Equilibrium Phase Behavior of a Continuous-Space Microphase Former

Periodic microphases universally emerge in systems for which short-range interparticle attraction is frustrated by long-range repulsion. The morphological richness of these phases makes them desirable material targets, but our relatively coarse understanding of even simple models hinders controlling their assembly. We report here the solution of the equilibrium phase behavior of a microscopic microphase former through specialized Monte Carlo simulations. The results for cluster crystal, cylindrical, double gyroid, and lamellar ordering qualitatively agree with a Landau-type free energy description and reveal the nontrivial interplay between cluster, gel, and microphase formation.

Figure 1

Two-step TI for the lamellar phase at $T=0.3$ and $\rho =0.4$. Projections on the $xz$ plane of the coarse-grained number density $\rho (\mathbf{r})$ and external field profiles $\mathcal{F}(\mathbf{r})$ for (a) $\rho =0$ with field $\mathcal{F}(\mathbf{r})={\mathcal{F}}_0\cos (2\pi z/\ell )$, where ${\mathcal{F}}_0=2$ and $\ell =5.25$, (b) $\rho =0.4$ with a field, and (c) $\rho =0.4$ without a field. Summing the TI results for the equation of state from (a) to (b) with those from the alchemical transformation in $\delta$ from (b) to (c) [46]—in (d) and (e), respectively—gives the free energy constrained to a given area density ${\varrho }_{\ell }=\rho \ell$ [46]. (f) From the minimum of a quadratic fit (dashed lines) to $f_{\mathrm{c}}$ (points), we obtain the equilibrium thermodynamic $f$ at $\ell ={\varrho }_{\ell }/\rho =5.05$ (large dot).

Figure 2

Summary (a) $T-\rho$ and (b) $p-T$ phase diagrams indicating the first-order (empty symbols) and the order-disorder transition (solid circle) as well as the cmc (solid triangles) and percolation (solid diamonds) lines. Errors are comparable to the symbol sizes, striped areas correspond to coexistence regimes, and lines are guides for the eye. Clausius-Clapeyron results for the slope of the coexistence line (dashed lines) validate the numerical results in (b). Sample average density profiles for the different phases are given in (a). The inset provides percolation and cmc lines to higher $T$. Three triple points can be identified (solid squares): (i) fluid-fcc-cluster-cylindrical coexistence at $T=0.410(5)$ and $p=0.051(1)$, (ii) fluid-cylindrical-lamellar coexistence at $T=0.491(4)$ and $p=0.285(2)$, and (iii) cylindrical-double gyroid-lamellar coexistence at $T=0.37(1)$ and $p=0.15(1)$.

Figure 3

(a) The equilibrium occupancy ${\varrho }_{\ell }$ of lamellae depends only weakly on temperature in this regime, where $T=0.3$ (dots), $T=0.35$ (squares), and $T=0.4$ (triangles). (b) Decay of $A(T)$ at $\rho =0.35$ (dashed line) fitted to an Ising critical form $A(T)\sim |1-T/0.54{|}^{{\beta }_{\mathrm{c}}}$ (solid line) with ${\beta }_{\mathrm{c}}=0.3264$ up to $T=0.545$. We estimate $T_{\mathrm{ODT}}=0.535(5)$. (c) The average cluster size at the cmc, ${\overline{n}}_{\mathrm{cmc}}$, decreases as $T$ increases (dots). The line is a guide for the eye. (d) Equilibrium fcc-cluster $n_{\mathcal{C}}$ (solid symbols) and average fluid cluster size $\overline{n}$ (empty symbols), at $T=0.35$ (squares) and 0.40 (triangles). Because $\overline{n}>n_{\mathcal{C}}$ at the cluster fluid-fcc-cluster crystal transition, most clusters in the fluid must shrink for crystallization to proceed.

Figure 4

Sample configuration slices after (a) 0, (b) 200, (c) 1600, and (d) 3200 MC sweeps of an initially equilibrated simulation with $N=2000$ at $T=0.45$ and $\rho =0.2$. The distinction between particles that remain within $0.5\sigma$ of their initial conditions (red) and those that have moved beyond it (blue) illustrates the network dynamics. Surface particles decorrelate more rapidly than core particles, and network nodes (yellow circles) reorganize more slowly than network edges.