes a mixture of active and passive particles. First, a microscopic derivation from dynamical density functional theory (DDFT) is presented that includes a systematic treatment of the relevant orientational degrees of freedom. Of particular interest is the construction of the nonlinear and coupling terms. This allows for interesting insights into the microscopic justification of phenomenological constructions used in PFC models for active particles and mixtures, the approximations required for obtaining them, and possible generalizations. Second, the derived model is investigated using linear stability analysis and nonlinear methods. It is found that the model allows for a rich nonlinear behavior with states ranging from steady periodic and localized states to various time-periodic states. The latter include standing, traveling, and modulated waves corresponding to spatially periodic and localized traveling, wiggling, and alternating peak patterns and their combinations.

Non-reciprocity induces resonances in a two-field Cahn–Hilliard model

“…Even though the conservation laws play an important role in the instabilities, this is similar to many other widely studied systems, e.g. RD models [37][38][39][40] and active phase-field crystal models [41][42][43]. Remarkably, the non-reciprocal interactions may also result in the transformation of a stationary large-scale instability (CH instability) typical for phase separation [44] into a stationary small-scale Turing-like instability with mass conservation (conserved-Turing instability).…”

“…The pitchfork bifurcation corresponds in the original model to a kind of drift-pitchfork bifurcation. Related localized patterns with superposed oscillation and drift are also found in a particular non-reciprocal CH model [36] and in active phase-field-crystal models [42,43]. Note that, due to the resonance, the described complex scenario differs from the basic codimension-one scenario where a steady state starts to drift at a drift-pitchfork [67] or drift-transcritical [14] bifurcation (all called travelling bifurcation in [58]).…”

“…Related localized patterns with a superposed oscillation on a larger scale are also found in a non-reciprocal CH model [36]. In active phase-field crystal models, related states are described as alternating localized or alternating periodic states [43].…”

“…In the original model, this Hopf bifurcation results in a state corresponding to a standing wave with a superposed travelling modulated wave (figure 3 h – j ). To our knowledge, such states have not yet been systematically studied in systems with conservation laws although some states of similar complexity are described for an active phase-field-crystal model for a mixture of active and passive particles [43]. The instability limits of both symmetric and asymmetric stationary solutions merge at the double zero singularity located at φ=34π,μ=14.…”

Perspective: New directions in dynamical density functional theory

“…Recent studies of PFC models have focused on active matter with [212][213][214] and without [215][216][217][218][219][220][221] inertia, amorphous solids [222], bifurcation diagrams [219,220,[223][224][225], colored noise [226], cubic terms [227], crystals [228], dislocation lines [229], electromigration [230], grain boundaries [231,232], mixtures [200,220,[233][234][235], nucleation [236], solidification [237,238], and stress tensors [239].…”