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基于双层耦合模型，先通过求解无黏附能情况下满足给定边界条件的欧拉-拉格朗日方程组，找到了约化面积差a稍大于1的内凹开口形状解，并发现以往Umeda和Suezaki（2005 Phys.Rev.E 71 011913）给出的杯形解是对应a1的另一支解，该支解在a趋于1时开口是外凸的.进而在无黏附能和有黏附能的情况下对开口膜泡的两支解进行了深入研究，发现在a=1附近这两支解之间有一个间隙，在该间隙内不存在开口解.随着黏附半径的增大，该间隙的位置较缓慢地向右移动.在a=1附近，在无黏附能时的闭合形状只有球形一个解，而在有黏附能的情况下，闭合形状在1附近的一个区间内都有解.在无黏附及有黏附情况下的计算结果都表明这两支开口解及闭合形状属于不同的分支，它们之间不能连续演化.在间隙右侧的这一支解随着a的增大可以通过自交形状连续地演化到开口哑铃形.在有黏附的情况下，在a参数空间，同一支解会发生折叠，即出现同一a值对应多个解（形状）的情况，这在以往双层耦合模型的计算中没有出现过.讨论了a对无黏附和有黏附开口膜泡的形状和能量的影响.
[1] Seifert U 1997 Adv. Phys. 46 13
[2] Alberts B, Bray D, Lewis J, Raff M, Roberts K, Watson J D 1983 Molecular Biology of the Cell (New York: Garlan Publishing) pp112-115
[3] Sternberg B, Grumpert J, Reinhardt G, Gawrisch K 1987 Biochim. Biophys. Acta 898 223
[4] Bernard A L, Guedeau-Boudeville M A, Jullien L, Meglio J M D 2000 Langmuir 16 6809
[5] Sackmann E 1996 Science 271 43
[6] Keller C A, Glasmstar K, Zhdanov V P, Kasemo B 2000 Phys. Rev. Lett. 84 5443
[7] Seifert U, Lipowsky R 1990 Phys. Rev. A 42 4768
[8] Capovilla R, Guven J 2002 Phys. Rev. E 66 041604
[9] Tordeux C, Fournier J B 2002 Phys. Rev. E 65 041912
[10] Rosso R, Virga E G 1999 Proc. R. Soc. Lond. A 455 4145
[11] Das S, Du Q 2008 Phys. Rev. E 77 011907
[12] Saitoh A, Takiguchi K, Tanaka Y, Hotani H 1998 Proc. Natl. Acad. Sci. USA 95 1026
[13] Capovilla R, Guven J, Santiago J A 2002 Phys. Rev. E 66 021607
[14] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[15] Umeda T, Suezaki Y 2005 Phys. Rev. E 71 011913
[16] Kang W B, Zhang S G, Wang Y, Mu Y R, Huang C 2011 Sci. China: Phys. Mech. Astron. 54 2243
[17] Huang C, Zhang S G 2013 J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 41 31 (in Chinese) [黄聪, 张劭光 2013 陕西师范大学学报 41 31]
[18] Kong X B, Zhang S G 2016 Acta Phys. Sin. 65 068701 (in Chinese) [孔祥波, 张劭光 2016 物理学报 65 068701]
[19] Ni D, Shi H J, Yin Y J 2005 Colloids Surf. B 46 162
[20] L C J, Yin Y J, Yin J 2009 Colloids Surf. B 74 380
[21] Canham P B 1970 J. Theoret. Biol. 26 61
[22] Helfrich W 1973 Z. Naturforsch. C 28 693
[23] Svetina S, Ottova-Lietmannova A, Glaser R 1982 J. Theor. Biol. 94 13
[24] Svetina S, Zeks B 1985 Biomed. Biochim. Acta 44 979
[25] Ouyang Z C, Helfrich W 1987 Phys. Rev. Lett. 59 2486
[26] Ouyang Z C, Helfrich W 1989 Phys. Rev. A 39 5280
[27] Miao L, Seifert U, Wortis M, Dbereiner H G 1994 Phys. Rev. E 49 5389
[28] Seifert U, Berndl K, Lipowsky R 1991 Phys. Rev. A 44 1182
[29] Jlicher F, Lipowsky R 1996 Phys. Rev. E 53 2670
[30] Yang P, Du Q, Tu Z C 2017 Phys. Rev. E 95 042403
-
[1] Seifert U 1997 Adv. Phys. 46 13
[2] Alberts B, Bray D, Lewis J, Raff M, Roberts K, Watson J D 1983 Molecular Biology of the Cell (New York: Garlan Publishing) pp112-115
[3] Sternberg B, Grumpert J, Reinhardt G, Gawrisch K 1987 Biochim. Biophys. Acta 898 223
[4] Bernard A L, Guedeau-Boudeville M A, Jullien L, Meglio J M D 2000 Langmuir 16 6809
[5] Sackmann E 1996 Science 271 43
[6] Keller C A, Glasmstar K, Zhdanov V P, Kasemo B 2000 Phys. Rev. Lett. 84 5443
[7] Seifert U, Lipowsky R 1990 Phys. Rev. A 42 4768
[8] Capovilla R, Guven J 2002 Phys. Rev. E 66 041604
[9] Tordeux C, Fournier J B 2002 Phys. Rev. E 65 041912
[10] Rosso R, Virga E G 1999 Proc. R. Soc. Lond. A 455 4145
[11] Das S, Du Q 2008 Phys. Rev. E 77 011907
[12] Saitoh A, Takiguchi K, Tanaka Y, Hotani H 1998 Proc. Natl. Acad. Sci. USA 95 1026
[13] Capovilla R, Guven J, Santiago J A 2002 Phys. Rev. E 66 021607
[14] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[15] Umeda T, Suezaki Y 2005 Phys. Rev. E 71 011913
[16] Kang W B, Zhang S G, Wang Y, Mu Y R, Huang C 2011 Sci. China: Phys. Mech. Astron. 54 2243
[17] Huang C, Zhang S G 2013 J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 41 31 (in Chinese) [黄聪, 张劭光 2013 陕西师范大学学报 41 31]
[18] Kong X B, Zhang S G 2016 Acta Phys. Sin. 65 068701 (in Chinese) [孔祥波, 张劭光 2016 物理学报 65 068701]
[19] Ni D, Shi H J, Yin Y J 2005 Colloids Surf. B 46 162
[20] L C J, Yin Y J, Yin J 2009 Colloids Surf. B 74 380
[21] Canham P B 1970 J. Theoret. Biol. 26 61
[22] Helfrich W 1973 Z. Naturforsch. C 28 693
[23] Svetina S, Ottova-Lietmannova A, Glaser R 1982 J. Theor. Biol. 94 13
[24] Svetina S, Zeks B 1985 Biomed. Biochim. Acta 44 979
[25] Ouyang Z C, Helfrich W 1987 Phys. Rev. Lett. 59 2486
[26] Ouyang Z C, Helfrich W 1989 Phys. Rev. A 39 5280
[27] Miao L, Seifert U, Wortis M, Dbereiner H G 1994 Phys. Rev. E 49 5389
[28] Seifert U, Berndl K, Lipowsky R 1991 Phys. Rev. A 44 1182
[29] Jlicher F, Lipowsky R 1996 Phys. Rev. E 53 2670
[30] Yang P, Du Q, Tu Z C 2017 Phys. Rev. E 95 042403
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