Gaussian Curvature and the Equilibrium among Bilayer Cylinders, Spheres, and Discs

In mixtures of cetyltrimethylammonium bromide (CTAB) and sodium perfluorooctanoate (FC7) in aqueous solution, novel bilayer cylinders with hemispherical end caps and open, flat discs coexist with spherical unilamellar vesicles, apparently at equilibrium. Such equilibrium among bilayer cylinders, sph...

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Published inProceedings of the National Academy of Sciences - PNAS Vol. 99; no. 24; pp. 15318 - 15322
Main Authors H.-T. Jung, Lee, S. Y., Kaler, E. W., Coldren, B., Zasadzinski, J. A.
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences 26.11.2002
National Acad Sciences
Subjects
Bending
Caprylates
Cetrimonium
Cetrimonium Compounds
Chemical engineering
Chemical reactions
Cryoelectron Microscopy
Cryogenic engineering
Curvature
Cylinders
Engineering
Entropy
Fluorocarbons
Lipid Bilayers
Molecules
Physical Sciences
Radius of a sphere
Radius of curvature
Size distribution
Solutions
Surfactants
Water
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Summary:In mixtures of cetyltrimethylammonium bromide (CTAB) and sodium perfluorooctanoate (FC7) in aqueous solution, novel bilayer cylinders with hemispherical end caps and open, flat discs coexist with spherical unilamellar vesicles, apparently at equilibrium. Such equilibrium among bilayer cylinders, spheres, and discs is only possible for systems with a spontaneous curvature, Ro, and a positive Gaussian curvature modulus,$\bar\kappa$. We have measured the size distributions of the spherical vesicles, cylinders, and discs by using cryo-electron microscopy; a simple analysis of this length distribution allows us to independently determine that the mean curvature modulus, κ≈ 5 ± 1 kBTand$\bar\kappa$≈ 2 ± 1 kBT. This is one of the few situations in which Ro, κ, and$\bar\kappa$can be determined from the same experiment. From a similar analysis of the disk size distribution, we find that the edges of the discs are likely stabilized by excess CTAB. The fraction of discs, spherical vesicles, and cylinders depends on the CTAB/FC7mole ratio: increasing CTAB favors discs, while decreasing CTAB favors cylinders. This control over aggregate shape with surfactant concentration may be useful for the design of templates for polymerization, mesoporous silicates, etc.
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The ideal gas entropy difference of three spheres relative to one sphere is 2 kB; however, this overestimates the entropy difference here, because there are additional orientational modes for a cylinder that contribute to the entropy. However, the entropy contribution is negligible in comparison to the errors in the experimental measurements.
To whom correspondence should be addressed. E-mail: gorilla@engineering.ucsb.edu.
Edited by Benjamin Widom, Cornell University, Ithaca, NY, and approved October 3, 2002
This paper was submitted directly (Track II) to the PNAS office.
The radius of the minimum curvature energy vesicle, r0, determined from the mean of the size distribution of the spherical vesicles, is related to the spontaneous curvature, R0, in Eq. 1 by R0 = \documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \begin{equation*}\left(\frac{2{\kappa}}{2{\kappa}+{\bar {{\kappa}}}}\right)\end{equation*}\end{document}r0. This can be derived by minimizing the energy density (terms in square brackets in Eq. 2) with respect to R. For our measured values of r0 = 23 nm, κ = 5 kBT, and κ̄ = 2 kBT, this gives a spontaneous curvature, R0 = 19–20 nm. R0 is smaller than the mean vesicle size because κ̄ is positive. A positive κ̄ means that the Gaussian curvature energy increases proportionally to the number of vesicles. Hence, the minimum energy vesicle (or cylinder) has a radius larger than the spontaneous curvature to minimize the total curvature energy. However, using the mean vesicle or cylinder diameter or the corrected value of the spontaneous curvature does not have an effect on the values we calculate for κ and κ̄ within experimental error.
Helfrich predicted a general sphere to ellipsoidal instability for R/Ro= 6 in ref. 1.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.242374499