Issue
J. Phys. II France
Volume 4, Number 8, August 1994
Page(s) 1333 - 1362
DOI https://doi.org/10.1051/jp2:1994203
References of J. Phys. II France 4 1333-1362
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  23. Vesicles with nonzero genus (e.g. toroidal vesicles) have been observed experimentally. See for example : Mutz M. and Bensimon D., Phys. Rev. A 43 (1991) 4525 [CrossRef] [PubMed] ; Fourcade B., Mutz M. and Bensimon D., Phys. Rev. Lett. 68 (1992) 2551 [CrossRef] [PubMed].
  24. In the case where an amphiphilic monolayer consists of one type of amphiphiles with a difference in sizes between the polar head and hydrocarbon tail (e.g., the shape of an amphiphilic molecule is just like a wedge), a spontaneous curvature may be introduced into the bending energy in equation (2.1). Such a term also enters into the total free energy through the coupling energy F3 [see Eqs. (2.10) and (2.11)].
  25. We consider a vesicle which is made out of two kinds of amphiphiles, where the elastic modulus may depend, in general, on the total composition of the membrane. However, for simplicity, we neglected such a composition dependence of the elastic modulus.
  26. Coxeter H. S. M., Introduction to Geometry (J. Wiley, New York, 1965) ; Visconti A., Introductory Differential Geometry for Physicists (World Scientific, Singapore, 1992).
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  31. Neglecting higher than 2nd order terms for $\tilde{p} < \tilde{p}_\textrm{c} = 3$ was checked numerically by taking into account higher order terms in $\tilde{c}_{n}$ and $\tilde\psi_{n}$ up to 6th order. Indeed, the resulting phase diagrams do not depend in any significant way on the 3rd and 4th order terms in $\tilde{c}_{n}$ and can be omitted when $\tilde{p} < 3$.
  32. Further increase in the external pressure leads to unphysical situation where the vesicle will intersect itself in an unphysical way. For example, a single-component vesicle on the n = 2 branch self-intersects at a pressure $\tilde{p} = 5.4$ as is explained in reference [29].
  33. See e.g., Milner S. T. and Safran S. A., Phys. Rev. A 36 (1987) 4371 [CrossRef] [PubMed].
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  37. In another approach, the threshold value of the spontaneous curvature (the critical spontaneous curvature) $\tilde{H}_\textrm{sp} > -\: 39/23$ is obtained by Deuling and Helfrich (see Ref. [2]). They evaluated this threshold value using a perturbative method. The difference in the two values of the critical spontaneous curvatures is discussed by Peterson M. A., J. Appl. Phys. 57 (1985) 1739 [CrossRef].
  38. For $\tilde{H}_\textrm{sp} = 0$, Jenkins used a somewhat different method and obtained such a first-order phase transition for $\tilde{p}_{1} = 5.69$ and the appearance of a metastable state at $\tilde{p}_{2} = 5.64$. These values are evaluated numerically by solving the Euler-Lagrange equation for the axisymmetric case. For more details (see Ref. [3]). Our obtained values for $\tilde{p}_{1}$ and $\tilde{p}_{2}$ are different from Jenkins's, the difference comes from the single mode approximation and the truncation of the free energy expansion.
  39. The superosition of cn cos $n\theta$ and sn sin $n\theta$, expressing a vesicle shape, can be expressed by $\sqrt{c^{2}_{n} + s^{2}_{n}}$ cos $(n\theta + \delta)$, $\delta$ being a phase shift. The order parameter is also expressed by $\sqrt{\psi^{2}_{tn} + \psi^{2}_{sn}}$ cos $\displaystyle\left(\frac{2\pi n}{L} s(\theta) + \delta^{\prime}\right)$, $\delta^{\prime}$ being phase shift. We use equations (3.21) and (3.22) in Sma, because a coherent state $(\delta = \delta^{\prime})$ of the shape and the order parameter is energetically lower than other incoherent states due to the coupling term F3.