Phase Separation on Bicontinuous Cubic Membranes: Symmetry Breaking, Reentrant, and Domain Faceting

We study the phase separation of binary lipid mixtures that form bicontinuous cubic phases. The competition between the nonuniform Gaussian membrane curvature and line tension leads to a very rich phase diagram, where we observe symmetry breaking of the membrane morphologies and reentrant phenomena due to the formation of bridges between segregated domains. Upon increasing the line tension contribution, we also find faceting of lipid domains that we explain using a simple argument based on the symmetry of the underlying surface and topology.

Figure 1

Visualizing the $P$ surface. Left panel: normalized Gaussian curvature field $(G(x)/G_{\min })$ [note that $G(x)\le 0$] on a single patch with the zero-curvature point $p$ at its center. Right panel: curvature induced formation of $A$-lipid domains (in green) in $k$ patches among the eight available denoted by $(\genfrac{}{}{0ex}{}{8}{k})$ (for $f_A=0.07$). For the sake of illustration we show them for $k=1$ (top) and $k=8$ (bottom).

Figure 2

Phase diagrams of the number of patches $k$ occupied by lipid $A$ domains among the eight available as a function of the area fraction $f_A$ and the ratio $J\ell /|\delta \kappa |$. $\ell$ is the lattice spacing of the cubic unit cell. For clarity, we show the phase diagrams (a) excluding (filled symbols) and (b) including (open symbols) the possibilities of bridge formations between lipid $A$ domains. In (c) are shown configurations with increasing energy from top to bottom for $f_A=0.75$ and $J\ell /|\delta \kappa |=0.5$. The boxed configuration with seven connected domains corresponds to the most stable phase.

Figure 3

Domain faceting. Shape of a lipid $A$ domain in the neighborhood of the zero-curvature point $p$ of a single hexagonal patch: (a) in the absence of line tension and (b) with high line tension ($J\ell /|\delta \kappa |>1$). Note that only half of a patch is represented. (c) Geodesic curvature as a function of the curvilinear coordinate $l$ in the absence of line tension (dashed) and with high line tension (solid).