The Phase Diagram Of The Double-opening And Single-opening Bubble Was Studied By The Relaxation Method

The study of the shapes of vesicles and the shape transformations between them is an important subject in the field of soft matter physics. Until 1998, A. Saitoh et al. successfully made holes on vesicles by adding talin molecules in the aqueous solution. From then on, study on the opening-up vesicles aroused more people's attention.A. Saitoh et al. observed the opening-up membrane vesicles with cup, tube and funnel shapes experimentally. In theory, the shape equation and boundary condition of the opening-up membrane vesicles are given by Capovilla et al. The shape equation and boundary condition of the spontaneous curvature model are also given by Tu et al. Umeda et al. for the first time obtained the adjacent spherical opening-up vesicles correspond with the experimental results in the framework of the area difference elastic model. The value of reduced relaxation area difference ??₀ corresponding to Umeda's solution is relatively small. They claimed that no opening-up solution is found for ??₀>1.23. Previous work of our group showed that there exist new solutions that are different from Umeda's solution, which are opening-up dumbbell shapes with one hole and 3-budding shape with one hole. Their work were carried out under the SC model by shooting method. They have not found opening-up dumbbell shapes with two holes. It is not clear whether there exist this branch of solution.In this thesis opening-up dumbbell shape with one hole and the opening-up 3-budding shape with one hole are first studied under the ADE model. It is certified that there exist opening-up vesicles for ??₀> 1.23.The central work of this thesis is to derive the phase diagram for opening-up vesicles. For this purpose, it is necessary to obtain as many solutions as possible. We speculate that there should exist opening-up dumbbells with two holes thinking of the fact that tube and funnel shapes are observed in experiment. In order to obtain this branch of solution, we make a thorough explore by using the relaxation method under ADE model. The opening-up dumbbell shapes with two holes are obtained at last by changing the topology structure of shapes with one hole. This is a new branch solution which is not known by the previous researchers.Based on this foundation, the shape transformations between opening-up dumbbell shapes with one hole and two holes are studied in detail.In order to study the transformation between our new shapes and the shapes obtained by Umeda et al., shapes with relatively small values of ??₀ is first studied. Umeda's solutions are all obtained. It is found that for increased ??₀, at which value the solution of opening-up dumbbell shapes may exist, the Umeda solution all transformed into closed spheres. Thus we only need to compare the energy of closed spheres with the opening-up dumbbell shapes. The transformations are exemplified in the case of ??₀= 0.76. We also repeated the calculation of Umeda et al. with their parameter. Our results are consistent with theirs completely, which confirms our calculation method.The influences of the parameters ??₀ and ??? on the opening-up diameter, the shape and energy are studied in detail which is exemplified for some parameters.Then the calculations are made systematically through the whole parametric space ????, ??₀? in order to determine the existing region of the opening-up dumbbell shapes. It is found that the ??₀ interval is [1.338,1.725] and the ??? interval is [0.002,1.61] for the single-hole opening-up dumbbell shape solution, while the ??₀ interval is [1.25,1.73] and the ??? interval is [0,1.61] for the double-hole opening-up dumbbell shape solution.Then the transformations between different branches are studied which will at last lead to the determination of the phase diagram. The transformations are depicted in the case of ??₀= 1.38. It is found that the stable region for the opening-up dumbbells with two holes is much larger than that with one hole.We also obtained a new branch of solution of opening-up 3-budding shapes with two holes in the ADE model. Then the influences of parameters ??₀ and ??? on the diameter, shape, energy and other aspects of the opening-up 3-budding shape with one hole and two-holes are studied. The existence interval of the solution of the opening-up 3-budding shapes with one hole and two holes are determined. The ??₀ interval is [1.77,2.11] and the ??? interval is [0,1.256] for the single hole opening-up 3-budding shape solution. The ??₀ interval is [1.68,2.11] and the ??? interval is [0,1.258] for the double holes opening-up 3-budding shape solution. The phase transition of the opening-up dumbbell shape is discussed in detail for the case of ??₀=1.9. A similar discontinuous phase transition can also occur when ???=0.9796 and 0.9962.In certain parametric region, the new opening-up vesicle shapes that we found have lower energy than the shapes that have been found in experiment. We believe that they should also be found in future experiment.