Two bilayers

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture · Macromolecules, membranes and cell architecture

Soft Matter at the Thermal Scale, Ch Membranes, fluctuations and soft matter in biology

Problem

Two bilayers, each of overall thickness d=5 nmd = 5\,\mathrm{nm} and bending modulus κb=10kBT\kappa_b = 10k_BT at T=298 KT = 298\,\mathrm{K} (take kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J}), face each other across water. Their van der Waals attraction comes from a Hamaker constant A=8×10−21 JA = 8\times10^{-21}\,\mathrm{J} acting between the two half-spaces on either side of each bilayer, and their Helfrich steric repulsion comes from confinement of thermal undulations. At a surface-to-surface spacing ds=4 nmd_s = 4\,\mathrm{nm}: ((a)) Calculate the Helfrich steric repulsion per unit area, Vrep=3π2128(kBT)2κbds2V_{\rm rep} = \dfrac{3\pi^2}{128}\dfrac{(k_BT)^2}{\kappa_b d_s^2}. ((b)) Calculate the van der Waals attraction per unit area between the two bilayer slabs by superposing single half-space interactions, W(h)=−A/(12πh2)W(h) = -A/(12\pi h^2), as ∣Wslab∣=∣W(ds)−2W(ds+d)+W(ds+2d)∣|W_{\rm slab}| = |W(d_s) - 2W(d_s+d) + W(d_s+2d)|. ((c)) State which energy is larger at ds=4 nmd_s = 4\,\mathrm{nm}. Then find the pressure each interaction exerts there, as minus the derivative of its energy per unit area with respect to dsd_s, and say which way the net force on the bilayers points at that spacing.

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