Spatially localized self-assembly driven by electrically charged phase separation

Mean field approach to first and second order phase transitions in ionic liquids

“…2(a)), in the sub-critical case (type I) these solutions are linearly unstable until reaching a fold, σ SN c , whereon they change direction and grow the instability onset of the homogeneous state. However, as in the Ohta-Kawasaki case [29], the large amplitude periodic solutions belonging to a branch portion after the fold (σ < σ SN , see figure) are stable on 2L = λ c = 2π/k c (exhibiting hysteresis) but become Eckhaus unstable on large domains; due to applications our interest here is in domains that are larger than the wavelength of the bifurcating states, i.e., L ≫ π/k c . Consequently, the hysteresis region σ c < σ < σ SN c , between the k c solutions and the uniform state is destroyed on large domains.…”

“…where p 0 = 1/(1+γ) is the uniform cation concentration, ǫ ≪ 1 measures the distance from the instability onset, and following [28,29]…”

“…Notably, although the model equations are gradient the coupling to Poisson equation damps the large scale mode [17,28,29], i.e., s(0) < 0. Since k 2 c , γ > 0, the instability may exist only for…”

“…The Ginzburg-Landau partial differential amplitude equation ( 7) can be used to approximate large scale modulations of the basic pattern cos(k c x) near the (phase transition) onset [28,29] but here for simplicity we assume A to be independent of space. Then…”