Interface geometry of binary mixtures on curved substrates

Motivated by recent experimental work on multicomponent lipid membranes supported by colloidal scaffolds, we report an exhaustive theoretical investigation of the equilibrium configurations of binary mixtures on curved substrates. Starting from the Jülicher-Lipowsky generalization of the Canham-Helfrich free energy to multicomponent membranes, we derive a number of exact relations governing the structure of an interface separating two lipid phases on arbitrarily shaped substrates and its stability. We then restrict our analysis to four classes of surfaces of both applied and conceptual interest: the sphere, axisymmetric surfaces, minimal surfaces, and developable surfaces. For each class we investigate how the structure of the geometry and topology of the interface is affected by the shape of the substrate and we make various testable predictions. Our work sheds light on the subtle interaction mechanism between membrane shape and its chemical composition and provides a solid framework for interpreting results from experiments on supported lipid bilayers.

Figure 1

The surface $\mathrm{\Sigma }$ is partitioned into (multiple) connected domains ${\mathrm{\Sigma }}_{+}$ and ${\mathrm{\Sigma }}_{-}$, separated by the linear interface $\mathrm{\Gamma }$. The tangent and normal two-vectors of the curve are $T^i$ and $N^i$, which together form a local basis for the tangent space of $\mathrm{\Sigma }$. We show their three-dimensional representation $\mathit{T}$ and $\mathit{N}$, along with the normal to the surface $\mathit{n}$. The three orthonormal vectors $\{\mathit{T},\mathit{N},\mathit{n}\}$ form the Darboux frame (or material frame) of $\mathrm{\Gamma }$.

Figure 2

Constant geodesic curvature curves on a generic surface. The black line is an unstable closed geodesic; its length can be easily shortened by a shift in any direction. Conversely, the blue line is a stable geodesic, lying along a region of negative $K$ and whose length can only be increased by fluctuations. The red curve is a closed CGC. Since this surface is axisymmetric, meridians and parallels are also principal directions. This dumbbell-shaped surface was taken from [14] and is also used to construct the phase diagram of Fig. 6.

Figure 3

Local effects on the shape of $\mathrm{\Gamma }$ of the various terms arising in Eq. (3). The four columns correspond to four different quadric surfaces, parametrized by Eq. (9), with various values of ${\kappa }_1$ and ${\kappa }_2$, as shown at the bottom. The first column correspond to a flat plane, the second to a parabolic cylinder, the third to a symmetric paraboloid, and the fourth to a hyperboloid. Each row corresponds to solutions of Eq. (3) where only one of the three terms on the right-hand side is different from zero, shown on the right. Different curve colors correspond to different values of the coupling constants. Pure blue lines always correspond to geodesic interfaces. Note that the legend on the right refers on the modulus of the couplings, while in the drawing both signs are considered. The scale bar for $L$, used in Eq. (5), is shown in the top left surface. All curves intersect at the point $x=y=0$, at the center of the surface. Notice how ${\eta }_k$ does not influence the interface on hyperboloids (it is an almost-minimal surface) and how ${\eta }_{\overline{k}}$ does not affect $\mathrm{\Gamma }$ on a cylinder, being a developable surface.

Figure 4

Parametrization of an axisymmetric surface. On the left, we show the radial profile $\left\{r\right(t),z(t\left)\right\}$ as a function of $t$, the arc-length parameter of the profile. The full surface is obtained by rotations along the $z$ axis parametrized by the angle $\varphi$, as in Eq. (19). We define $\psi =\psi \left(t\right)$ to be the angle between ${\mathit{k}}_1$, the meridian principal direction, and the horizontal plane. On the right, we show how the curve $\mathrm{\Gamma }$ on $\mathrm{\Sigma }$ is parametrized in its own arc length $s$; its unit tangent vector $\mathit{T}$ makes an angle $\theta =\theta \left(s\right)$ with ${\mathit{k}}_1$. Notice that the orientation of the principal directions is not fixed a priori; we choose it to match the one of $\{\mathit{T},\mathit{N}\}$.

Figure 5

Solutions of Eq. (24) on a corrugated cylinder. (a) Interfaces with varying $\mathrm{\Delta }\lambda$ are CGC lines, with shapes which clearly resembles circles. (b) Curves with varying ${\eta }_k$. Interfaces are closed but the shape departs significantly from a circle. (c) Varying the coupling with the Gaussian curvature. Closed curves are possible only for high values of ${\eta }_{\overline{k}}$ and do not encompass multiple corrugations. In all three panels the pure blue vertical line has zero coupling and thus corresponds to a vertical geodesic. The scale bar on the top left shows the value chosen for $L$ in Eq. (5).

Figure 6

Phase diagrams of minimal configurations for the dumbbell-shaped particle of Fig. 2, for varying $\phi$. (a) Effect of ${\eta }_k$ while keeping ${\eta }_{\overline{k}}=0$. (b) Effect of ${\eta }_{\overline{k}}$ while keeping ${\eta }_k=0$. In both panels, different colors correspond to different minimal energy configurations: Type I (light green) consist of two domains and one interface; types ${\text{II}}_{\pm }$ (light blue) and type ${\text{III}}_{\pm }$ (light red) have two interfaces and three domains. In the insets, hard and soft phases are respectively depicted in green and magenta. All interfaces are CGC parallels. We set $L=\sqrt{A_{\mathrm{\Sigma }}}$.

Figure 7

Interfaces on the Schwarz $P$ surface. (a) We highlight in yellow the boundary of the fundamental patch. By properly gluing 48 copies of this patch one can construct a full unit cell, here drawn by aid of Surface Evolver [46]. (b) The interface (30) is actually solved for a curve in $\mathbb{C}$, and solutions are shown as blue-green curves, with blue corresponding to geodesics. The yellow lines highlight the region $\mathcal{D}$ [defined in Eq. (C22)] which is mapped onto the fundamental patch in (a). The gray contour shades in the background show the value of the Gaussian curvature $K$ [see Eq. (C14)] as a function of $z$. The value of the curvature is rescaled so that the range shown is $[0,-1]$. Note that zeros of $K$ correspond to poles of $f\left(z\right)$. All solutions have identical initial conditions: They start off from the same point on the boundary of the patch and have initial tangent vector $T^i\left(0\right)$ such that $\mathit{T}\left(0\right)$ is pointing horizontally in the embedded surface, as is clear from (a). Note that, in principle, all complex curves extend without problem outside $\mathcal{D}$, but cannot be mapped onto the surface. (c) Evaluation of Eq. (31) along the solutions displayed in (a) and (b). Each curve parameter $s$ is rescaled so that it spans the interval [0,1]. The vertical axis uses arbitrary units, since only the sign of the stability factor contains relevant information. By increasing the modulus of ${\eta }_{\overline{k}}$ we see that interfaces become progressively more unstable; namely, only ${\eta }_{\overline{k}}=0$ (geodesic) and ${\eta }_{\overline{k}}=-0.3$ seem to allow locally stable interfaces.

Figure 8

Layers $z_{+,0}$ and $z_{-,1}$ [see Eq. (C25)] of the nodal approximation of the $P$ surface form together a planar square lattice of unit cells, four of which are displayed. In the upper layer we show some solutions of Eq. (28). The color coding of the solid lines is the same as that of Fig. 7, while the dashed lines have ${\eta }_{\overline{k}}$ rescaled by a factor $1/10$. We evaluated Eq. (29) for the solutions and found that, for the solid curves, only the geodesic and the ${\eta }_{\overline{k}}=-0.3$ are stable, similarly to what was found in Fig. 7 for the exact solution. We postulate that for the latter value of ${\eta }_{\overline{k}}$ it is possible to have the formation of stable, finite-size soft domains on the $P$ surface. The stability of the dashed lines is much more $s$ dependent. The first three curves seems to be stable: They correspond to noncontractible closed interfaces encompassing several unit cells.

Figure 9

Analogy between mean curvature and magnetic field. The top panel shows a developable surface with translational invariance and sinusoidal profile height $z\left(x\right)=L_{z}sinx/L_x$. The red curve is a generic interface with $\mathrm{\Delta }\lambda L_x=\sigma /2$ and ${\eta }_k=1/10$, obtained with initial conditions $x_0=0$ and ${\theta }_0=\pi /2$. This closed curve is analogous to the planar trajectory of a charged particle in an $x$-dependent axial magnetic field $\mathit{B}=B_z\widehat{\mathit{z}}$, which oscillates between the values $\mathrm{\Delta }\lambda /\sigma$ and $\mathrm{\Delta }\lambda /\sigma +{\eta }_k{L}_x^2/4L_z^4$ with spatial periodicity of $\pi L_x$. The tangent $\mathit{T}$ is mapped to the planar velocity $\mathit{v}$ and the normal $\mathit{N}$ is mapped to the in-plane normal $\tilde{\mathit{N}}$.

Figure 10

Using the same sinusoidal surface geometry of Fig. 9, we display closed interfaces which minimize Eq. (35) for varying ${\eta }_k$. Each interface is chosen such that the area of the enclosed domain is equal to $L_x^2$ [and we set $L=L_x$ in Eq. (5)]. The left panel shows ${\mathrm{\Sigma }}_{-}$ domains, i.e., soft domains surrounded by a LO background. The right panel shows ${\mathrm{\Sigma }}_{+}$ domains, i.e., hard domains surrounded by a LD background. Gray tones in the background follow the same color scheme of Fig. 9, indicating the magnitude of $\mathit{B}$. While for each ${\eta }_k$ there are multiple solutions which have the same area, only the ones minimizing Eq. (35) are displayed here.