Phase separation with non-local interactions

Abstract

Phase separation in complex systems is a ubiquitous phenomenon. While simple theories predict coarsening until only macroscopically large phases remain, concrete models often exhibit patterns with finite length scales. To unify such models, we here propose a general field-theoretic model that combines phase separation with non-local interactions. Our analysis reveals that long-range interactions generally suppress coarsening, whereas systems with non-local short-range interactions additionally exhibit a continuous phase transition to patterned phases. Only the latter system allows for the coexistence of homogeneous and patterned phases, which we explain by mapping to the conserved Swift-Hohenberg model. Taken together, our generic model reveals an underlying framework that describes similar phenomena observed in many complex phase-separating systems.

Figures (2)

• Figure 1: Short-range interactions induce transition from homogeneous phase coexistence to regular patterns. (A) Dispersion relation $\omega(q)$ as a function of wave number $q$ for $\xi<\xi_c \equiv h\sqrt{\kappa/\epsilon}$ (dashed) and $\xi>\xi_c$ (solid) for various $f"$ putting the system outside, at, and inside the spinodal boundary (colors). The dominant wave number $q_*$ (stars, black for $\xi>\xi_c$ and red otherwise) is finite inside the spinodal region; at the spinodal boundary (black lines) $q_*=q_\mathrm{s}$ continuously increases from $q_\mathrm{s}=0$ to $q_\mathrm{s}\neq 0$ as $\xi$ crosses $\xi_c$. (B) Dominant wave number $q_*$ (Appendix \ref{['app:qstar']}) as a function of $\xi/\xi_c$ and $f"$ for fixed $\lambda/h=1.4$. Along the spinodal (solid black line) $q_\mathrm{s}$ transitions continuously from $q_\mathrm{s}=0$ to $q_\mathrm{s}>0$ at $\xi=\xi_c$. Macroscopic modes with $q\to 0$ are unstable below the dashed line, corresponding to the yellow lines in (A). (C) $q_*$ as a function of overall density $\bar{\phi}$ and interaction strength $\chi$ for $\xi/\xi_c = 1.6$. (A--C) Model parameters are $\kappa=\chi$, $\lambda/h=1.4$, with $f$ given by Eq. \ref{['eq:FloryF']}.
• Figure 2: Patterned phase can coexist with homogeneous phase for short-range interactions. Density fields $\phi(\boldsymbol{x})$ from 2D numerical integration (Appendix \ref{['app:simulations']}) of Eqs. \ref{['eq:freeEnergy']}--\ref{['eq:modelB']} with short-range interactions given by Eq. \ref{['eq:kerAn']} for $\xi>\xi_c$ (first three panels) and $\xi<\xi_c$ (lower right panel). (B) Density correlations in Fourier space for different wave numbers $q$ for each of the three snapshots in (A) with $\xi>\xi_c$. (C) Coexisting densities (red symbols) as function of $\bar{\phi}$ and $\chi$ are approximated by effective binodal (black line, without non-local interactions), indicating a first order phase transition in two-dimensions. Inset shows that changing $\bar{\phi}$ at fixed $\chi=0.1$ does not affect coexisting densities. Symbols indicate parameter values in panel (A). (A--C) Parameters are as in Fig. \ref{['fig:qDyn_f0']}, except for the lower right snapshot in (A) which has $\xi/\xi_c=0.9$.