Surface charge relaxation and the pearling instability of charged surfactant tubes

The pearling instability of bilayer surfactant tubes was recently observed during the collapse of fluid monolayers of binary mixtures of Dimyristoylphosphocholine (DMPC): Palmitoyloleoylphosphoglycerol (POPG) and Dipalmitoylphosphocholine (DPPC):POPG surfactants. It can be explained by a Rayleigh-like instability under the action of the bilayer surface tension. The magnitude of surface tension is dictated by the electrostatic interaction between charged surfactants. Relaxation of charged molecules is proposed here as an additional mechanism driving the instability. We find the functional dependence of the electrostatic surface tension and relaxation energies on the screening length ${\kappa }^{-1}$ explicitly. Relaxation lowers the cost of bending a tube into pearls making the cylindrical tube even more unstable. It is known that for the weak screening case in which the tube radius is smaller than the screening length of the solution, this effect is important. However, for the case of strong screening it is negligible. For the experiments mentioned, the situation is marginal. In this case, we show that the effect of relaxation remains small. It contributes about 20% to the total electrostatic energy.

Figure 1

Two fluorescence micrographs taken within a few seconds of each other, showing a surfactant tube undergoing a pearling instability. The top end of the tube is attached to the 7DMPC:3POPG monolayer from which it grew. The scale bar corresponds to $20\mu \mathrm{m}$.

Figure 2

(Color online) A cylindrical tube is deformed radially with wave vector $k$.

Figure 3

The absolute value of the ratio between the electrostatic energy gain due to the relaxation of charged molecules to the total electrostatic energy when a tube deforms into a pearling structure as a function of the wave vector of deformation, $k{R}_0$ for $\kappa R_0=1$. Four different values of $\kappa R_0=0.1$, 0.5, 1.0, and 2.0, are used. Lighter curve corresponds to higher $\kappa R_0$. The divergencies observed near $k{R}_0\sim 1$ arise due to the vanishing of $\Delta E_e$ owing to Rayleigh instability.