J. Phys. II France
Volume 3, Numéro 7, July 1993
Page(s) 971 - 997
References of J. Phys. II France 3 971-997
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  24. The assumption of symmetric equilibrium values of the $\varphi$-field does not impose any restrictions on our model in 2d. If $\varphi_{\mathrm{A}}$ $\neq$ $\varphi_{\mathrm{B}}$, one can redefine $\varphi$ by $\varphi-(\varphi_{\mathrm{A}} + \varphi_{\mathrm{B}})$/2. Since $\int^{L}_{0}$ $\dot{\theta}$(s) ds = 2$\pi$, one can easily confirm that the extra terms arising from this transformation give only constant contributions to the free energy.
  25. The origin s = 0 should not coincide with any of the domain boundaries because we assumed continuity conditions at s = 0 and s = L.
  26. There always exists at least one axis of mirror symmetry for any n $\geq$ 1.
  27. The continuity condition for $\dot{\theta}$(s) $\equiv$ d$\theta$(s)/dtextits at s = 0 is automatically satisfied under equation (3.10) due to the mirror symmetry.
  28. We do not believe that F1 + F3 is exactly constant in such situations, but the actual deviation from a constant is numerically very small. For the case of $\Delta$P = 0, one can confirm from equation (3.16) that F1 + F3 is actually constant. When $\Delta$P $\neq$ 0, such a property can be obtained only numerically.
  29. If the outer pressure difference $\Delta$P becomes larger than a certain critical value, a perfectly circular shape is no longer stable and the vesicle undergoes a large deformation like a biconcave red blood cell. In this paper, we totally neglect such possibilities. However, if $\Lambda$ = 0, the phase separation is independent of the shape and the most stable mode is n = 1. [See Seifert U., Berndl K. and Lipowsky R., Phys. Rev. A 44 (1991) 1182 [CrossRef] [PubMed].].
  30. The Gauss-Bonnet Theorem still holds for our two component vesicles although the Gaussian curvature changes discontinuously at every domain wall. (We assumed same Gaussian bending moduli for both domains). As the vesicle shape itself is smooth (not cusp-like), the Gaussian curvature does not diverge but only changes discontinuously at the domain walls. Therefore, the contribution from such a discontinuity to the elastic energy is negligible because the width of the domain wall region is taken to be smaller than any other length in the problem in the strong segregation limit.
  31. One can confirm this from equation (4.3), which leads to: F1 + F3 = $\displaystyle\frac{\pi \kappa}{4} \int^{S_{o}}_{0}$ x(s) $\displaystyle\left(\dot{\mathrm{\theta}}(s) + \frac{sin \theta(s)}{x(s)} + \frac{2\Lambda}{\kappa} \varphi(s)\right)^{2}$ + const. = $\frac{\kappa}{8}S_{M}C^{2}$ + const.