Effective interactions between fluid membranes

A self-consistent theory is proposed for the general problem of interacting undulating fluid membranes subject to the constraint that they do not interpenetrate. We implement the steric constraint via an exact functional integral representation, and through the use of a saddle-point approximation transform it into a novel effective steric potential. The steric potential is found to consist of two contributions: one generated by zero mode fluctuations of the membranes, and the other by thermal bending fluctuations. For membranes of cross-sectional area $S$, we find that the bending fluctuation part scales with the inter-membrane separation $d$ as $d^{-2}$ for $d \ll \sqrt{S}$, but crosses over to $d^{-4}$ scaling for $d \gg \sqrt{S}$, whereas the zero mode part of the steric potential always scales as $d^{-2}$. For membranes interacting exclusively via the steric potential, we obtain closed-form expressions for the effective interaction potential and for the rms undulation amplitude $\sigma$, which becomes small at low temperatures $T$ and/or large bending stiffnesses $\kappa$. Moreover, $\sigma$ scales as $d$ for $d \ll \sqrt{S}$, but saturates at $\sqrt{k_{{\rm B}} T S/\kappa}$ for $d \gg \sqrt{S}$. In addition, using variational Gaussian theory, we apply our self-consistent treatment to study inter-membrane interactions subject to three different types of potential: (i)~the Moreira-Netz potential for a pair of strongly charged membranes with an intervening solution of multivalent counterions, (ii)~an attractive square well, (iii)~the Morse potential, and (iv)~a combination of hydration and van der Waals interactions.