Pattern formation and self-assembly driven by competing interactions

Abstract

Colloidal fluids interacting via effective potentials which are attractive at the short range and repulsive at the long range have long been raising considerable attention because such an instance provides a simple mechanism leading to pattern formation even for isotropic interactions. If the competition between attraction and repulsion is strong enough, the gas-liquid phase transition is suppressed, and replaced by the formation of mesophases, i.e., inhomogeneous phases displaying periodic density modulations whose length, although being larger than the particle size, cannot nevertheless be considered macroscopic. We describe a fully numerical implementation of density-functional theory in three dimensions, tailored to periodic phases. The results for the equilibrium phase diagram of the model are compared with those already obtained in previous investigations for the present system as well as for other systems which form mesophases. The phase diagram which we find shows a strong similarity with that of block copolymer melts, in which self-assembly also results from frustration of a macroscopic phase separation. As the inhomogeneous region is swept by increasing the density from the low-density side, one encounters clusters, bars, lamellae, inverted bars, and inverted clusters. Moreover, a bicontinuous gyroid phase consisting of two intertwined percolating networks is predicted in a narrow domain between the bar and lamellar phases.

Combined density functional and Brazovskii theories for systems with spontaneous inhomogeneities

“…25 high T the L phase in simulations is stable at significantly higher density (ρ ≈ 0.39) than in MF (ρ ≈ 0.25). In a more accurate DFT, a phase diagram qualitatively similar to the one shown in panel (b) was obtained [2,3]. In particular, at high T the sequence of phases is D,C,H,L,IH,IC,D, instead of D,L,D obtained in simulations.…”

“…For sufficiently low temperatures, the sequence of structures for increasing volume fraction of the particles is: disordered (D phase), cluster crystal with a cubic symmetry (C phase), hexagonally ordered cylindrical clusters (H phase), parallel layers (L phase), hexagonally ordered cylindrical voids (IH phase), cubic crystal of spherical voids (IC phase), and again the disordered phase. In addition, a gyroid G phase is stable between the H and L phases, and the IG phase is stable between the L and IH phases for some temperature range [1,3,13,20]. Theories of meanfield nature predict the above sequence of phases for the whole range of T < T L , where above T L only the disordered phase (no periodic structure) is stable for the whole range of the volume fractions [1,3,11].…”

“…In addition, a gyroid G phase is stable between the H and L phases, and the IG phase is stable between the L and IH phases for some temperature range [1,3,13,20]. Theories of meanfield nature predict the above sequence of phases for the whole range of T < T L , where above T L only the disordered phase (no periodic structure) is stable for the whole range of the volume fractions [1,3,11]. Simulations, however, show that the C phase looses stability at T = T C , and for T C < T < T H the disordered phase coexists with the hexagonal phase.…”

Mesoscopic theory for systems with competing interactions near a confining wall

“…The sequence of the ordered phases at low T is the same in block copolymer systems [16] and in systems containing particles interacting with the effective SALR potential that is attractive at short-and repulsive at large separations [4,[8][9][10]. This universal behavior follows from the fact that on a qualitative level both systems can be described by the Landau-Brazovskii functional (LB) [4,6].…”

Evolution of infinite populations of immigrants: Micro- and mesoscopic description

“…By (4. 19) we have that q t,1 = k t ; hence, it remains to prove that there exist T > 0 and ϑ > ϑ 0 such that the convergence stated in (2.42) does hold. For ϑ 0 as just discussed, let δ(ϑ 0 ) and τ (ϑ 0 ) be as in (3.63) and (3.64), respectively.…”