Effective interactions between fluid membranes

Bing-Sui Lu and Rudolf Podgornik

Phys. Rev. E 92, 022112 – Published 7 August 2015

Abstract

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 intermembrane separation $d$ as $d^{- 2}$ for $d ≪ \sqrt{S}$ but crosses over to $d^{- 4}$ scaling for $d ≫ \sqrt{S}$ , 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 $σ$ , which becomes small at low temperatures $T$ and/or large bending stiffnesses $κ$ . Moreover, $σ$ scales as $d$ for $d ≪ \sqrt{S}$ but saturates at $\sqrt{k_{B} T S / κ}$ for $d ≫ \sqrt{S}$ . In addition, using variational Gaussian theory, we apply our self-consistent treatment to study intermembrane interactions subject to different types of potentials: (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.

Received 4 May 2015
Revised 23 July 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022112

Authors & Affiliations

Bing-Sui Lu^* and Rudolf Podgornik^†

Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia and Department of Theoretical Physics, J. Stefan Institute, 1000 Ljubljana, Slovenia

^*bing-sui.lu@fmf.uni-lj.si
^†rudolf.podgornik@fmf.uni-lj.si

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Vol. 92, Iss. 2 — August 2015

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Figure 1
Comparison of plots of $Θ_{λ} (x)$ generated using the exact function [Eq. (2.9); behavior is described by the blue-gray circles, yellow squares, and green diamonds for $λ = 0.01 a, 0.1 a$ , and $0.3 a$ , respectively, where $a$ is a microscopic length scale] and the saddle-point approximation [Eq. (2.13); behavior is displayed by the blue, red dashed, and green dot-dashed lines for $λ = 0.01 a, 0.1 a$ and $0.3 a$ , respectively]. The agreement is quite good, with the accuracy improving for smaller values of $λ$ .
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Figure 2
Steric interaction: Behavior of the dimensionless separation ${\tilde{ℓ}}^{*}$ as a function of dimensionless osmotic pressure $\tilde{p}$ and (inset) behavior of dimensionless rms fluctuation amplitude $\tilde{σ}$ as a function of ${\tilde{ℓ}}^{*}$ , where ${\tilde{ℓ}}^{*} \approx 〈 ℓ_{0} 〉 / d^{*}$ is the equilibrium separation in the saddle-point approximation, $\tilde{σ} \equiv σ / d^{*}, \tilde{p} \equiv β P {(d^{*})}^{3}$ , and $d^{*} \equiv \sqrt{3 c} {(k_{B} T)}^{1 / 4} \sqrt{S} / {(4 κ)}^{1 / 4}$ .
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Figure 3
Moreira-Netz potential. Behavior of ${\tilde{ℓ}}^{*} \equiv ℓ_{0}^{*} / μ_{0}$ (where $ℓ_{0}^{*}$ is the saddle-point approximation to the equilibrium intermembrane separation and $μ_{0} \equiv 1 / 2 π ℓ_{B} σ_{s}$ ) as a function of rescaled osmotic pressure $\tilde{p} \equiv β P / 4 π ℓ_{B} σ_{s}^{2}$ [given by Eq. (3.32)] and (inset) behavior of the rescaled rms fluctuation amplitude $\tilde{σ} \equiv σ / μ_{0}$ as a function of ${\tilde{ℓ}}^{*}$ [defined in Eq. (3.29)] for $t = 0.1, c = 1$ , and ${\tilde{σ}}_{s} = 1$ , studied for the following three cases: (i) $q = 2$ (blue), (ii) $q = 3$ (green dashed); and (iii) $q = 4$ (red dot-dashed). For comparison, we have displayed the behavior of the disjoining osmotic pressure (horizontal axis) due to counterions between two fixed charged plates as a function of the interplate separation (vertical axis) for counterion valences $q = 2$ (black dotted), $q = 3$ (cyan dot-dot-dashed), and $q = 4$ (orange dot-dashed-dashed) [obtained by differentiating Eq. (3.24)], and also shown (cf. inset) is the behavior of $\tilde{σ}$ of a membrane interacting only sterically with a hard wall [black; cf. Eq. (2.26)].
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Figure 4
Attractive square well. Behavior of external osmotic pressure $\tilde{p}$ (scaled in units of $k_{B} T / b$ ) versus average separation ${\tilde{ℓ}}^{*}$ for $t = 0.1$ and $c = 1$ , and the following four binding strengths: (i) ${\tilde{v}}_{0} = 0$ (black), (ii) ${\tilde{v}}_{0} = - 0.04$ (blue dashed), (iii) ${\tilde{v}}_{0} = - 0.06$ (red dotted), and (iv) ${\tilde{v}}_{0} = - 0.08$ (green dot-dashed). Inset: Behavior of dimensionless rms fluctuation $\tilde{σ} \equiv σ / b$ (where $b$ is the well width) versus dimensionless separation ${\tilde{ℓ}}^{*} \equiv ℓ_{0}^{*} / b$ for the same values of $t, c$ , and the above four binding strengths.
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Figure 5
Morse potential. Behavior of ${\tilde{ℓ}}^{*} \equiv κ_{D} ℓ_{0}^{*}$ (where $ℓ_{0}^{*}$ is the saddle-point approximation to ${〈 ℓ_{0} 〉}_{1}$ ) as a function of external osmotic pressure $\tilde{p} \equiv β P κ_{D}^{- 3}$ and (inset) behavior of rms fluctuation amplitude $\tilde{σ} \equiv κ_{D} σ$ as a function of separation ${\tilde{ℓ}}^{*}$ for $c = 1, κ = 10 k_{B} T, {\tilde{ℓ}}^{*} ≪ \sqrt{S}$ , and the following four cases: (i) $u_{1} = u_{2} = 0$ (black), (ii) $u_{1} = 1, u_{2} = 0.5$ (blue, dashed), (iii) $u_{1} = u_{2} = 1$ (red, dotted), and (iv) $u_{1} = 1, u_{2} = 2$ (green, dot-dashed).
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Figure 6
Hydration repulsion and van der Waals attraction. Behavior of dimensionless free energy $\tilde{f} \equiv β λ_{H}^{2} f_{var}$ (with $P = 0$ ) and (inset) dimensionless external osmotic pressure $\tilde{p}$ with respect to rescaled separation ${\tilde{ℓ}}^{*} \equiv ℓ_{0}^{*} / λ_{H}$ for ${\tilde{A}}_{H} = β A_{H} λ_{H}^{2} = 4.83, \tilde{w} = β W / 12 π, \tilde{δ} = δ / λ_{H} = 13.3, t = k_{B} T / K = 0.0248, T = 270$ K, $c = 0.255$ , and the following Hamaker strengths: (i) $\tilde{w} = 0.04$ (black), (ii) $\tilde{w} = 0.04902$ (orange dash-dash-dotted), (iii) $\tilde{w} = 0.053$ (red dot-dashed), (iv) $\tilde{w} = 0.061$ (blue disks), and (v) $\tilde{w} = 0.068$ (green diamonds).
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