Intrinsic Contact Angle of Aqueous Phases at Membranes and Vesicles

Biomimetic membranes exposed to several aqueous phases are studied theoretically. Specific examples are provided by lipid vesicles and phase separating polymer solutions. The theory reveals the existence of an intrinsic contact angle, ${\theta }_{\mathrm{in}}$, between the membranes and the aqueous phases. An explicit relation is derived by which this intrinsic angle can be determined from the effective contact angles as measured by optical microscopy. A detailed comparison between theory and experiment provides strong evidence that the intrinsic contact angle ${\theta }_{\mathrm{in}}$ represents a material parameter, in contrast to the effective contact angles.

Figure 1

(a) Cross-section of a vesicle enclosing one $\alpha$ (top) and one $\beta$ (bottom) droplet suspended in the exterior solution $\gamma$. When viewed with optical resolution, the vesicle shape exhibits a sharp kink along the contact line ($\odot$) and can be characterized by three effective contact angles ${\theta }_{\alpha }$, ${\theta }_{\beta }$, and ${\theta }_{\gamma }$ with ${\theta }_{\alpha }+{\theta }_{\beta }+{\theta }_{\gamma }=2\pi$. These contact angles are related to the three tensions ${\Sigma }_{\alpha \beta }$, ${\Sigma }_{\alpha \gamma }$, and ${\Sigma }_{\beta \gamma }$ via the force balance along the contact line; and (b) Enlarged view close to the contact line: Intrinsic contact angle ${\theta }_{\mathrm{in}}$ between the two planes that are tangential to the $\alpha \beta$ interface and to the smoothly curved vesicle membrane, respectively.

Figure 2

Axisymmetric vesicle containing one $\alpha$ and one $\beta$ droplet. The coordinate along the symmetry axis is denoted by $z$, the distance from this axis by $\rho$, the arc length by $s$, and the local tilt angle by $\psi$. All of these variables depend on the contour parameter $t$ with $t_0=0\le t\le t_2$. The contact line is located at $t=t_1$, the gray area corresponds to the spherical cap with volume $\Delta V_{\beta }$ and tangent angle $\theta$.

Figure 3

(a) Effective and (b) intrinsic contact angles in degrees as a function of the reduced volume $v$ for volume ratio $V_{\alpha }/V_{\beta }=0.33\pm 0.06$ and essentially fixed polymer composition of the $\alpha$ and $\beta$ droplets. In (a), the effective contact angles ${\theta }_{\alpha }$, ${\theta }_{\beta }$, and ${\theta }_{\gamma }$ as measured for 63 different vesicles are shown in different colors (scattered symbols) and compared with the theoretical calculations (full lines); and (c),(d) Comparison between calculated vesicle shapes (left) and phase contrast vesicle images (right) for two different values of reduced volume $v$. The scale bars are $20\mu \mathrm{m}$.