Active Cahn-Hilliard theory for non-equilibrium phase separation: Quantitative macroscopic predictions and a microscopic derivation

Phase-separating active systems can display phenomenology that is impossible in equilibrium. The binodal densities are not solely determined by a bulk (effective) free energy, but also affected by gradient terms, while capillary waves and Ostwald processes are determined by three distinct interfacial tensions. These and related phenomena were so far explained at continuum level using a top-down minimal theory (Active Model B+). This theory, by Taylor-expanding in the scalar order parameter (or density), effectively assumes that phase separation is weak, which is not true across most of the phase diagram. Here we develop a quantitative account of active phase separation, by introducing the active counterpart of Cahn–Hilliard theory, constructing the density current from all possible terms with up to four spatial derivatives without Taylor-expanding in the density. From this O(∇4) theory, we show how to compute binodals and interfacial tensions for arbitrary choices of the five density-dependent ‘coefficient functions’ that specify the theory (replacing the four constant coefficients of Active Model B+). We further show how a well-studied particle model for active phase separation, involving thermal quorum-sensing active particles (tQSAPs), yields a fully specified example of the O(∇4) theory upon coarse-graining. We find that to coarse-grain consistently at O(∇4) requires a novel procedure, based on multiple-scale analysis, to systematically eliminate fast-evolving orientational moments. Using this, we calculate from microscopic physics all five coefficient functions of the active Cahn–Hilliard theory for tQSAPs. We identify contributions that were missed in previous continuum theories, and show how neglecting them becomes justified only in the limit of large quorum-sensing range parameter γ. Comparison with particle-based simulations of tQSAPs shows that, both qualitatively and quantitatively, our O(∇4) theory improves on previous continuum models, predicting not only that binodal densities depend on γ, but also—unexpectedly— that phase boundaries become re-entrant at small thermal diffusivity. A further surprising prediction of the wider theory is that in certain regimes of parameter space, bulk phase separation exhibits an instability in which, at O(∇4) level, the interfacial width shrinks to zero. We link this to the runaway effects of active pumping terms that cause positive feedback of a sharpening interface.