Adhesion of cylindrical colloids to the surface of a membrane

We study the system of cylindrical colloids adhering to an originally flat fluid membrane, on the basis of a full treatment of the Helfrich model. Our approach allows for numerical calculation of the free energy in both shallow- and deep-wrapping conformations. We show that the free energy of two cylinders adhering to the same side of a membrane has two branches corresponding to shallow and deep wrapping and that the system of two cylinders adhering to opposite sides of a membrane can undergo a first-order phase transition between two membrane-mediated attractive states.

Figure 1

(Color online) Sketch of the cross section of three systems considered. The curves represent a cross-section view of the membrane profile. Plot (a) shows a single cylinder adhered to a membrane, where the wrapping angle $\alpha$ and tangent angle $\psi$ are also defined. Plots (b) and (c) show two cylinders adhered to a membrane, where the surface-to-surface distance $D$, wrapping angle $\gamma$, and crossing angle $\theta$ are also specified.

Figure 2

(Color online) Adhesion of a single cylinder to a membrane. The phase diagram for the system of a cylinder adhering to a membrane as a function of the reduced surface tension and adhesion energy is shown in plot (a). The illustrations in (b) and (c) are sketches for a typical partial wrapping state and typical closure state, respectively.

Figure 3

(Color online) Adhesion of two cylinders to the same side of a membrane. The intercylinder free energy difference, $\Delta {\tilde{F}}_I^{\prime }$ in Eq. (20), as a function of the surface-to-surface distance between the cylinders, $\tilde{D}$, is shown in plots (a)–(c) for $\tilde{\sigma }=1.0$, 0.1, and 0.01. Two branches of the free energy are shown using solid and long-dashed curves. Squares represent the results from a perturbation theory in Ref. [20]. Circles represent the locations where two branches cross, hence the locations of first-order transitions. Sketches (d)–(f) are three typical configurations of the system.

Figure 4

Phase diagrams of the shallow-to-deep phase transition for $\tilde{w}=0.8$, $\tilde{w}=0.9$, and $\tilde{w}=1.0$ as a function of surface-to-surface distance $\tilde{D}$ and reduced tension $\tilde{\sigma }$. The area below a solid transition line corresponds to the shallow configuration, and the area above a transition line corresponds to deep wrapping. First-order transition solid lines terminate at a second-order point specified by a square on the plots.

Figure 5

Dependence of the intercylinder free energy [in Eq. (20)] on the wrapping angle $\gamma$, when the distance between the adhered cylinders is fixed. (a) The free energy is plotted for $\tilde{\sigma }=0.1$, $\tilde{w}=1.069$, and $\tilde{D}=1.8,3.0,4.1$ from top to bottom. (b) The free energy is plotted for $\tilde{\sigma }=0.1$, $\tilde{w}=1.1$, and $\tilde{D}=1.0,1.84,3.0$ from top to bottom. The open circles denote double energy minima, and solid circles single energy minimum. The shaded area in (b) corresponds to the closure state.

Figure 6

(Color online) Adhesion of two parallel cylinders to opposite sides of a membrane. Plots (a), (b), and (c) show the free energy dependence [Eq. (23) after minimization] on the surface-to-surface distance $\tilde{D}$ between the cylinders for a several given sets of $\tilde{w}$ and $\tilde{\sigma }$. The location of a terminal minimum near $\tilde{D}=-4$ is also shown by a circle. Overlaying on the curves are squares, which represent the results calculated from Ref. [20], agreeing well with our results in weak adhesion. The configuration in (d) is a sketch of a typical shallow adsorption profile and the configuration in (e) is a sketch for a typical wrapping state where the positions of the cylinders are exchanged from those in (d).

Figure 7

Phase diagram of the system where two cylinders adhere to opposite sides of a membrane. The dashed line represents the same second-order adsorption transition shown in Fig. 2A and the solid curve represents a first-order swapping transition, where the system undergoes a transition from a state with the membrane weakly wrapping two parallel cylinders [plot (d) in Fig. 6] to a state with the membrane fully wrapping both cylinders (the positions of cylinders are exchanged in the horizontal direction [plot (e) in Fig. 6]).